Centripetal (angular) acceleration
3. Magnetic forces This force is discussed in Topic 6
Then of course we have the example of objects constrained to move in circles by strings or wires attached to the object.
2.4.4 SOLVE
PROBLEMSINVOLVING
CIRCULARMOTION
Example
A model airplane of mass 0.25 kg has a control wire of length 10.0 m attached to it. Whilst held in the hand of the controller it flies in a horizontal circle with a speed of 20 m s–1. Calculate the tension in the wire.
Solution
The tension in the wire provides the centripetal force and is equal to m v ____ r 2 which in this situation is equal to
0.25 × 400 _________ 10 = 10 N.
An interesting situation arises when we have circular motion in a vertical plane. Consider a situation in which you attach a length of string to an object of mass m and then whirl it in a vertical circle of radius r. If the speed of the object is v at its lowest point then the tension in the string at this point will be mg + m v ____ r 2 and at the highest point the tension will be mg – m v ____ r 2
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Lowest point:
mg T
Figure 258 Lowest Point
The resultant force on mass, m / kg, throughout its motion in a circle (as long as the speed is constant) is always mv___ r 2.
Taking the positive direction to be towards the centre of the circle, at its lowest point, the resultant force is provided by the expression T – mg, so that
mv2 r --- = T–mg That is, T mv2 r ---+mg = Highest point: mg T
Figure 259 Highest Point
Again, we have that the resultant force on mass, m / kg, throughout its motion in a circle (as long as the speed is constant) is always mv2.
r
---
Taking the positive direction to be towards the centre of the circle, at its highest point, the resultant force is provided by the expression T + mg, so that
, i.e., mv2 r --- = T+mg T mv2 r ---–mg = Example
A “wall of death” motorcyclist rides his motorcycle in a vertical circle of radius 20 m. Calculate the minimum speed that he must have at the top of the circle in order to complete the loop.
Solution
mg R
Let R be the reaction force on the bike, then we need to use the expression mv2
r
--- = R+mg (when the bike is at
its highest point).
However, the bike must always make contact with the track, that is, we must have that R ≥ 0.
Now, re-arranging the expression we have that
R mv2 r
---–mg
= , however, as R ≥ 0, we have that
mv2 r --- mg 0 mv2 r ---≥mg ⇒ ≥
– . So, the minimum speed will
be given by g v2 r
---
= since at any lower speed mg will be greater than mv2
r
--- and the motor bike will leave the track.
So in this case the speed will be 14 m s-1.
It is also worthwhile noting that in circular motion with constant speed, there is no change in kinetic energy. This is because the speed, v, is constant and so the expression for the kinetic energy, 1-2 mv2, is also always constant (anywhere
along its motion). Another way to look at this is that since the force acts at right angles to the particle then no work is done on the particle by the force.
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Exercise 2.4
1. An object is travelling at a constant speed of 40 ms-1 in a circular path of radius 80 m.
Calculate the acceleration of the object?
2. A body of mass 5.0 kg, lying on a horizontal smooth table attached to an inextensible string of length 0.35 m, while the other end of the string is fixed to the table. The mass is whirled at a constant speed of 2.0 m s-1. Calculate
(a) the centripetal acceleration. (b) the tension in the string. (c) the period of motion.
3. The radius of the path of an object in uniform circular motion is doubled. The centripetal force needed if its speed remains the same is
A. half as great as before. B. the same as before. C. twice as great as before. D. four times as great as before.
4. A car rounds a curve of radius 70 m at a speed of 12 m s-1 on a level road. Calculate its centripetal
acceleration?
5. A 500 g sphere is hung from an inextensible string 1.25 m long and swung around to form a ‘conical pendulum’. The sphere moves in a circular horizontal path of radius 0.75 m. Determine the tension in the string.
1.25 m
0.75 m
6. Determine the maximum (constant) speed at which a car can safely round a circular curve of radius 50 m on a horizontal road if the coefficient of static friction (μ) between the tyres and the road is 0.7. (Use g = 10 m s–2). (HINT: if the normal
reaction is N, the relationship is F = μ N)
7. A sphere of mass m, attached to an inextensible string as shown in the diagram is released from rest at an angle θ with the vertical. When the sphere passes through its lowest point, show that the tension in the string is given by mg (3 - 2cosθ)
m kg l
θ
8. A 3 kg mass attached to a string 6 m long is to be swung in a circle at a constant speed making one complete revolution in 1.25 s. Determine the value that the breaking strain that the string must not exceed if the string is not to break when the circular motion is in
(a) a horizontal plane? (b) a vertical plane?
9. A mass, m kg, is released from point A, down a
smooth inclined plane and once it reaches point B, it completes the circular motion, via the smooth circular track B to C to D and then back through B, which is connected to the end of the incline and has a radius a / m. 4a A B C D a
Determine the normal reaction of the track on the mass
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Topic 2.1
1. Sketch the distance-time, speed-time, velocity- time and acceleration-time for a free fall parachutist from the time that she leaves the aeroplane to the time that she lands on the ground. (Remember that she does not open a parachute until some time after jumping from the aeroplane).
2. The graph below shows the idealised velocity-time graph for a car pulling away from one set of traffic lights until it is stopped by the next set.
sp ee d / m s –1 time / 0 20 40 60 80 100 120 140 0 5 10 15 20 25 Calculate the
i. acceleration of the car between 0 and 20 s and between 120 and 130 s.
ii. total distance travelled by the car during braking.
iii. total distance between the traffic lights.
3. A person drops a stone down a water well and hears a splash 2.0 s after it leaves his hand. Determine the depth of the well (g = 10 m s-2)
4. A girl throws a stone vertically upwards. The stone leaves her hand with a speed of 15.0 m s–1.
Determine (i) the maximum height reached by the stone and (ii) how long the time it takes to return to the ground after leaving her hand (g = 10 m s-2)
Topic 2.2
1. An object is thrown through the air. Ignoring air resistance, draw a free body diagram of the forces acting on the ball whilst it is in flight.
2. An object of weight 50 N is suspended vertically by two strings as shown
60°
The strings are of the same length and the angle between them is 60°. Draw a free body diagram of the forces acting on the object. Calculate the tension in the strings.
3. When a person stands on bathroom scales the scale reads 60 kg. Suppose the person stands on the same scales when in an elevator (lift). The elevator accelerates upwards at 2.0 m s-2.
Determine the new reading on the scale.
4. The diagram shows two blocks connected by a string that passes over a pulley.
B
A
Block A has a mass of 2.0 kg and block B a mass of 4.0 kg and rests on a smooth table.
Determine the acceleration of the two blocks?
5. Here are four statements about a book resting on a table.
A. The book exerts a force on the table. B. The table exerts a force on the book. C. the book exerts a force on the Earth. D. the Earth exerts a force on the book.
MISCELLANEOUS QUESTIONS ON TOPIC 2
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Which forces form a pair of forces as described by Newton’s Third Law?
6. When a golfer strikes a golf-ball it is in contact with the club head for about 1 ms and the ball leaves the club head with a speed of about
70 m s-1. If the mass of the ball is 50 g estimate the
maximum accelerating force exerted on the golf ball, stating any assumptions that you make.
7 A ball of mass 0.1 kg is dropped from a height of 2.0 m onto a hard surface. It rebounds to a height of 1.5 m and it is in contact with the surface for 0.05 s. Calculate the
i. speed with which it strikes the surface. ii. speed with which it leaves the surface. iii. change in momentum of the ball. iv. impulse given to the ball on contact with
the surface.
v. average force that the surface exerts on the ball.
8 A bullet of mass 0.02 kg is fired into a block of wood of mass 1.5 kg resting on a horizontal table. The block moves off with an initial speed of 8.0 m s-1. Estimate the speed with which the bullet
strikes the block.
9 The bullet in question 8 is fired from a rifle of mass 2.5 kg. Assuming that the bullet leaves the barrel of the rifle with the speed calculated above, find the recoil speed of the rifle if it is free to move. In reality the rifle is held and for a certain person the rifle recoils a distance of 0.12 m. Determine the average force that the person exerts on the rifle?
Topic 2.3
1. The diagram shows a pile driver that is used to
drive a metal bar (the pile) into the ground. pile driver
pile
ground
A particular pile driver has a mass of 500 kg and
it falls through a height of 2.5 m before striking the top of the pile. It stays in contact with the pile and drives it a distance of 0.40 m into the ground. Calculate the average force exerted by the ground on the pile by using
i. energy considerations
ii. the equations of uniform motion and Newton’s Second Law. (assume that the mass of the pile driver is much greater than the mass of the pile.)
2. A man slides a box of mass 50 kg at constant speed up an inclined slope to a height of 2.0 m. The slope makes an angle of 30° with the horizontal and it takes him 4 s to reach the height of 2.0 m and a constant frictional force of 250 N acts on the block.
Calculate
i. the work the man does against friction ii. the work the man does against gravity iii. the efficiency of the “man-slope machine” iv. the power the man develops to push the
block up the slope
3. This question is about calculating the power output of a car engine. Here is some data about a car that travels along a level road at a speed of 25 m s-1.
Fuel consumption = 0.20 litre km-1
Calorific value of the fuel = 5.0 × 106 J litre-1
Engine efficiency 50%
Determine:
i. the rate at which the engine consumes fuel ii. the rate at which the fuel supplies energy iii. the power output of the engine
iv. the power used to overcome the frictional forces acting on the car
v. the average frictional force acting on the car.
Explain why:
(i) the power supplied by the engine is not all used to overcome friction
(ii) the fuel consumption increases as the speed of the car increases