**FINAL EXAM REVIEW PROBLEMS**

**UNIFORM ACCELERATION**

1. At the instant when the traffic light turns green, an automobile starts with a constant acceleration of 1.8
m/s2_{. At the same instant a truck travelling with a constant speed of 8.5 m/s overtakes and passes the }

automobile.

(a) How far beyond the starting point will the automobile overtake the truck? (b) How fast will the car be travelling at that instant?

2. A subway train starts from rest at a station and accelerates at the rate of 2.0 m/s2_{ for 10 s. It runs at a }

constant speed for 30 s, and then decelerates at 2.4 m/s2_{ until it stops at the next station. Find the total }

distance between the stations and the average speed of the train.

3. A boy in a wagon starts down a hill 90 m long with an initial velocity of 1.2 m/s, reaching the bottom of the hill in 0.50 min. Calculate his acceleration (assumed to be uniform) and his velocity at the bottom of the hill.

4. An object starts from rest and accelerates at 3.0 m/s2_{ for 4.0 s. Its velocity remains constant for 7.0 s, }

and it finally comes to rest with uniform deceleration after another 5.0 s. Find the following: (a) the displacement for each stage of the motion

(b) the average velocity over the whole time interval.

5. A Formula One car accelerates from rest uniformly at 2.40 m/s2_{ for 15.0 s, then moves with a uniform }

velocity for 200 s, and finally decelerates uniformly at 3.60 m/s2_{ until it comes to a stop. How long }

will the car be in motion, how far will it travel, and what will be its average speed?

6. A turtle is moving with a constant acceleration along a straight ditch. He starts his stopwatch as he passes a fence post and notes that it takes him 10 s to reach a pine tree 10 m farther along the ditch. As he passes the pine tree, his speed is 1.2 m/s. How far was he from the fence post when he started from rest?

7. Two trains, one travelling at 100 km/h and the other at 128 km/h, are headed towards one another
along a straight, level track. When the trains are 1.2 km apart, each engineer simultaneously sees the
other’s train and applies the brakes. Both trains have equal, constant decelerations of 0.9 m/s2_{. Will }

there be a collision?

8. A parachutist jumps from a height of 3.1 × 103_{ m and falls freely for 10 s. She then opens her }

parachute, and for the next 20 s slows down with an acceleration of –4.5 m/s2_{. After that, she falls the }

rest of the distance to the ground at a uniform velocity. (a) What is her velocity just before the parachute opens? (b) At what altitude does the parachute open?

(c) What is the velocity of the parachutist, just before she strikes the ground? (d) Calculate the time required for the whole descent.

(e) From what height would she have to fall freely in order to strike the ground with the same velocity as she does when wearing a parachute? (This is how parachutists are trained.)

9. A stone is dropped into the water from a bridge 44 m above the water. Another stone is thrown vertically downward 1.0 s after the first was dropped. Both stones strike the water at the same time. What was the initial velocity of the second stone?

10. A student is determined to test the law of gravity for himself. He walks off a skyscraper 320 m high,
stopwatch in hand, and starts his free fall (zero initial velocity). Five seconds later, Superman arrives at
the scene and dives off the roof to save the student. What must Superman’s initial velocity be in order
to catch the student just before the student reaches the ground? (Assume that Superman’s acceleration
*is that of any freely falling body (i.e., g = 9.8 m/s*2_{.)}

11. A convertible with its top down drove towards the entrance of an underground garage with a velocity of 24 km/h. A window cleaner on a scaffold directly above the entrance accidentally kicked a bucket of water off a moving scaffold. The scaffold at that moment was 9.0 m vertically higher than the top of the car and the car was 12 m from the entrance. The scaffold was moving up at 1.5 m/s. Did the driver get wet?

12. An efficient parcel service wants to speed up its deliveries by dropping parcels into moving trucks. An employee is positioned on an overpass directly above a straight, level road to drop the parcels into the trucks at just the right time. One day, a delivery truck starts from rest and drives along the road with a constant acceleration of . A package is released at the correct instant to land in the truck. If the overpass was 30 m above the truck and the truck started from a position 100 m from the point of impact, how long after the truck started did the employee wait before dropping the parcel?

13. A rally driver completes three consecutive sections of a straight rally course as follows: section 1 (10 km) in 7.50 min, section 2 (18 km) in 14.40 min, and section 3 (9.8 km) in 5.80 min. What was the average velocity for the three sections? (Assume no stop between sections.)

14. A car accelerates uniformly from rest at the rate of 2.0 m/s2_{ for 6.0 s. It then maintains a constant }

velocity for 0.50 min. Finally, the brakes are applied, and the vehicles slows down at a uniform rate and comes to rest in 5.0 s. Find the following.

(a) the maximum velocity of the car (b) the total displacement

**VECTORS**

1. A train moving at a constant speed of 100 km/h travels east for 40 min, then 30º east of north for 20 min, and finally west for 30 min. What is the train’s average velocity for the trip?

2. A man walks 600 m [E47°N], then 500 m [N38°W], then 300 m [W29°S], and finally 400 m [S13°E]. Find his resultant displacement.

3. The current in a river moves at 2.0 m/s [S]. How fast and in what direction must a swimmer move through the water in order to have a resultant velocity relative to the river bank of

(a) 3.6 m/s[S] (b) 3.6 m/s[N] (c) 3.6 m/s[E]

4. A ball is thrown from the top of a building with a speed of 20 m/s and at a downward angle of 30° to the horizontal, as shown. What are the horizontal and vertical components of the ball’s initial velocity?

5. A boat travelling at 3.0 m/s through the water keeps its bow pointing north across a stream that flows west at 5.0 m/s. What is the resultant velocity of the boat with respect to the shore?

6. A dog walks at 1.6 m/s on the deck of a boat that is travelling north at 7.6 m/s with respect to the water.

(a) What is the velocity of the dog with respect to the water if it walks towards the bow? (b) What is the velocity of the dog with respect to the water if it walks towards the stern?

(c) What is the velocity of the dog with respect to the water if it walks towards the east rail, at right angles to the boat’s keel?

7. An airplane maintains a heading due west at an air speed of 900 km/h. It is flying through a hurricane with winds of 300 km/h, from the northeast.

(a) What is the plane’s ground speed?

(b) In which direction is the plane moving relative to the ground?

(c) How long would it take the plane to fly from one city to another 500 km away, along the path in (b)?

8. Two boathouses are located on a river, 1.0 km apart on the same shore. Two men make round trips from one boathouse to the other, and back. One man paddles a canoe at a velocity of 4.0 km/h relative to the water, and the other walks along the shore at a constant velocity of 4.0 km/h. The current in the river is 2.0 km/h in the starting direction of the canoeist.

(a) How much sooner than the walker does the canoeist reach the second boathouse? (b) How long does it take each to make the round trip?

9. A 70 m wide river flows at 0.80 m/s. A girl swims across it at 1.4 m/s relative to the water. (a) What is the least time she requires to cross the river?

(b) How far downstream will she be when she lands on the opposite shore?

(c) At what angle to the shore would she have to aim, in order to arrive at a point directly opposite the starting point?

10. A pilot maintains a heading due west with an air speed of 240 km/h. After flying for 30 min, he finds himself over a town that he knows is 150 km west and 40 km south of his starting point.

(a) What is the wind velocity, in magnitude and direction?

(b) What heading should he now maintain, with the same air speed, to follow a course due west from the town?

11. The navigator of an airplane plans a flight from one airport to another 1200 km away, in one direction 30° east of north. The weather office informs him of a prevailing wind from the west, of 80 km/h. The pilot wants to maintain an air speed of 300 km/h.

(a) What heading should the navigator give the pilot? (b) How long will the flight take?

(c) How much time did the wind save?

12. A puck sliding across the ice at 20 m/s [E] is struck by a stick and moves at 30 m/s, at an angle of 120° to its original path. Find its change in velocity.

13. Two boys are at point X on one side of a river, 40 m wide and having a current of 1.0 m/s, flowing as shown. Simultaneously, they dive into the water in an attempt to reach point Y, directly opposite X. Both swim at 2.0 m/s relative to the water, but one directs himself so that his net motion corresponds to XY, while the other keeps his body perpendicular to the current and consequently lands at point Z. After landing, he runs along the shore to point Y at a speed of 6.0 m/s. Which boy arrives at Y first, and by how much time does he beat the other?

14. The easterly and northerly components of a car’s velocity are 24 m/s and 30 m/s, respectively. In what direction and with what speed is the car moving? In other words, what is the car’s velocity?

15. A cannon fires a cannonball with a speed of 100 m/s at an angle of 20° above the horizontal. What are the horizontal and vertical components of the initial velocity of the cannonball?

16. A football player is running at a constant speed in a straight line up the field at an angle of 15° to the sidelines. The coach notices that it takes the player 4.0 s to get from the 25 m line to the goal line. How fast is the player running?

17. A girl swims at 3.0 m/s across a swimming pool at an angle of 30° to the side of the pool, as shown. What are the components of her swimming velocity in each of the following directions?

(a) across the pool (b) along the pool’s edge

18. The pilot of a light plane heads due north at an air speed of 400 km/h. A wind is blowing from the west at 60 km/h.

(a) What is the plane’s velocity with respect to the ground?

(b) How far off course would the plane be after 2.5 h, if the pilot had hoped to travel due north but had forgotten to check the wind velocity?

19. A canoeist paddles “north” across a river at 3.0 m/s. (The canoe is always kept pointed at right angles to the river.) The river is flowing east at 4.0 m/s and is 100 m wide.

(a) What is the velocity of the canoe relative to the river bank? (b) Calculate the time required to cross the river.

(c) How far downstream is the landing point from the starting point?

20. A pilot wishes to make a flight of 300 km [NE] in 45 min. On checking with the meteorological office, she finds that there will be a wind of 80 km/h from the north for the entire flight. What heading and airspeed must she use for the flight?

21. A helicopter travelling horizontally at 150 km/h [E] executes a gradual turn, and eventually is moving at 120 km/h [S]. If the turn takes 50 s to complete, what is the average acceleration of the helicopter? 22. A clock has a second hand that is 12 cm long. Find each of the following.

(a) the average speed of the tip of the second hand

(b) its instantaneous velocity as it passes the 6 and the 9 on the clock face (c) its average velocity in moving from the 3 to the 12 on the clock face Note: the circumference of a circle is .

1. A small boy pulls his wagon, of mass 24 kg, giving it a horizontal acceleration of 1.5 m/s2_{. If the }

wagon’s handle makes an angle of 40° with the ground while the boy is pulling on it, and there is a frictional force of 6.0 N opposing the wagon’s motion, with what force is he pulling on the handle of the wagon?

2. Forces of 2.0 N and 1.0 N act on an object of mass 5.0 kg, as shown in the diagram. (a) Calculate the net force acting on the object.

(b) What is the acceleration of the object?

3. Two girls, one of mass 40 kg and the other of mass 60 kg, are standing side by side in the middle of a frozen pond. One pushes the other with a force of 360 N for 0.10 s. The ice is essentially frictionless. (a) What is each girl’s acceleration?

(b) What velocity will each girl acquire in the 0.10 s that the force is acting? (c) How far will each girl move during the same time period?

4. A motorist has a reaction time of 0.60 s. (Reaction time is the interval between seeing a danger and applying the brakes.) While driving at 72 km/h, he sees a child run suddenly onto the road, 40 m in front of his car. If the mass of the car is 1000 kg and the average horizontal force supplied during braking is 8000 N, will he be able to stop in time to avoid hitting the child?

5. A child’s wagon experiences a frictional force of 73 N whenever it is in motion, regardless of the load
it is carrying. An applied horizontal force of 128 N causes the wagon to accelerate at 5.0 m/s2_{. The }

same applied force, with a child on the wagon, causes it to accelerate at 1.0 m/s2_{. What is the mass of }

*6. A net force of 8.0 N gives a mass m*1 and acceleration of 2.0 m/s2* and a mass m*2, an acceleration of 4.0

m/s2_{. What acceleration would the force give the two masses if they were fastened together?}

7. Two girls pull a sled across a field of snow, as shown in the diagram. A third girl pulls backward with a 2.0 N force. If the mass of the sled is 10 kg, determine its instantaneous acceleration.

8. A plane takes off from a level runway with two gliders in tow, one behind the other. The first glider has a mass of 1600 kg and the second a mass of 800 kg. The frictional drag may be assumed as constant and equal to 2000 N on each glider. The towrope between the first glider and the plane can withstand a tension of 10 000 N.

(a) If a velocity of 40 m/s is required for takeoff, how long a runway is needed? (b) How strong must the towrope between the two gliders be?

9. A baby carriage with a mass of 50 kg is being pushed along a rough sidewalk with an applied horizontal force of 200 N, and it has a constant velocity of 3.0 m/s.

(a) What other horizontal force is acting on the carriage, and what is the magnitude of that force? (b) What value of applied horizontal force would be required to accelerate the carriage from rest to 7.0 m/s in 2.0 s?

10. A bullet of mass 20 g strikes a fixed block of wood at a speed of 320 m/s. The bullet embeds itself in the block of wood, penetrating to a depth of 6.0 cm. Calculate the average net force acting on the bullet while it is being brought to rest.

11. An elevator, complete with contents, has a mass of 2000 kg. By drawing free-body diagrams and by
*performing the necessary calculations, determine the value of T (the tension in the elevator cable) in *
the following:

(a) when the elevator is at rest

(b) when the elevator is moving toward at a constant velocity of 2.0 m/s (c) when the elevator is moving downward at a constant velocity of 2.0 m/s (d) when the elevator is accelerating upward at 1.0 m/s2

(e) when the elevator is accelerating downward at 1.0 m/s2

12. A 2.0 kg mass, placed on a smooth, level table, is attached by a light string passing over a frictionless pulley to a 5.0 kg mass hanging freely over the edge of the table, as illustrated. Calculate

(a) the tension in the string

(b) the acceleration of the 2.0 kg mass.

13. Two spheres of masses 1.5 kg and 3.0 kg are tied together by a light string looped over a frictionless pulley. They are allowed to hang freely. What will be the acceleration of each mass? Assume that up is positive and down is negative.

14. A 40 kg block on a level, frictionless table is connected to a 15 kg mass by a rope passing over a frictionless pulley. What will be the acceleration of the 15 kg mass when it is released?

15. A 3.0 kg mass is attached to a 5.0 kg mass by a strong string that passes over a frictionless pulley. When the masses are allowed to hang freely, what will be

(b) the magnitude of the tension in the string

16. A 20 kg box is dragged across a level floor with a force of 100 N. The force is applied at an angle of 40° above the horizontal. If the coefficient of kinetic friction is 0.32, what is the acceleration of the box?

17. A boy on a toboggan is sliding down a snow-covered hillside. The boy and toboggan together have a mass of 50 kg, and the slope is at an angle of 30° to the horizontal. Find the boy’s acceleration considering the following.

(a) if there is no friction

(b) if the coefficient of kinetic friction is 0.15

18. A cart with a mass of 2.0 kg is pulled across a level desk by a horizontal force of 4.0 N. If the coefficient of kinetic friction is 0.12, what is the acceleration of the cart?

19. A girl pushes a light snow shovel at a uniform velocity across a sidewalk. If the handle of the shovel is inclined at 55° to the horizontal and she pushes along the handle with a force of 100 N, what is the force of friction? What is the coefficient of kinetic friction?

20. A 10 kg block of ice slides down a ramp 20 m long, inclined at 10° to the horizontal. (a) If the ramp is frictionless, what is the acceleration of the block of ice?

(b) If the coefficient of kinetic friction is 0.10, how long will it take the block to slide down the ramp, if it starts from rest?

Note: The kinetic friction is implied.

21. A skier has just begun descending a 20° slope. Assuming that the coefficient of kinetic friction is 0.10, calculate

(a) the acceleration of the skier (b) his final velocity after 8.0 s

23. Jane wishes to quickly scale a slender vine to visit Tarzan in his treetop hut. The vine is known to safely support the combined weight of Tarzan, Jane, and Cheetah. Tarzan has twice the mass of Jane, who has twice the mass of Cheetah. If the vine is 60 m long, what minimum time should Jane allow for the climb?

24. A boy pushing a 20 kg lawn mower exerts a force of 100 N along the handle. If the handle is elevated 37° to the horizontal, determine

(a) the component of the applied force that pushes the lawn mower forward (b) the acceleration of the lawn mower, if the frictional force is 60 N

(c) the component of the applied force that pushes the lawn mower vertically toward the ground (d) the gravitational force exerted on the mower

(e) the total downward force of the mower on the ground, when pushed (f) the normal force exerted on the mower by the ground

(g) the effective coefficient of kinetic friction.

25. An 80 kg man is standing in an elevator on a set of spring scales calibrated in newtons. Suppose the
elevator accelerates downward at 3.0 m/s2_{. What reading will the scales have?}

26. An empty elevator of mass 2.7 × 103_{ kg is pulled upward by a cable with an upward acceleration of }

1.2 m/s2_{.}

(a) What is the tension in the cable?

(b) What would the tension be if the elevator were accelerating downward at 1.2 m/s2_{?}

(c) Does the direction of motion of the elevator matter in (a) and (b)? 27. A fish hangs from a spring scale supported from the roof of an elevator.

(a) If the elevator has an upward acceleration of 1.2 m/s2_{ and the scale reads 200 N, what is the true }

force of gravity on the fish?

(b) Under what circumstances will the scale read 150 N? (c) What will the scale read if the elevator cable breaks?

28. For each of the following systems:

(ii) find the tension in the string(s),

(iii) find the rate at which the masses will accelerate, and

(iv) calculate how far each mass will move in 1.2 s, if the system starts from rest. (Assume that both the surfaces and the pulleys are frictionless.)

(a)

(b)

(c)

29. Two masses are connected by a light cord over a frictionless pulley, as illustrated. The coefficient of kinetic friction is 0.18.

(a) What is the acceleration of the system? (b) What is the tension in the cord?

30. What is the acceleration of the system illustrated, if the coefficient of kinetic friction is 0.20? Assume that it starts.

31. Tarzan (mass 100 kg) holds one end of an ideal vine (infinitely strong, completely flexible, but having zero mass). The vine runs horizontally to the edge of a cliff, then vertically to where Jane (mass 50 kg) is hanging on, above a river filled with hungry crocodiles. A sudden sleet storm has removed all friction. Assuming that Tarzan hangs on, what is his acceleration towards the cliff edge?

32. A 70 kg hockey player coasts along the ice on steel skates. If the coefficient of kinetic friction is 0.010. (a) What is the force of friction?

(b) How long will it take him to coast to a stop, if he is travelling at 1.0 m/s?

33. A small 10 kg cardboard box is thrown across a level floor. It slides a distance of 6.0 m, stopping in 2.2 s. Determine the coefficient of friction between the box and the floor.

34. A 0.5 kg wooden block is placed on top of a 1.0 kg wooden block. The coefficient of static friction between the two blocks is 0.35. The coefficient of kinetic friction between the lower block and the level table is 0.20. What is the maximum horizontal force that can be applied to the lower block without the upper block slipping?

35. A boy pulls a 50 kg crate across a level floor with a force of 200 N. If the force acts at an angle of 30° up from the horizontal, and the coefficient of kinetic friction is 0.30, determine the following.

(a) the normal force exerted on the crate by the floor

(b) the horizontal frictional force exerted on the crate by the floor (c) the acceleration of the crate

1. A helicopter is rising vertically at a uniform velocity of 14.7 m/s. When it is 196 m from the ground, a ball is projected from it with a horizontal velocity of 8.5 m/s with respect to the helicopter. Calculate the following.

(a) when the ball will reach the ground (b) where it will hit the ground

(c) what its velocity will be when it hits the ground

2. A stone thrown horizontally from the top of a tall building takes 7.56 s to reach the street. How high is the building?

3. A bullet is projected horizontally at 300 m/s from a height of 1.5 m. Ignoring air resistance, calculate how far it travels horizontally before it hits the ground. (Assume that the ground is level.)

4. An object is projected horizontally with a velocity of 30 m/s. It takes 4.0 s to reach the ground. Neglecting air resistance, determine the following.

(a) the height at which the object was projected

(b) the magnitude of the resultant velocity, just before the object strikes the ground 5. A bomber in level flight, flying at 92.0 m/s, releases a bomb at a height of 1950 m.

(a) How long is it before the bomb strikes Earth? (b) How far does it travel horizontally?

(c) What are the horizontal and vertical components of its velocity when it strikes? (d) What is its velocity of impact?

6. A cannonball shot horizontally from the top of a cliff with an initial velocity of 425 m/s is aimed towards a schooner on the ocean below. If the cliff is 78 m above the ocean surface, calculate the following:

(a) the time for the cannonball to reach the water (b) the horizontal displacement of the cannonball

(c) the velocity of the cannonball just before it strikes the water

7. A balloon is rising at a vertical velocity of 4.9 m/s. At the same time, it is drifting horizontally with a velocity of 1.6 m/s. If a bottle is released from the balloon when it is 9.8 m above the ground,

determine the following.

(a) the time it takes for the bottle to reach the ground

(b) the horizontal displacement of the bottle from the balloon

8.A cannonball is fired with a velocity of 100 m/s at 25° above the horizontal. Determine how far away it lands on level ground.

9. A ball is thrown horizontally from a window at 10 m/s and hits the ground 5.0 s later. What is the height of the window and how far from the base of the building does the ball hit?

10. A cannon is fired at 30° above the horizontal with a velocity of 200 m/s from the edge of a cliff 125 m high. Calculate where the cannonball lands on the level plain below.

11. A shell is fired horizontally from a powerful gun, located 44 mm above a horizontal plane, with a muzzle speed of 245 m/s.

(a) How long does the shell remain in the air? (b) What is its range?

(c) What is the magnitude of the vertical component of its velocity as it strikes the target?

12. A bomber, diving at an angle of 53° with the vertical, releases a bomb at an altitude of 730 m. The bomb hits the ground 5.0 s after being released.

(a) What was the velocity of the bomber?

(b) How far did the bomb travel horizontally during its flight?

(c) What were the horizontal and vertical components of its velocity just before striking the ground? 13. A driver, accelerating too quickly on a horizontal bridge, skids, crashes through the bridge railing, and

lands in the river 20.0 m below the level of the bridge roadway. The police find that the car is not vertically below the break in the railing, but is 53.6 m beyond it horizontally.

(a) Determine the speed of the car before the crash, in km/h.

(b) What properties of falling bodies did you assume in making your calculation in (a)? (c) State whether your answer in (a) is an overestimate or an underestimate, and why.

14. An artillery gun is fired so that its shell has a vertical component of a velocity of 210 m/s and a horizontal component of 360 m/s. If the target is at the same level as the gun, and air friction is neglected,

(a) how long will the shell stay in the air?

(b) how far down-range will the shell hit the target?

15. A baseball, thrown from shortstop position to first base, travels 32 m horizontally, rises 3.0 mm, and falls 3.0 m. Find the initial velocity of the ball.

16. A player kicks a football with an initial velocity of 15 m/s at an angle of 42° above the horizontal. A second player standing at a distance of 30 m from the first, in the direction of the kick, starts running to meet the ball at the instant it is kicked. How fast must he run in order to catch the ball before it hits the ground?

CIRCULAR MOTION

1. A 3.5 kg steel ball is swung at a constant speed in a vertical circle of radius 1.2 m, on the end of a light, rigid steel rod, as illustrated. If the ball has a frequency of 1.0 Hz, calculate the tension in the rod due to the mass at the top (A) and at the bottom (B) positions.

2. The pilot of an airplane, which has been diving at a speed of 540 km/h, pulls out of the dive at constant speed.

(a) What is the minimum radius of the plane’s circular path in order that the acceleration of the pilot at
*the lowest point will not exceed 7g?*

(b) What force is applied on an 80 kg pilot by the plane seat at the lowest point of the pull-out?

3. In the Bohr model of the hydrogen atom, the electron revolves around the nucleus. If the radius of the
orbit is 5.3 × 10–11_{ m and the electron makes 6.6 ×}_{ 10}15_{ r/s, find the following.}

(a) the acceleration of the electron.

(b) the centripetal force acting on the electron (This force is due to the attraction between the

positively charged nucleus and the negatively charged electron.) The mass of the electron is 9.1 × 10–
31_{ kg.}

4. When you whirl a ball on a cord in a vertical circle, you find a critical speed at the top for which the tension in the cord is zero. This is because the force of gravity on the object itself supplies the

necessary centripetal force. How slowly can you swing a 2.5 kg ball like this so that it will just follow a circle with a radius of 1.5 m?

5. An object of mass 3.0 kg is whirled around in a vertical circle of radius 1.3 m with a constant velocity of 6.0 m/s. Calculate the maximum and minimum tension in the string.

6. Snoopy is flying his vintage war plane in a “loop the loop” path chasing the Red Baron. His

instruments tell him the plane is level (at the bottom of the loop) and travelling with a speed of 180 km/h. He is sitting on a set of bathroom scales, and notes that they read four times the normal force of gravity on him. What is the radius of the loop? Answer in metres.

**Answer Section**
**PROBLEM**
1. ANS:
(a)
(b)
(c)

(d)

REF: K/U OBJ: 2.2 LOC: FMV.02 KEY: FOP 5.2, p.158 MSC: P

2. ANS:

(a) Using Newton’s Law of Motion,

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.3, p.159 MSC: SP

3. ANS:
*For m*1:

*the net force on m*1 is

*The acceleration of m*1 is

*For m*2:

*the net force on m*2 is

The acceleration of the masses have the same magnitude, but opposite directions. Thus

*The acceleration of each mass is the same. Substituting for m*1 the acceleration is

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.3, p.160 MSC: SP

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.3, p.161 MSC: P

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.3, p.161 MSC: P

Now

Taking the vertical components,

But , since is less than . Thus

As a result, the force of friction is

The net horizontal force on the box is

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.4, p.164 MSC: SP

7. ANS: (a)

REF: K/U OBJ: 2.3, 2.4 LOC: FMV.02 KEY: FOP 5.4, p.165 MSC: SP

8. ANS:

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.4, p.166 MSC: P

*Since v is constant, *
But

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.4, p.166 MSC: P

(a)

(b)

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.4, p.166 MSC: P

11. ANS:

(b)

Note that the acceleration and the final speed of the skier do not depend on the mass of the skier. REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.4, p.166

MSC: P 12. ANS:

(a) Consider first the vertical component of the ball’s motion.

The solutions of the equation are

The time taken to reach the ground is 8.0 s, since the negative solution has no meaning in this problem.

(b)

(c) The vertical component of the velocity is

The horizontal component of the velocity is 8.5 m/s. Therefore the resultant velocity is the vector sum of the vertical and horizontal components as follows:

The velocity of impact is 64 m/s [82° below the horizontal].

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.8, p.173 MSC: SP

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.8, p.174 MSC: P

14. ANS:

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.8, p.174 MSC: P

15. ANS:

(b)

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.8, p.174 MSC: P 16. ANS: (a) (b) (c) (d)

REF: K/U, MC OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.8, p.174 MSC: P 17. ANS: (a) (b) (c)

REF: K/U, MC OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.8, p.174 MSC: P

18. ANS:

(a)

(Negative answer has no meaning in this question.)

(b) Since the balloon and the object are both moving with horizontal velocity of 1.6 m/s, there will be no horizontal displacement of the object from the balloon. (Relative to the ground there will be a horizontal displacement.)

REF: K/U, C OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.8, p.175 MSC: P

Solution 1:

Solution 2:

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.9, p.178 MSC: P

At the bottom, where the two forces act in opposite directions,

Taking vertical components,

This is the maximum value for the tension in the rod, pulling up. At the top, where the two forces act downward,

Taking vertical components,

Alternate Solution:

At the bottom, the rod must support the ball’s weight as well as provide the centripetal force necessary to make it move in a circle.

At the top, the ball’s weight provided part of the centripetal force; the rod provides the rest.

REF: K/U OBJ: 3.2 LOC: FM1.04 KEY: FOP 5.10, p.182 MSC: SP

21. ANS:

(a)

REF: K/U, MC OBJ: 3.2 LOC: FM1.04 KEY: FOP 5.10, p.183 MSC: P

22. ANS:

REF: K/U, MC OBJ: 2.2 LOC: FMV.02 KEY: FOP 5.13, p.197 MSC: P

23. ANS:

Mass of Jane Mass of Tarzan Mass of Cheetah Maximum tension

Maximum acceleration of Jane would be:

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.13, p.198 MSC: P

24. ANS: (a) (b) Horizontal Components: (c) (d) (e)

(f) The upward force of the ground is (Newton’s Third Law).

(g)

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.13, p.198 MSC: P

25. ANS:

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.13, p.198 MSC: P

(b)

(c) No. The net force depends only on the direction of the acceleration and not on the direction of motion.

REF: K/U, C OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.13, p.198 MSC: P

27. ANS: (a)

(b)

(c) If the cable breaks, there is no force on the object by the scale.

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.13, p.198 MSC: P

28. ANS: (a)

(i)

(iii)

(iv)

(b)

(ii)

(iii)

(iv)

(c)

1. 2. 3.

(ii) From (1):

*Note! Do part (iii) first to get a.*

From (3):

(iii) The sum of (1) and (2), and (3) is:

(iv)

MSC: P 29. ANS:

(a)

Note: part (b) is done before part (a)

(a)

(c) For no motion, and,

and,

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.13, p.199 MSC: P

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.13, p.199 MSC: P

31. ANS:

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.13, p.199 MSC: P

32. ANS:

(b)

REF: K/U OBJ: 2.4 LOC: FMV.01 KEY: FOP 5.13, p.200 MSC: P

REF: K/U OBJ: 2.4 LOC: FMV.01 KEY: FOP 5.13, p.200 MSC: P

34. ANS:

Consider top block. When

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.13, p.200 MSC: P

35. ANS:

(since )

(b)

(c) Taking horizontal components:

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.13, p.200 MSC: P

36. ANS:

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.13, p.201 MSC: P

REF: K/U, MC OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.13, p.201 MSC: P

(a)

(b)

(c)

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.13, p.201 MSC: P

(a)

(b)

(c)

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.13, p.201 MSC: P

(a)

(b) no air resistance no friction car and bridge

no reduction in speed because of guardrail, etc. vertical motion independent of horizontal motion

(c) Underestimate, since car would have travelled farther horizontally without the above. REF: K/U, MC, I OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.13, p.201 MSC: P

41. ANS:

**or **

(b)

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.13, p.202 MSC: P

42. ANS:

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.13, p.202 MSC: P

43. ANS:

Vertical time of flight:

REF: K/U, MC OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.13, p.202 MSC: P

44. ANS:

(a)

(b)

REF: K/U OBJ: 3.2 LOC: FM1.04 KEY: FOP 5.13, p.202 MSC: P

45. ANS:

*At the top, T = 0*

REF: K/U OBJ: 3.2 LOC: FM1.04 KEY: FOP 5.13, p.203 MSC: P

46. ANS:

Note the mass of the ball is immaterial. At the top,

At the bottom,

REF: K/U OBJ: 3.2 LOC: FM1.04 KEY: FOP 5.13, p.203 MSC: P

47. ANS:

REF: K/U OBJ: 3.2 LOC: FM1.04 KEY: FOP 5.13, p.203 MSC: P

48. ANS:

REF: K/U OBJ: 2.4 LOC: FMV.01 KEY: FOP 5.13, p.200 MSC: P

**e**

1. An elevator, complete with contents, has a mass of 2000 kg. By drawing free-body diagrams and by
*performing the necessary calculations, determine the value of T (the tension in the elevator cable) in *
the following:

(a) when the elevator is at rest

(b) when the elevator is moving toward at a constant velocity of 2.0 m/s (c) when the elevator is moving downward at a constant velocity of 2.0 m/s (d) when the elevator is accelerating upward at 1.0 m/s2

(e) when the elevator is accelerating downward at 1.0 m/s2

2. A 2.0 kg mass, placed on a smooth, level table, is attached by a light string passing over a frictionless pulley to a 5.0 kg mass hanging freely over the edge of the table, as illustrated. Calculate

(a) the tension in the string

(b) the acceleration of the 2.0 kg mass.

3. Two spheres of masses 1.5 kg and 3.0 kg are tied together by a light string looped over a frictionless pulley. They are allowed to hang freely. What will be the acceleration of each mass? Assume that up is positive and down is negative.

4. A 40 kg block on a level, frictionless table is connected to a 15 kg mass by a rope passing over a frictionless pulley. What will be the acceleration of the 15 kg mass when it is released?

5. A 3.0 kg mass is attached to a 5.0 kg mass by a strong string that passes over a frictionless pulley. When the masses are allowed to hang freely, what will be

(b) the magnitude of the tension in the string

6. A 20 kg box is dragged across a level floor with a force of 100 N. The force is applied at an angle of 40° above the horizontal. If the coefficient of kinetic friction is 0.32, what is the acceleration of the box?

7. A boy on a toboggan is sliding down a snow-covered hillside. The boy and toboggan together have a mass of 50 kg, and the slope is at an angle of 30° to the horizontal. Find the boy’s acceleration considering the following.

(a) if there is no friction

(b) if the coefficient of kinetic friction is 0.15

8. A cart with a mass of 2.0 kg is pulled across a level desk by a horizontal force of 4.0 N. If the coefficient of kinetic friction is 0.12, what is the acceleration of the cart?

9. A girl pushes a light snow shovel at a uniform velocity across a sidewalk. If the handle of the shovel is inclined at 55° to the horizontal and she pushes along the handle with a force of 100 N, what is the force of friction? What is the coefficient of kinetic friction?

10. A 10 kg block of ice slides down a ramp 20 m long, inclined at 10° to the horizontal. (a) If the ramp is frictionless, what is the acceleration of the block of ice?

(b) If the coefficient of kinetic friction is 0.10, how long will it take the block to slide down the ramp, if it starts from rest?

11. A skier has just begun descending a 20° slope. Assuming that the coefficient of kinetic friction is 0.10, calculate

(a) the acceleration of the skier (b) his final velocity after 8.0 s

12. A helicopter is rising vertically at a uniform velocity of 14.7 m/s. When it is 196 m from the ground, a ball is projected from it with a horizontal velocity of 8.5 m/s with respect to the helicopter. Calculate the following.

(a) when the ball will reach the ground (b) where it will hit the ground

(c) what its velocity will be when it hits the ground

13. A stone thrown horizontally from the top of a tall building takes 7.56 s to reach the street. How high is the building?

14. A bullet is projected horizontally at 300 m/s from a height of 1.5 m. Ignoring air resistance, calculate how far it travels horizontally before it hits the ground. (Assume that the ground is level.)

15. An object is projected horizontally with a velocity of 30 m/s. It takes 4.0 s to reach the ground. Neglecting air resistance, determine the following.

(a) the height at which the object was projected

(b) the magnitude of the resultant velocity, just before the object strikes the ground 16. A bomber in level flight, flying at 92.0 m/s, releases a bomb at a height of 1950 m.

(a) How long is it before the bomb strikes Earth? (b) How far does it travel horizontally?

(c) What are the horizontal and vertical components of its velocity when it strikes? (d) What is its velocity of impact?

17. A cannonball shot horizontally from the top of a cliff with an initial velocity of 425 m/s is aimed towards a schooner on the ocean below. If the cliff is 78 m above the ocean surface, calculate the following:

(a) the time for the cannonball to reach the water (b) the horizontal displacement of the cannonball

(c) the velocity of the cannonball just before it strikes the water

18. A balloon is rising at a vertical velocity of 4.9 m/s. At the same time, it is drifting horizontally with a velocity of 1.6 m/s. If a bottle is released from the balloon when it is 9.8 m above the ground,

determine the following.

(a) the time it takes for the bottle to reach the ground

(b) the horizontal displacement of the bottle from the balloon

19. A cannonball is fired with a velocity of 100 m/s at 25° above the horizontal. Determine how far away it lands on level ground.

20. A 3.5 kg steel ball is swung at a constant speed in a vertical circle of radius 1.2 m, on the end of a light, rigid steel rod, as illustrated. If the ball has a frequency of 1.0 Hz, calculate the tension in the rod due to the mass at the top (A) and at the bottom (B) positions.

21. The pilot of an airplane, which has been diving at a speed of 540 km/h, pulls out of the dive at constant speed.

(a) What is the minimum radius of the plane’s circular path in order that the acceleration of the pilot at
*the lowest point will not exceed 7g?*

(b) What force is applied on an 80 kg pilot by the plane seat at the lowest point of the pull-out?
22. An aerospace scientist has designed a rocket with a mass of 1.0 × 103_{ kg. He wants it to accelerate }

straight up with an initial acceleration of 21 m/s2_{. What thrust must the rocket engine develop?}

23. Jane wishes to quickly scale a slender vine to visit Tarzan in his treetop hut. The vine is known to safely support the combined weight of Tarzan, Jane, and Cheetah. Tarzan has twice the mass of Jane, who has twice the mass of Cheetah. If the vine is 60 m long, what minimum time should Jane allow for the climb?

24. A boy pushing a 20 kg lawn mower exerts a force of 100 N along the handle. If the handle is elevated 37° to the horizontal, determine

(a) the component of the applied force that pushes the lawn mower forward (b) the acceleration of the lawn mower, if the frictional force is 60 N

(c) the component of the applied force that pushes the lawn mower vertically toward the ground (d) the gravitational force exerted on the mower

(e) the total downward force of the mower on the ground, when pushed (f) the normal force exerted on the mower by the ground

(g) the effective coefficient of kinetic friction.

25. An 80 kg man is standing in an elevator on a set of spring scales calibrated in newtons. Suppose the
elevator accelerates downward at 3.0 m/s2_{. What reading will the scales have?}

26. An empty elevator of mass 2.7 × 103_{ kg is pulled upward by a cable with an upward acceleration of }

1.2 m/s2_{.}

(b) What would the tension be if the elevator were accelerating downward at 1.2 m/s2_{?}

(c) Does the direction of motion of the elevator matter in (a) and (b)? 27. A fish hangs from a spring scale supported from the roof of an elevator.

(a) If the elevator has an upward acceleration of 1.2 m/s2_{ and the scale reads 200 N, what is the true }

force of gravity on the fish?

(b) Under what circumstances will the scale read 150 N? (c) What will the scale read if the elevator cable breaks? 28. For each of the following systems:

(i) draw a free-body diagram for each mass. (ii) find the tension in the string(s),

(iii) find the rate at which the masses will accelerate, and

(iv) calculate how far each mass will move in 1.2 s, if the system starts from rest. (Assume that both the surfaces and the pulleys are frictionless.)

(a)

(b)

(c)

29. Two masses are connected by a light cord over a frictionless pulley, as illustrated. The coefficient of kinetic friction is 0.18.

(a) What is the acceleration of the system? (b) What is the tension in the cord?

(c) Assume that the system starts to move. For what values of will it not start?

30. What is the acceleration of the system illustrated, if the coefficient of kinetic friction is 0.20? Assume that it starts.

31. Tarzan (mass 100 kg) holds one end of an ideal vine (infinitely strong, completely flexible, but having zero mass). The vine runs horizontally to the edge of a cliff, then vertically to where Jane (mass 50 kg) is hanging on, above a river filled with hungry crocodiles. A sudden sleet storm has removed all friction. Assuming that Tarzan hangs on, what is his acceleration towards the cliff edge?

32. A 70 kg hockey player coasts along the ice on steel skates. If the coefficient of kinetic friction is 0.010. (a) What is the force of friction?

(b) How long will it take him to coast to a stop, if he is travelling at 1.0 m/s?

33. A small 10 kg cardboard box is thrown across a level floor. It slides a distance of 6.0 m, stopping in 2.2 s. Determine the coefficient of friction between the box and the floor.

34. A 0.5 kg wooden block is placed on top of a 1.0 kg wooden block. The coefficient of static friction between the two blocks is 0.35. The coefficient of kinetic friction between the lower block and the level table is 0.20. What is the maximum horizontal force that can be applied to the lower block without the upper block slipping?

35. A boy pulls a 50 kg crate across a level floor with a force of 200 N. If the force acts at an angle of 30° up from the horizontal, and the coefficient of kinetic friction is 0.30, determine the following.

(a) the normal force exerted on the crate by the floor

(b) the horizontal frictional force exerted on the crate by the floor (c) the acceleration of the crate

36. A ball is thrown horizontally from a window at 10 m/s and hits the ground 5.0 s later. What is the height of the window and how far from the base of the building does the ball hit?

37. A cannon is fired at 30° above the horizontal with a velocity of 200 m/s from the edge of a cliff 125 m high. Calculate where the cannonball lands on the level plain below.

38. A shell is fired horizontally from a powerful gun, located 44 mm above a horizontal plane, with a muzzle speed of 245 m/s.

(a) How long does the shell remain in the air? (b) What is its range?

(c) What is the magnitude of the vertical component of its velocity as it strikes the target?

39. A bomber, diving at an angle of 53° with the vertical, releases a bomb at an altitude of 730 m. The bomb hits the ground 5.0 s after being released.

(a) What was the velocity of the bomber?

(b) How far did the bomb travel horizontally during its flight?

(c) What were the horizontal and vertical components of its velocity just before striking the ground? 40. A driver, accelerating too quickly on a horizontal bridge, skids, crashes through the bridge railing, and

lands in the river 20.0 m below the level of the bridge roadway. The police find that the car is not vertically below the break in the railing, but is 53.6 m beyond it horizontally.

(a) Determine the speed of the car before the crash, in km/h.

(b) What properties of falling bodies did you assume in making your calculation in (a)? (c) State whether your answer in (a) is an overestimate or an underestimate, and why.

41. An artillery gun is fired so that its shell has a vertical component of a velocity of 210 m/s and a horizontal component of 360 m/s. If the target is at the same level as the gun, and air friction is neglected,

(a) how long will the shell stay in the air?

(b) how far down-range will the shell hit the target?

42. A baseball, thrown from shortstop position to first base, travels 32 m horizontally, rises 3.0 mm, and falls 3.0 m. Find the initial velocity of the ball.

43. A player kicks a football with an initial velocity of 15 m/s at an angle of 42° above the horizontal. A second player standing at a distance of 30 m from the first, in the direction of the kick, starts running to meet the ball at the instant it is kicked. How fast must he run in order to catch the ball before it hits the ground?

44. In the Bohr model of the hydrogen atom, the electron revolves around the nucleus. If the radius of the
orbit is 5.3 × 10–11_{ m and the electron makes 6.6 ×}_{ 10}15_{ r/s, find the following.}

(a) the acceleration of the electron.

(b) the centripetal force acting on the electron (This force is due to the attraction between the

positively charged nucleus and the negatively charged electron.) The mass of the electron is 9.1 × 10–
31_{ kg.}

45. When you whirl a ball on a cord in a vertical circle, you find a critical speed at the top for which the tension in the cord is zero. This is because the force of gravity on the object itself supplies the

necessary centripetal force. How slowly can you swing a 2.5 kg ball like this so that it will just follow a circle with a radius of 1.5 m?

46. An object of mass 3.0 kg is whirled around in a vertical circle of radius 1.3 m with a constant velocity of 6.0 m/s. Calculate the maximum and minimum tension in the string.

47. Snoopy is flying his vintage war plane in a “loop the loop” path chasing the Red Baron. His

instruments tell him the plane is level (at the bottom of the loop) and travelling with a speed of 180 km/h. He is sitting on a set of bathroom scales, and notes that they read four times the normal force of gravity on him. What is the radius of the loop? Answer in metres.

48. A 10 kg box is pulled across a level floor, where the coefficient of kinetic friction is 0.35. What
horizontal force is required for an acceleration of 2.0 m/s2_{?}

**e**

**Problem**

1. Forces as given below are acting on a common point. Using two different solutions for each combination of forces, find the additional force required to maintain static equilibrium. (a) 160 N [east], 120 N [west]

(b) 200 N [east], 160 N [north] (c) 100 N [N45°E], 150 N [W]

(d) 6.0 N [N30°E], 10.0 N [N45°W], 12.0 N [W10°S]

2. A helium balloon is attached to the middle of a light fishing-line anchored at both ends, as illustrated. If the upward force of the balloon is 0.05 N, find the tension in the fishing-line.

3. A 0.50 kg block of cheese sits on a level table, as shown. The coefficient of static friction is 0.60. Three strings are tied together in a knot at K. Kc is horizontal and fastened to the cheese. Kw angles up to the wall at 30° to the horizontal. Km hangs vertically, supporting a mouse. What is the maximum mass of the mouse, if the cheese and the mouse remain in equilibrium?

4. Three children are playing in the snow with a sleigh. One child exerts a maximum force of 40 N and another 58 N, on ropes, to just get the sleigh to move with the third child seated in it. The angles the horizontal ropes make with the direction of travel are 30° and 20°, respectively. If the combined mass of the seated child and the sleigh is 75 kg, and the sleigh moves in a straight line, what is the

coefficient of friction between the sleigh and the snow?

5. A 2.0 kg wooden block is attached to a 0.50 kg mass by a string passing through a frictionless pulley, as illustrated.

(a) If the mass of 0.50 kg provides the minimum force required to just get the block to move, what is the coefficient of friction?

(b)

If the same plane surface is now inclined 20° to the horizontal, what minimum mass, attached to the string, will just get the block moving?

6. A 2.5 kg block rests on a plane inclined at 15° to the horizontal. If the coefficient of static friction for the surface is 0.30, will the block slide down the plane?

7. The CN Tower, opened in Toronto in 1976, is still the tallest free-standing structure in the world.
Made of reinforced concrete, it soars 553.2 m from the base of the tower to the highest antenna on its
communication mast. A massive foundation plunges an additional 16.7 m below the base, to solid
rock. The foundation has threefold symmetry, with the three corners lying on a circle of radius 55.0 m.
The total tower mass is 1.18 × 108_{ kg, and the centre of gravity is located about 75 m above the }

bottom of the foundation. For the purposes of the problem given below, do not consider the effects of the earth and rock around the foundation, but assume the tower and foundation to be resting on horizontal bedrock.

(a) A blimp fastens a cable to the uppermost tip of the tower, and pulls horizontally so that the tower pivots on two corners of the foundation. So long as nothing bends, breaks, or slides, what tension is needed to just start the tower tipping? Convert your answer to a fraction of the tower’s mass.

(b) Suppose the blimp succeeds in starting the tower tipping, and carefully pulls it over to balance at the critical tipping angle: calculate this angle.

8. A 10 kg mass is held in a hand, as illustrated. The mass of the forearm is 2.0 kg, and its centre of gravity is located 14 cm from the joint. The biceps muscle is attached to the radius bone at 4.0 cm from the joint, and the mass is 32 cm from the joint.

(a) Find the tension in the biceps muscle.

(b) Find the tension in the biceps muscle if the forearm forms an angle of 45° with the horizontal. (c) What is the reaction force of the radius on the humerus, at the joint, when the forearm is outstretched, as in (a)?

9. Given the dimensions on the diagram of the forearm holding a mass of 5.4 kg, find the following. (a) the tension in the biceps muscle

(b) the reactive force of the radius on the humerus, at the joint

10. A pioneer farmer hitches a horse and an ox to a stump in order to remove it from a field. The horse
pulls with a force of 1.2 × 104_{ N [W], while the ox pulls with a force of 1.6 ×}_{ 10}4_{ N [W30°N]. What }

is the resultant force on the stump?

11. A patient is in neck traction with skull calipers, as illustrated. Find the maximum value of the

suspended mass that will cause no tension or other force in the neck, given that mass of the head is 4.2 kg and the coefficient of friction between the bed and the head is 0.20.

12. A 0.60 g spider hangs on its thread from the branch of a tree. A horizontal wind blows the spider and the thread to an angle of 35° from the vertical. Find the force of the wind on the spider and the tension in the thread.

13. To keep Robin from being captured, Batman tosses him out of a third-storey window, knowing that a 17.0 m rope hangs slack between hooks of equal height on adjacent buildings 13.0 m apart. Robin grabs the rope and hangs on at a point 5.0 m from one end. Assuming that Robin’s mass is 45.0 kg and the rope withstands the initial impulse, what is the tension in each part of the rope when equilibrium is established?

14. A load of 250 kg is supported by two steel cables, as shown in the diagram. (a) Find the tensions in the cables.

(b) If both cables have a diameter of 2.0 mm, how much does each cable stretch?

15. A spring whose force constant is 48 N/m has a 0.25 kg mass suspended from it. What is the extension of the spring?

16. Find the tension in the quadriceps tendon for the situation shown, assuming that the weight of the leg is ignored.

**e**

**Answer Section**
**PROBLEM**

1. ANS:

(a) 160 N [E] + 120 N [W] = 40 N [E] or 160 N [E] – 120 N [E] = 40 N [E] (b)

(d)

REF: K/U OBJ: 2.2 LOC: FMV.02 KEY: FOP 6.1, p.208 MSC: P

2. ANS:

Note that the tension in each string is about 6 times the balloon force.

REF: K/U OBJ: 2.2 LOC: FMV.02 KEY: FOP 6.1, p.209 MSC: P

Since the knot is in equilibrium, we know that

Taking horizontal components, with right as positive, this vector equation gives us

Taking vertical components, with up as positive, gives us

REF: K/U OBJ: 2.4 LOC: FMV.01 KEY: FOP 6.2, p.211 MSC: SP

4. ANS:

REF: K/U OBJ: 2.4 LOC: FMV.01 KEY: FOP 6.2, p.212 MSC: P

5. ANS: (a)

(b)

The mass required is

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 6.2, p.213 MSC: P

6. ANS:

Alternate Solution:

Therefore 15° is less than angle of repose and object will not slide.

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 6.2, p.213 MSC: P

7. ANS:

(a) The centre of gravity of the tower will be directly above the centre of the foundation, which is located one-half the circle radius from the pivot line (see diagram).

This is a large force, but it is only about 5% of the force of gravity on the tower, as seen below.

(b) The critical angle will occur when the centre of gravity of the tower is vertically above the supporting edge, as illustrated.

At the critical tipping angle,

Thus, the tower will fall back to its upright position if the tipping angle is less than 20°. REF: K/U KEY: FOP 6.5, p.228 MSC: SP

8. ANS: (a)

(b) When the elbow forms an angle of 45°, the forces act at 45° to the forearm, as illustrated. The torque equation in (a) will be the same, except that each factor will be multiplied by

sin 45°. The result will be the same—a tension in the biceps muscle of 8.5 × 102_{ N. It follows that the }

tension in the biceps muscle will be the same for all angles of the elbow, when the same mass is held. (c) The net force on the forearm is zero, since there is equilibrium; that is,

Take vertical components, with up as positive.

The humerus bone pushes down on the forearm at the elbow.

REF: K/U, MC KEY: FOP 6.9, p.244 MSC: SP 9. ANS:

*(a) Taking torques about P:*

Thus

Taking the vertical components,

REF: K/U, MC KEY: FOP 6.11, p.256 MSC: P 10. ANS:

Using Components:

REF: K/U OBJ: 2.2 LOC: FMV.02 KEY: FOP 6.11, p.256 MSC: P

11. ANS:

*Components of F:*

Taking vertical components:

Taking horizontal components:

But,

REF: K/U, MC OBJ: 2.2 LOC: FMV.02 KEY: FOP 6.11, p.256 MSC: P

Taking vertical components:

Taking horizontal components:

REF: K/U OBJ: 2.2 LOC: FMV.02 KEY: FOP 6.11, p.257 MSC: P

13. ANS:

Taking vertical components:

Substitute (1) in (2):

REF: K/U OBJ: 2.2 LOC: FMV.02 KEY: FOP 6.11, p.257 MSC: P

14. ANS:

(a) Taking horizontal components:

REF: K/U, MC KEY: FOP 6.11, p.262 MSC: P 15. ANS:

If the spring obeys Hooke’s Law,

REF: K/U OBJ: 4.5 LOC: EM1.08 KEY: FOP 6.6, p.232 MSC: SP

16. ANS:

Taking moments about P:

Horizontal components:

Vertical components:

Note the direction of is really below the horizontal.

**e**
**Answer Section**
**PROBLEM**
1. ANS:
(a)
(b)
(c)

(d)

REF: K/U OBJ: 2.2 LOC: FMV.02 KEY: FOP 5.2, p.158 MSC: P

2. ANS:

(a) Using Newton’s Law of Motion,

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.3, p.159 MSC: SP

3. ANS:
*For m*1:

*the net force on m*1 is

*The acceleration of m*1 is

*For m*2:

*the net force on m*2 is

The acceleration of the masses have the same magnitude, but opposite directions. Thus

*The acceleration of each mass is the same. Substituting for m*1 the acceleration is

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.3, p.160 MSC: SP

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.3, p.161 MSC: P

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.3, p.161 MSC: P

Now

Taking the vertical components,

But , since is less than . Thus

As a result, the force of friction is

The net horizontal force on the box is

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.4, p.164 MSC: SP

7. ANS: (a)

REF: K/U OBJ: 2.3, 2.4 LOC: FMV.02 KEY: FOP 5.4, p.165 MSC: SP

8. ANS:

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.4, p.166 MSC: P

*Since v is constant, *
But

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.4, p.166 MSC: P

(a)

(b)

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.4, p.166 MSC: P

11. ANS:

(b)

Note that the acceleration and the final speed of the skier do not depend on the mass of the skier. REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.4, p.166

MSC: P 12. ANS:

(a) Consider first the vertical component of the ball’s motion.

The solutions of the equation are

The time taken to reach the ground is 8.0 s, since the negative solution has no meaning in this problem.

(b)

(c) The vertical component of the velocity is

The horizontal component of the velocity is 8.5 m/s. Therefore the resultant velocity is the vector sum of the vertical and horizontal components as follows:

The velocity of impact is 64 m/s [82° below the horizontal].

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.8, p.173 MSC: SP

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.8, p.174 MSC: P

14. ANS:

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.8, p.174 MSC: P

15. ANS:

(b)

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.8, p.174 MSC: P 16. ANS: (a) (b) (c) (d)

REF: K/U, MC OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.8, p.174 MSC: P 17. ANS: (a) (b) (c)

REF: K/U, MC OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.8, p.174 MSC: P

18. ANS:

(a)

(Negative answer has no meaning in this question.)

(b) Since the balloon and the object are both moving with horizontal velocity of 1.6 m/s, there will be no horizontal displacement of the object from the balloon. (Relative to the ground there will be a horizontal displacement.)

REF: K/U, C OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.8, p.175 MSC: P

Solution 1:

Solution 2:

REF: K/U OBJ: 1.4 LOC: FM1.03 KEY: FOP 5.9, p.178 MSC: P

At the bottom, where the two forces act in opposite directions,

Taking vertical components,

This is the maximum value for the tension in the rod, pulling up. At the top, where the two forces act downward,

Taking vertical components,

Alternate Solution:

At the bottom, the rod must support the ball’s weight as well as provide the centripetal force necessary to make it move in a circle.

At the top, the ball’s weight provided part of the centripetal force; the rod provides the rest.

REF: K/U OBJ: 3.2 LOC: FM1.04 KEY: FOP 5.10, p.182 MSC: SP

21. ANS:

(a)

REF: K/U, MC OBJ: 3.2 LOC: FM1.04 KEY: FOP 5.10, p.183 MSC: P

22. ANS:

REF: K/U, MC OBJ: 2.2 LOC: FMV.02 KEY: FOP 5.13, p.197 MSC: P

23. ANS:

Mass of Jane Mass of Tarzan Mass of Cheetah Maximum tension

Maximum acceleration of Jane would be:

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.13, p.198 MSC: P

24. ANS: (a) (b) Horizontal Components: (c) (d) (e)

(f) The upward force of the ground is (Newton’s Third Law).

(g)

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.13, p.198 MSC: P

25. ANS:

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.13, p.198 MSC: P

(b)

(c) No. The net force depends only on the direction of the acceleration and not on the direction of motion.

REF: K/U, C OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.13, p.198 MSC: P

27. ANS: (a)

(b)

(c) If the cable breaks, there is no force on the object by the scale.

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.13, p.198 MSC: P

28. ANS: (a)

(i)

(iii)

(iv)

(b)

(ii)

(iii)

(iv)

(c)

1. 2. 3.

(ii) From (1):

*Note! Do part (iii) first to get a.*

From (3):

(iii) The sum of (1) and (2), and (3) is:

(iv)

MSC: P 29. ANS:

(a)

Note: part (b) is done before part (a)

(a)

(c) For no motion, and,

and,

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.13, p.199 MSC: P

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.13, p.199 MSC: P

31. ANS:

REF: K/U OBJ: 2.3 LOC: FMV.02 KEY: FOP 5.13, p.199 MSC: P

32. ANS:

(b)

REF: K/U OBJ: 2.4 LOC: FMV.01 KEY: FOP 5.13, p.200 MSC: P

REF: K/U OBJ: 2.4 LOC: FMV.01 KEY: FOP 5.13, p.200 MSC: P

34. ANS:

Consider top block. When

REF: K/U OBJ: 2.4 LOC: FMV.02 KEY: FOP 5.13, p.200 MSC: P

35. ANS: