I. NEWTONIAN MECHANICS UNIT 1. INTRODUCTION
Physics Skills
UNIT 2. VECTORS AND EQUILIBRIUM
Vector Addition, Static and Rotational Equilibrium UNIT 3. KINEMATICS
Kinematics in One-Dimension Kinematics in Two-Dimensions UNIT 4. DYNAMICS
Newton's Second and Third Laws Friction
Uniform Circular Motion and Gravitation UNIT 5. CONSERVATION LAWS
Conservation of Energy Conservation of Momentum UNIT 6. SIMPLE HARMONIC MOTION
Spring Mass Systems and Simple Pendulum II. FLUID MECHANICS AND THERMAL PHYSICS
UNIT 7. FLUIDS
Buoyancy, Fluid Flow Continuity and Bernoulli's Equation UNIT 8. THERMODYNAMICS
Temperature and Kinetic Theory, Heat Transfer The Laws of Thermodynamics
III. WAVES AND OPTICS
UNIT 9. WAVES AND SOUND Mechanical Waves and Sound
Boundary Behavior and Wave Phenomena UNIT 10. GEOMETRIC AND PHYSICAL OPTICS
IV. ELECTRICITY AND MAGNETISM UNIT 11. ELECTROSTATICS
Electric Force, Field and Potential UNIT 12. ELECTRICITY
Electric Current, DC Circuits, Kirchhoff's Rules UNIT 13. MAGNETISM
Magnetic Force and Field Electromagnetic Induction V. ATOMIC AND NUCLEAR PHYSICS
UNIT 14. MODERN PHYSICS
Quantum Theory, Photoelectric Effect, Atomic Energy Levels
UNIT I INTRODUCTION WHAT IS PHYSICS?
Physics is the most basic of the experimental sciences. Physics provides the foundation for other scientific and technical disciplines. As a science that deals with matter and energy, physics explores a wide variety of fields.
The five major areas in Physics that we will study in the course are: 1. Newtonian Mechanics
Kinematics: The study of how objects move.
Dynamics: The study of the causes of motion: Newton's Laws. Circular motion and Gravitation
Conservations Laws: Energy and Momentum Oscillations
2. Fluid Mechanics and Thermodynamics
Fluid Mechanics: Fluids at rest and fluids in motion.
Thermodynamics: The relationship between heat and other properties. 3. Electricity and Magnetism
Electrostatics: Electric charge, field and potential. DC Circuits: Electric current and direct-current circuits. Electromagnetism
4. Waves and Optics
Wave motion.
Geometric Optics: reflection, refraction and optical devices. Physical Optics: Study of interference and diffraction phenomena. 5. Modern Physics
Atomic and Nuclear Physics. UNITS, STANDARDS AND THE SI SYSTEM The base units that will be used in this course are:
meter (m): One meter is equal to the path length traveled by light in vacuum during a time interval of 1/299,792,458 of a second.
kilogram (kg): One kilogram is the mass of a Platinum-Iridium cylinder kept at the International Bureau of Weights and Measures in Paris.
second (s): One second is the time occupied by 9,192,631,770 vibrations of the light (of a specified wavelength) emitted by a Cesium-133 atom.
SYSTEME INTERNATIONAL
The scientific community follows the Systeme International (SI), a program of weights and measures based on the metric system:
Quantity Unit SI symbol
Length meter m
Mass kilogram kg
Time second s
Electric current ampere A
Thermodynamic temperature Kelvin K
Amount of substance mole mol
Luminous intensity candela cd
SI PREFIXES
Power Prefix
Abbreviation Power Prefix Abbreviation
1012 tera- T 10-2 centi- c
109 giga- G 10-3 milli- m
106 mega- M 10-6 micro- μ
103 kilo- k 10-9 nano- n
10-12 pico- p
MATHEMATICAL NOTATION
Many mathematical symbols will be used throughout this course. = denotes equality of two quantities
denotes a proportionality < means is less than and > means greater than
means that two quantities are approximately equal to each other
x (read as “delta x”) indicates the change in the quantity x
PHYSICS SKILLS GRAPHING TECHNIQUES
Frequently an investigation will involve finding out how changing one quantity affects the value of another. The quantity that is deliberately manipulated is called the independent variable. The quantity that changes as a result of the independent variable is called the dependent variable. The relationship between the independent and dependent variables may not be obvious from simply looking at the written data. However, if one quantity is plotted against the other, the resulting graph gives evidence of what sort of relationship, if any, exists between the variables. When plotting a graph, take the following steps.
1. Identify the independent and dependent variables.
2. Choose your scale carefully. Make your graph as large as possible by spreading out the data on each axis. Let each space stand for a convenient amount. For example, choosing three spaces equal to ten is not convenient because each space does not divide evenly into ten. Choosing five spaces equal to ten would be better. Each axis must show the numbers you have chosen as your scale. However, to avoid a cluttered appearance, you do not need to number every space.
3. All graphs do not go through the origin (0,0). Think about your experiment and decide if the data would logically include a (0,0) point. For example, if a cart is at rest when you start the timer, then your graph of speed versus time would go through the origin. If the cart is already in motion when you start the timer, your graph will not go through the origin.
4. Plot the independent variable on the horizontal (x) axis and the dependent variable on the vertical (y) axis. Plot each data point. Darken the data points.
5. If the data points appear to lie roughly in a straight line, draw the best straight line you can with a ruler and a sharp pencil. Have the line go through as many points as possible with approximately the same number of points above the line as below. Never connect the dots. If the points do not form a straight line, draw the best smooth curve possible.
6. Title your graph. The title should dearly state the purpose of the graph and include the independent and dependent variables.
7. Label each axis with the name of the variable and the unit. Using a ruler, darken the lines representing each axis. The graph shown on the next page was prepared using good graphing techniques. Go back and check each of
The first column refers to the y-axis and the second column to the x-axis 1. 2. Volume (mL) Pressure (torr) 800 100 400 200 200 400 133 600 114 700 100 800 80 1000
Position (m) Time (s)
0 0 5 1 20 2 45 3 80 4 125 5
What type of curve did you obtain?
PART V. INTERPRETING GRAPHS
In laboratory investigations, you generally control one variable and measure the effect it has on another variable while you hold all other factors constant. For example, you might vary the force on a cart and measure its acceleration while you keep the mass of the cart constant. After the data are collected, you then make a graph of acceleration versus force, using the techniques for good graphing. The graph gives you a better understanding of the relationship between the two variables.
There are three relationships that occur frequently in physics:
Graph A: If the dependent variable varies directly with the independent variable, the graph will be a straight line
Graph B: If y varies inversely with x, the graph will be a hyperbola. Graph C. If y varies directly with the square of x, the graph is a parabola.
Speed (m/s) Time (s)
0 0 20 1 45 2 60 3 84 4 105 5
Reading from the graph between data points is called interpolation. Reading from the graph beyond the limits of your experimentally determined data points is called extrapolation. Extrapolation must be used with caution because you cannot be sure that the relationship between the variables remains the same beyond the limits of your investigation.
1. Suppose you recorded the following data during a study of the relationship of force and acceleration. Prepare a graph showing these data.
a. Describe the relationship between force and acceleration as shown by the graph.
b. What is the slope of the graph? Remember to include units with your slope. (1 N: 1 kg.m/s2)
c. What physical quantity does the slope represent?
d. Write an equation for the line.
e. What is the value of the force for an acceleration of 15 m/s2?
f. What is the acceleration when the force is 50.0 N?
Force (N)
Acceleration (m/s2)
10 6.0
20 12.5
30 19.0
2. The following data show the distance an object travels in certain time periods. Prepare a graph showing these data.
a. Describe the relationship between x and y and write a general equation for the curve.
b. Linearize the graph by plotting d vs. t2. Complete the table with the values.
e. What is the slope of the graph? Remember to include units with your slope. Position (m) Time (s)
0 0 2 1 8 2 18 3 32 4 50 5
Position (m) Time2(s2)
GENERAL PROBLEM SOLVING STRATEGY Solving problems require three major steps:
I. Prepare II. Solve III. Assess I. PREPARE
The "Prepare" step of a solution is where you identify important elements of the problem and collect information you will need to solve it.
It's tempting to jump right to the "solve" step, but a skilled problem solver will spend the most time on this step, the preparation. Preparation includes:
1. Identifying the Physics Principle(s):
Read the problem carefully and identify what is the underlying physics principle of the problem, then, write down the principle using an acronym such as the ones in the following list. If the problem has several steps, write down the principle(s) as appropriate.
Newton's First Law (N1L) Newton's Second Law (N2L)
Kinematics in One-Dimension (K1D) Kinematics in Two-Dimension (K2D) Conservation of Energy (COE) Conservation of Momentum (COM) Work-Energy Theorem (WE) Simple Harmonic Motion (SHM) First Law of Thermodynamics (T1L) 2. Data: Given and Unknown
Make a table of the quantities whose values you can determine from the problem statement or that can be found quickly with simple geometry or unit conversions. Any relevant constants should be written here. All units should be consistent with the SI values (i.e. kg, m, s). All unit conversion should take place in this section.
Also, identify the quantity or quantities that will allow you to answer the question. 3. Sketch, graph, FBD
In many cases, this is the most important part of a problem. The picture lets you model the problem and identify the important elements. As you add information to your picture, the outline of the solution will take shape. If appropriate, select a coordinate axis.
If the quantities involved are vectors, be sure to draw an arrow with the tip of the arrow clearly indicating the direction of the vector.
II. SOLVE
The "Solve" step of a solution is where you actually do the math or reasoning necessary to arrive to the solution needed. This is the part of the problem-solving strategy that you likely think of when you think of "solving problems". But don't make the mistake starting here! If you just choose an equation and plug in numbers, you will likely go wrong and will waste time trying to figure out why. The "Prepare" step will help you be certain you understand the problem before you start putting numbers in equations.
Solving the problem includes:
4. Equation (always solve for unknown)
Write the relevant equation or equations that will allow you to solve for the unknown. Be sure to always solve the equation for the unknown instead of just a 'plug and chug' approach.
5. Substitution
Once you have solved the equation algebraically, substitute the appropriate values. 6. Answer with Units
Write down the answer with the appropriate units. Remember that 'naked' numbers make no sense in Physics!
III. ASSESS
The "Assess" step of your solution is very important. When you have an answer, you should check to see if it makes sense.
7. Check the Answer Ask yourself:
- Does my solution answer the question that was asked?
Make sure that you have addressed all parts of the question and clearly written down your solutions.
- Does my answer have the correct units and number of significant figures? - Does the value I computed make physical sense?
- Can I estimate what the answer should be to check my solution?
UNIT II
VECTORS AND EQUILIBRIUM VECTORS AND SCALARS
A scalar quantity has only magnitude and is completely specified by a number and a unit. Examples are mass (a stone has a mass of 2 kg), volume (1.5 L), and frequency (60 Hz). Scalar quantities of the same kind are added by using ordinary arithmetic.
A vector quantity has both magnitude and direction. Examples are displacement (an airplane has flown 200 km to the southwest), velocity (a car is moving at 60 km/h to the north), and force (a person applies an upward force of 25 N to a package). When vector quantities are added, their directions must be taken into account.
A vector is represented by an arrowed line whose length is proportional to the vector quantity and whose direction indicates the direction of the vector quantity. Examples of vectors:
VECTOR COMPONENTS
A vector in two dimensions may be resolved into two component vectors acting along any two mutually perpendicular directions. The figure shows the vector F and its x and y vector
components:
2.1 Draw and calculate the components of the vector F (250 N, 235o)
VECTOR ADDITION: COMPONENT METHOD
To add two or more vectors A, B, C,… by the component method, follow this procedure: 1. Resolve the initial vectors into components x and y.
2. Add the components in the x direction to give Σx and add the components in the y direction to give Σy . That is, the magnitudes of Σx and Σy are given by, respectively:
Σx = Ax + Bx + Cx…
Σy = Ay + By + Cy…
Fx = F cos
3. Calculate the magnitude and direction of the resultant R from its components by using the
Pythagorean theorem: and
CW: VECTORS
FORCE
An object that experiences a push or a pull has a force exerted on it. We will consider the forces exerted ON the object. The object is called the system. The world around the object that exerts forces on it is called the environment.
A force has both magnitude and direction. Force is a vector quantity and therefore it can be represented with a free-body-diagram.
CONTACT VERSUS FIELD FORCES
Forces exerted by the environment on a system can be divided into two types:
- Contact Forces: act on an object only by touching it. Examples: tension, friction. - Field Forces or Long-Range Forces: are exerted without contact. Example: gravitational, magnetic, electric.
FREE-BODY-DIAGRAMS
A free-body-diagram (FBD) is a vector diagram that shows all the forces that act on an object whose motion is being studied.
2.4 Draw a FBD for each situation below following these directions: - Choose a coordinate system defining the positive direction of motion.
- Replace the object by a dot and locate it in the center of the coordinate system. - Draw arrows to represent the forces acting on the system.
1. Object lies motionless.
2. Object slides at constant speed without friction
4. Object slides without friction.
5. Static friction prevents sliding.
6. An object is suspended from the ceiling.
8. The object is motionless.
9. The object is motionless.
11. The object is pushed by a force applied downward at an angle.
.
12. The object is falling (no air resistance).
13. The object is falling at constant (terminal) velocity.
NEWTON'S FIRST LAW OF MOTION Newton's first law of motion is often stated as:
"An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force." There are two parts to this statement:
It is the natural tendency of objects to resist changes in their state of motion. This tendency to resist changes in their state of motion is described as inertia. The inertia of a body is related to the amount of matter it contains. A quantitative measure of inertia is mass. The SI unit of mass is the kilogram (kg).
The weight (FG) of an object is given in newtons (N).
FIRST CONDITION FOR EQUILIBRIUM
A body is in translational equilibrium if and only if the vector sum of the forces acting upon it is zero.
Σ Fx = 0 Σ Fy = 0
2.4 A block of weight 50 N hangs from a cord that is knotted to two other cords, A and B
CW: FIRST LAW
3. Find the tension in the ropes A and B for the Figure. (A = 1410 N, B = 1150 N)
6. A 441 N crate is being pulled by a rope, up a frictionless inclined plane, which meets the horizontal at an angle of 35˚.
a. Find the magnitude of the normal force (FN) acting on the crate. (361.2 N)
TORQUE
Torque is a measure of a force's ability to rotate an object. It depends on the magnitude of the applied force and on the length of the moment arm, according to the following equation:
τ = F r Units: N.m r is measured perpendicular to the line of action of the force F
Sign Convention:
Torque will be positive if F tends to produce counterclockwise rotation. Torque will be negative if F tends to produce clockwise rotation.
ROTATIONAL EQUILIBRIUM
An object is in rotational equilibrium when the sum of the forces and torques acting on it is zero. First Condition of Equilibrium: Σ Fx = 0 and Σ Fy = 0 (translational equilibrium)
Second Condition of Equilibrium: Σ τ = 0 (rotational equilibrium)
By choosing the axis of rotation at the point of application of an unknown force, problems may be simplified.
CENTER OF MASS
The terms "center of mass" and "center of gravity" are used synonymously to represent the average position of all the mass that makes up the object. For example, a symmetrical object, such as a ball, has its center of mass at its geometrical center.
CW: EQUILIBRIUM
7. Assume that the weight of the bar is negligible. Find forces A and F. (26.7 N, 106.7 N)
8. An 8-m steel metal beam weighs 2400 N and is supported 3 m from the right end. a. If a 9000-N weight is placed on the right end, what force must be exerted at the left end to balance the system? (4920 N)
b. What is the magnitude of the support force F? (6480 N)
A
9000 N
F
4 m 1 m 3 m
9. Find the forces F1, F2 and F3 such that the system is in equilibrium.
(212.5 lb, 253.6 lb, 83.9 lb)
UNIT 3 KINEMATICS MOTION
An object is in motion if its position changes. The mathematical description of motion is called kinematics. The simplest kind of motion an object can experience is uniform motion in a straight line. The object experiences translational motion if it is moving without rotating.
Any description of motion takes place in a coordinate system that allows us to track the position of an object. One-dimensional motion means that objects are only free to move back and forth along a single line. As a coordinate system for one-dimensional motion, choose this line to be an x-axis together with a specified origin and positive and negative directions. DISTANCE AND DISPLACEMENT
The primary difference between the two is that the distance an object travels tells you nothing about the direction of travel, while displacement tells you precisely how far, and in what direction, from its initial position an object is located. Distance is the total length of travel and displacement is the net length of travel accounting for direction.
3.1 You leave your home and drive 4.83 km North on Preston Rd. to go to the grocery store. After shopping, you go back home by traveling South on Preston Rd.
a. What distance do you travel during this trip?
b. What is your displacement?
SPEED
If an object takes a time interval t to travel a distance x, then the average speed of the object is given by:
Units: m/s
3.2 A ship steams at an average speed of 30 km/h. a. What is the speed in m/s?
b. How far in km does it travel in a day?
c. How many hours does it take to travel 500 km?
Average velocity is the displacement divided by the amount of time it took to undergo that displacement. The difference between average speed and average velocity is that average speed relates to the distance traveled while average velocity relates to the displacement.
3.3 A car travels north at 100 km/h for 2 h, at 75 km/h for the next 2 h, and finally turns south at 80 km/h for 1 h. What is the car’s average speed and average velocity for the entire journey?
3.4 Give a qualitative description of the motion depicted in the following x-versus-t graphs:
a. b.
c. d.
The slope of the position-time graphs yields the average velocity. When the velocity is constant, the average velocity over any time interval is equal to the instantaneous velocity at any time.
t x
t x
t t
3.5 Give a qualitative description of the motion depicted in the following v-versus-t graphs:
a. b.
The area under the velocity-time curve yields the displacement. ACCELERATION
Acceleration happens when: - An object's velocity increases - An object's velocity decreases - An object changes direction
Acceleration is the rate of change of velocity. The change in velocity is the final velocity vf minus the initial velocity vo.
The acceleration of an object is given by: Units: m/s2
EQUATIONS FOR MOTION UNDER CONSTANT ACCELERATION:
3.6 An object starts from rest with a constant acceleration of 8 m/s2 along a straight line. a. Find the speed at the end of 5 s
b. Find the average speed for the 5 s interval, and
c. Find the distance traveled in the 5 s
t v
3.7 A truck's speed increases uniformly from 15 km/h to 60 km/h in 20 s. a. Determine the average speed
b. Determine the acceleration
c. Determine the distance traveled, all in units of meters and seconds.
3.8 A skier starts from rest and slides 9.0 m down a slope in 3.0 s. In what time after starting will the skier acquire a speed of 24 m/s? Assume that the acceleration is constant.
3.9 A car moving at 30 m/s slows uniformly to a speed of 10 m/s in a time of 5.0 s. a. Determine the acceleration of the car and
3.10 The speed of a train is reduced from 15 m/s to 7.0 m/s while traveling a distance of 90 m. a. Calculate the acceleration.
b. How much farther will the train travel before coming to rest, provided the acceleration remains constant?
3.11 A drag racer starts from rest and accelerates at 7.40 m/s2. How far has it traveled in 1.00 s, 2.00 s, and 3.00 s? Graph the results in a position versus time graph.
This example illustrates one of the key features of accelerated motion; position varies directly with the square of the time.
3.12 Draw qualitative graphs of x-versus-t, v-versus-t, and a-versus-t.
t (s)
vo = 0
vo = 0
x = 0 x = 0
x v a
t t
t
x v a
t t
vo = 0
x = 0
x = 0
vo ≠ 0
x v a
t t
t
x v a
t t
ACCELERATION AND VELOCITY
If the sign of the velocity and the acceleration is the same then the object is speeding up. If the sign of the velocity and the acceleration is the opposite then the object is slowing down.
GRAPHICAL ANALYSIS OF MOTION
Graphical interpretations for motion along a straight line (the x-axis) are as follows: the slope of the tangent of an x-versus-t graph yields the instantaneous velocity, the slope of the v-versus-t graph yields the average acceleration,
the area under the v-versus-t graph gives the displacement, the area under the a-versus-t graph gives the change in velocity.
3.13. A boat moves slowly inside a marina with a constant speed of 1.50 m/s. As soon as it leaves the marina, it accelerates at 2.40 m/s2.
a. How fast is the boat moving after accelerating for 5.00 s?
b. Plot a graph of velocity versus time. x = 0
vo = 0
x v a
t t
t
c. Find how far has the boat traveled in this time by using the graph. Clearly show your work.
CW: VELOCITY AND ACCELERATION
1. A moving object increases its speed uniformly from 200 to 400 cm/s in 2 min. a. What is the average speed? (300 cm/s)
b. How far did it travel in the 2 min? (3.6 x 10 4 cm)
2. A train accelerates from rest at 1.4 m/s2. After covering a distance of 67 m, the train then travels at a constant velocity for 4.0 s. At that instant, the train is braked to a stop in 6.0 s. a. What is the total distance traveled? (163 m)
b. How much total time was required? (19.8 s)
3. You are driving your new sports car at a speed of 90 km/h when you suddenly see a dog step into the road 50 m ahead. You hit the brakes hard to get a maximum deceleration of 7.5 m/s2. How far will you go before stopping? Can you avoid hitting the dog? Justify your answer. (42 m)
5. A speeder doing 17.9 m/s in a 10 m/s zone approaches a parked police car. The instant the speeder passes the police car; the police begin their pursuit. If the speeder maintains a constant velocity, and the police accelerates with a constant acceleration of 4.5 m/s2.
a. How long does it take for the police car to catch the speeder? (7.96 s)
b. How far have the two cars traveled in this time? (143 m)
c. What is the velocity of the police car when it catches the speeder? (35.8 m/s)
6. Plot the corresponding graph:
a. x v
t
t
b. x v
t
t
t
t
d. v x
t
t
a. Uniform motion in the positive direction? ______ b. Uniform motion in the negative direction? __________ c. Rest? __________
d. Positive acceleration with positive velocity? __________ e. Negative velocity with positive acceleration? __________ f. Negative acceleration with positive velocity? __________ g. Negative velocity with negative acceleration? __________ h. Zero velocity and negative acceleration? __________ i. Positive acceleration and zero velocity? __________
a. Determine the sprinter's constant acceleration during the first 2 s. (5 m/s2)
b. Determine the sprinters velocity after 2 s have elapsed. (10 m/s)
c. Determine the total time needed to run the full 100 meters. (11 s)
d. Plot a neat graph of the displacement vs. time for the sprinter.
g = 9.8 m/s2
A body falling from rest in a vacuum thus has a velocity of 9.8 m/sat the end of the first second, 19.6 m/s at the end of the next second, and so forth. The farther the body falls, the faster it moves. A body in free fall has the same downward acceleration whether it starts from rest or has an initial velocity in some direction.
The presence of air affects the motion of falling bodies partly through buoyancy and partly through air resistance. Thus two different objects falling in air from the same height will not, in general, reach the ground at exactly the same time. Because air resistance increases with
velocity, eventually a falling body reaches a terminal velocity that depends on its mass, size, shape, and it cannot fall any faster than that.
FREE FALL
When air resistance can be neglected, a falling body has the constant acceleration g, and the equations for uniformly accelerated motion apply. Just substitute a for g.
Sign Convention for direction of motion:
If the object is thrown downward then g = 9.8 m/s2 If the object is thrown upward then g = - 9.8 m/s2
There are a few facts concerning free fall motion that you can use in analyzing situations. These facts can be deduced from the four equations for motion with constant acceleration.
When an object launched vertically upward reaches the top of its path (its maximum height), its instantaneous velocity is zero, even though its acceleration continues to be 9.8 m/s2 downward.
An object launched upward from a given height takes an equal amount of time to reach the top of its path as it takes to fall from the top of its path back to the height from which it was launched.
The velocity an object has at a given height, on its way up, is equal and opposite to the velocity it will have at that same height on its way back down.
b. How long does it take to reach the ground?
3.15 A stone is thrown straight upward and it rises to a height of 20 m. With what speed was it thrown?
3.16 A stone is thrown straight upward with a speed of 20 m/s. It is caught on its way down at a point 5.0 m above where it was thrown.
a. How fast was it going when it was caught?
b. How long did the trip take?
b. The time taken to reach that height,
c. Its velocity 30 s after it is throw, and
d. Its velocity when the ball's height is 100 m.
3.18 A rock is thrown vertically upward with a velocity of 20 m/s from the edge of a bridge 42 m above a river. How long does the rock stay in the air?
10. At Six Flags, Free Fall riders seated in a gondola are taken to the top of a 10-story tower. Then the gondola is dropped 30 m down a vertical track that curves near the bottom, where the gondola slows to a stop.
a. How long does it take to fall from top to bottom? (2.5 s)
b. What maximum speed is reached? (24 m/s)
11. A baseball thrown vertically up from the roof of a tall building has an initial velocity of 20 m/s. a. Calculate the time required to reach its maximum height. (2.04 s)
b. Find the maximum height it reaches (20.4 m)
c. Determine its position and velocity after 1.5 s. (19 m, 5.3 m/s)
d. Determine its position and velocity after 5 s. (-23 m, -29 m/s)
An object launched into space without motive power of its own is called a projectile.
If we neglect air resistance, the only force acting on a projectile is its weight, which causes its path to deviate from a straight line. The projectile has a constant horizontal velocity and a vertical velocity that changes uniformly under the influence of gravity.
HORIZONTAL PROJECTION
If an object is projected horizontally, its motion can best be described by considering its horizontal and vertical motion separately.
3.19 A cannonball is projected horizontally with an initial velocity of 120 m/s from the top of a cliff 250 m above a lake.
a. In what time will it strike the water at the foot of the cliff?
b. What is the x-distance from the foot of the cliff to the point of impact in the lake?
c. What are the horizontal and vertical components of its final velocity?
d. What is the final velocity at the point of impact and its direction?
3.20 A person standing on a cliff throws a stone with a horizontal velocity of 15.0 m/s and the stone hits the ground 47 m from the base of the cliff. How high is the cliff?
CW: PROJECTILE MOTION I
12. A box of supplies is dropped from an airplane that is located a vertical distance of 340 m above a lake. If the plane has a horizontal velocity of 70 m/s, what horizontal distance will the box travel before striking the water? (581 m)
14. Two tall buildings are 15 m apart. A ball is thrown horizontally from the roof of the first building 75 m from the ground. With what horizontal velocity must the ball be thrown if it is to enter a window of the second building 40 m from the ground? Draw a sketch first! (5.7 m/s)
15. A stone is thrown horizontally from the top of a building with an initial velocity of 200 m/s. At the same instant another stone is dropped from rest.
a. Find the position and velocity of the second stone after 3 s. (44.1 m, 29.4 m/s)
c. How far has it traveled vertically? (44.1 m)
d. What are the horizontal and vertical components of its velocity after 3 s? (200 m/s, 29.4 m/s)
16. A bomber flying with a horizontal velocity of 500 km/h releases a bomb. Six seconds later the bomb strikes the ocean below.
a. At what altitude was the plane flying? (176.4 m)
b. How far did the bomb travel horizontally? (833.3 m)
c. What are the magnitude and direction of its final velocity? (150.7 m/s, 23)
PROJECTILE MOTION AT AN ANGLE
Problem solution:
1. Upward direction is positive. Acceleration (g) is downward thus negative. 2. Resolve the initial velocity vo into its x and y components:
and
3. The horizontal and vertical components of its position at any instant is given by:
and
4. The horizontal and vertical components of its velocity at any instant are given by:
and
5. The final position and velocity can then be obtained from their components as a resultant.
3.21 An artillery shell is fired with an initial velocity of 100 m/s at an angle of 30 above the horizontal. Find:
b. The time required to reach its maximum height and
c. The horizontal distance (a.k.a. range R).
3.22 A baseball is thrown with an initial velocity of 120 m/s at an angle of 40above the horizontal. How far from the throwing point will the baseball attain its original level?
b. What was the maximum height achieved by the ball?
3.24 a. Find the range of a gun which fires a shell with muzzle velocity vo at an angle θ.
b. What is the angle at which the maximum range is possible?
c. Find the angle of elevation θ of a gun that fires a shell with muzzle velocity of 120 m/s and hits a target on the same level but 1300 m distant.
CW: PROJECTILE MOTION II
b. How long will it take this projectile to reach the highest point in its trajectory? (5.6 s)
c. How long will this projectile be in the air? (11.22 s)
d. What will be the velocity of this projectile at the highest point? (78.6 m/s)
e. What will be the velocity of this projectile as it reaches the ground? (55 m/s)
f. How high will this projectile be at the highest point of its trajectory? (154 m)
g. What will be the range of this projectile? (885 m)
b. How much time is it in the air? (2.04 s)
c. What is its horizontal range? (35.3 m)
8. An arrow is shot into the air with a velocity of 120 m/s at an angle of 37
a. What are the horizontal and vertical components of its initial velocity? (95.8 m/s, 72.2 m/s)
b. What is the position (x and y) after 2 s? (192 m, 124.8 m)
c. What are the magnitude and direction of the resultant velocity after 2 s? (109.2 m/s, 28.7)
b. What horizontal distance did it travel? (14 m)
20. An arrow was shot at an angle of 55º with respect to the horizontal. The arrow landed at a horizontal distance of 875 m. Find the velocity of the arrow at the top of its path. (55 m/s)
If there is a net force acting on an object, the object will have an acceleration and the object's velocity will change. How much acceleration will be produced by a given force?
Newton's Second Law states that for a particular force, the acceleration of an object is
proportional to the net force and inversely proportional to the mass of the object. The direction of the force is the same as that of the acceleration. In equation form:
or F = ma
In the SI system, the unit for force is the newton (N): A newton is that net force which, when applied to a 1-kg mass, gives it an acceleration of 1 m/s2.
The second law of motion is the key to understanding the behavior of moving bodies since it links cause (force) and effect (acceleration) in a definite way.
THIRD LAW OF MOTION
According to Newton's third law of motion, when one body exerts a force on another body, the second body exerts on the first an equal force in opposite direction.
The Third Law of Motion applies to two different forces on two different objects: "The action force one object exerts on the other, and the equal but opposite reaction force the second object exerts on the first." Action and reaction forces never balance out because they act on different objects.
WEIGHT
The weight of a body is the gravitational force with which the Earth attracts the body.
Weight (a vector quantity) is different from mass (a scalar quantity). The weight of a body varies with its location near the Earth (or other astronomical object), whereas its mass is the same everywhere in the universe. The weight of a body is the force that causes it to be accelerated downward with the acceleration of gravity g. From the Second Law of Motion:
FG = mg Units: Newton (N)
Where the acceleration due to gravity g = 9.8 m/s2
THE NORMAL FORCE
A normal force is a force exerted by one surface on another in a direction perpendicular to the surface of contact.
Note: The gravitational force and the normal force are not an action-reaction pair.
4.2 A cord passing over a frictionless pulley has a 7 kg mass hanging from one end and a 9 kg mass hanging from the other. (This arrangement is called Atwood's machine).
a. Find the acceleration of the masses.
b. Find the tension of the cord
APPARENT WEIGHT
4.3 What will a spring scale read for the weight of a 75 kg man in an elevator that moves: a. With constant upward speed of 5 m/s
b. With constant downward speed of 5 m/s
c. With upward acceleration of 0.25 g
d. With downward acceleration of 0.25 g
e. In free fall?
1. A student whose normal weight is 500 N stands on a scale in an elevator and records the scale reading as a function of time. The data are shown in the graph above. At time t = 0, the elevator is at displacement x = 0 with velocity v = 0. Assume that the positive directions for displacement, velocity, and acceleration are upward.
a. On the diagram below, draw and label all of the forces on the student at t = 8 seconds.
b. Calculate the acceleration a of the elevator for each 5-second interval. (all m/s2: 0, 4, 0, -4)
a (m/s2)
ii. Plot the acceleration as a function of time on the following graph.
c. Determine the velocity v of the elevator at the end of each 5-second interval. (all m/s: 0, 20, 20, 0)
v (m/s)
ii. Plot the velocity as a function of time on the following graph.
d. Determine the displacement x of the elevator above the starting point at the end of each 5-second interval. (all m: 0, 50, 150, 200)
x (m)
2. A crane is used to hoist a load of mass m1 = 500 kg. The load is suspended by a cable from a hook of mass m2 = 50 kg, as shown in the diagram above. The load is lifted upward at a constant acceleration of 2 m/s2.
a. On the diagrams below draw and label the forces acting on the hook and the forces acting on the load as they accelerate upward.
3. A helicopter holding a 70-kg package suspended from a rope 5.0 m long accelerates upward at a rate of 5.2 m/s2. Neglect air resistance on the package.
a. On the diagram below, draw and label all of the forces acting on the package.
b. Determine the tension in the rope. (1050 N)
4. The vertical position of an elevator as a function of time is shown below.
a. From the graph determine the velocity of the elevator for each interval of time. (All in m/s: 1.5 m/s, 0, -2.4 m/s)
b. On the grid below, graph the velocity of the elevator as a function of time.
ii. On the box below that represents the elevator, draw a vector to represent the direction of this average acceleration.
c. Suppose that there is a passenger of mass 70 kg in the elevator. Calculate the apparent weight of the passenger at time t = 4 s. (686 N)
FRICTION: STATIC AND KINETIC FRICTION
When two surfaces are in direct contact and one surface either slides or attempts to slide across the other, a force, called friction, that opposes the motion (or attempted motion) is generated between the surfaces. The origin of friction is based on the microscopic structure of the surfaces involved.
Static friction occurs between surfaces at rest relative to each other. When an increasing force is applied to a book resting on a table, for instance, the force of static friction at first increases as well to prevent motion. In a given situation, static friction has a certain maximum value called starting friction. When the force applied to the book is greater than the starting friction, the book begins to move across the table. The kinetic friction (or sliding friction) that occurs afterward is usually less than the starting friction, so less force is needed to keep the book moving than to start it moving.
COEFFICIENT OF FRICTION
The frictional force between two surfaces depends on the normal (perpendicular) force FN pressing them together and on the natures of the surfaces. This factor is expressed quantitatively in the coefficient of friction (mu) whose value depends on the materials in contact.
The frictional force is experimentally found to be:
4.4 A horizontal force of 140 N is needed to pull a 60.0 kg box across the horizontal floor at constant speed. What is the coefficient of friction between floor and box?
4.5 A 70-kg box is slid along the floor by a horizontal 400 N force. Find the acceleration of the box if the value of between the box and the floor is 0.50.
4.7 A box sits on an incline that makes an angle of 30˚ with the horizontal. Find the acceleration of the box down the incline if = 0.30.
4.8 Two blocks m1 (300 g) and m2 (500 g), are pushed by a force F. If = 0.40. a. What must be the value of F if the blocks are to have an acceleration of 200 cm/s2?
4.9 An object mA = 25 kg rests on a tabletop. A rope attached to it passes over a light pulley and is attached to a mass mB = 15 kg. The coefficient of friction between the table and block A is = 0.20. a. What is the acceleration of the 25 kg block?
b. What is the tension in the string?
a. On the figure below, draw and label all the forces on block ml.
Express your answers to each of the following in terms of ml, m2, g, , and f.
b. Determine the coefficient of kinetic friction between the inclined plane and block 1.
d. The string between blocks 1 and 2 is now cut. Determine the acceleration of block 1 while it is on the inclined plane.
CW: FORCES II. Include a neat annotated FBD on every problem.
5. A force of 490 N pulls a 30 kg block horizontally across the floor. If = 0.1, find the acceleration of the block. (15.3 m/s2)
6. A 30 kg mass rests on a 37 inclined plane. The = 0.30. A push FP is applied parallel to the plane and directed up the plane causing the mass to accelerate at 3 m/s2.
a. What is the normal force? (234.8 N)
c. What is the magnitude of the push FP? (337.3 N)
d. What is the resultant force up the plane? (90 N)
7. A 49 N block rests on a tabletop. A rope attached to it passes over a light frictionless pulley and is attached to a mass of 2.0 kg. If = 0.20
a. Find the acceleration of the hanging mass. (1.4 m/s2)
8. A skier has just begun descending a 30 slope. If the = 0.10 calculate her acceleration (3.92 m/s2)
9. A force of 400 N pushes on a 25-kg box at an angle θ of 50 causing the box to accelerate at 0.5 m/s2. Find the coefficient of friction between the box and the floor. (0.44)
10.A 441 N sled is sitting on an icy horizontal surface that has a =0.155. A rope is attached to the front end of the sled at an angle of 28˚ between the rope and the horizontal. A force F is applied to the rope so as to pull the sled a constant speed i.e. a = 0.
a. What is the magnitude of the force F applied to the rope of the sled? (71.5 N)
b. What is the magnitude of the frictional force Ff acting on this sled? (63.15 N)
11.
a. Calculate the acceleration of the 10-kilogram block in case I. (3 m/s2)
b. Calculate the acceleration of the 10-kilogram block in case II. (2 m/s2)
c. Calculate the tension of the string in case II. (40 N)
The coefficient of friction between the 10 kg block and the table is 0.2. Use g = 10 m/s2.
Uniform circular motion is motion in which there is no change in speed, only a change in direction.
CENTRIPETAL ACCELERATION
An object experiencing uniform circular motion is continually accelerating. The position and velocity of a particle moving in a circular path of radius r are shown at two instants in the figure. When the particle is at point A, its velocity is represented by vector v1. After a time interval t, its velocity is represented by the vector v2. The acceleration is given by:
The change in velocity v is represented graphically in the figure below:
The centripetal acceleration is given by: Units: m/s2
v is the linear speed of a particle moving in a circular path of radius r.
The term centripetal means that the acceleration is always directed toward the center. The velocity and the acceleration are not necessarily in the same direction; v points in the direction of motion which tangential to the circle. v and a are perpendicular at every point.
Another useful parameter in engineering problems is the rotational speed, expressed in
revolutions per minute (rpm), revolutions per second (rev/s) or Hz (s-1). This quantity is called the frequency f of rotation and is given by the reciprocal of the period.
4.11 A 2 kg object is tied to the end of a cord and whirled in a horizontal circle of radius 2 m. If the body makes three complete revolutions every second, determine its linear speed and its centripetal acceleration.
4.12 A ball is whirled at the end of a string in a horizontal circle 60 cm in radius at the rate of 0.5 Hz. Find the ball's centripetal acceleration.
CENTRIPETAL FORCE
The inward force necessary to maintain uniform circular motion is defined as centripetal force. From Newton's Second Law, the centripetal force is given by:
Units: Newtons (N)
4.13 a. A car makes a turn, what force is required to keep it in circular motion?
b. A bucket of water is tied to a string and spun in a circle, what force is required to keep it in circular motion?
c. The moon orbits the Earth, what force is required to keep it in circular motion?
4.14 A 75 g toy airplane is fastened to one end of a 44 cm string while the other end is held fixed at the ceiling. The plane whirls in a horizontal circle with an angle to the vertical of 30˚.
a. What provides the centripetal force?
b. Calculate the speed of the plane
4.15 In a Rotor ride at the amusement park the room radius is 4.6 m and the rotation frequency is 0.5 revolutions per second when the floor drops out. What is the minimum coefficient of static friction so that the people will not slip down?
CW: CENTRIPETAL FORCE
12. A 2 kg object is tied to a cord and swung in a horizontal circle of radius 1 m with a frequency of 80 rpm. Determine:
b. The centripetal acceleration, (69.8 m/s2)
c. The centripetal force. (139.6 N)
d. What happens if the cord breaks?
13. a. What must the speed of a satellite be just above the surface of the earth if it is to travel in a circular orbit about the earth? (Rearth= 4000 mi, 1 mi = 1609 m) (7941.8 m/s)
b. What is the nature of the centripetal force in this case?
14. A 25 cm diameter record turns on a record player at 78 rpm. A bug rests on the record 2.5 cm from the outside edge.
a. If the bug's mass is 0.01 kg, what force acts on him? (6.67x10-2 N)
b. What exerts this force?
15. An object of mass M on a string is whirled with increasing speed in a horizontal circle, as shown above. When the string breaks, the object has speed vo and the circular path has radius R and is a height h above the ground. Neglect air friction.
a. Determine the following, expressing all answers in terms of h, vo, and g.
i. The time required for the object to hit the ground after the string break. ( )
ii. The horizontal distance the object travels from the time the string breaks until it hits the ground.
( )
b. On the figure below, draw and label all the forces acting on the object when it is in the position shown in the diagram.
.
c. Determine the tension in the string just before the string breaks. Express your answer in terms of M, R, vo, and g.
MOTION IN A VERTICAL CIRCLE
At the lowest point in the loop the centripetal force is:
When a body moves in a vertical circle at the end of a string, the tension FT in the string varies with the body's position. The centripetal force Fc on the body at any point is the vector sum of FT and the
component of the body's weight toward the center of the circle.
4.16 A string 0.5 m long is used to whirl a 1-kg stone in a vertical circle at a velocity of 5 m/s. a. What is the tension in the string when the stone is at the top of the circle and
b. When the stone is at the bottom of the circle?
4.17 A ball of mass M is attached to a string of length R
and negligible mass. The ball moves clockwise in a vertical circle, as shown. When the ball is at point P, the string is horizontal. Point Q is at the bottom of the circle and point Z is at the top of the circle. Air resistance is negligible. Express all algebraic answers in terms of the given quantities and fundamental constants.
b. Derive an expression for vmin the minimum speed the ball can have at point Z without leaving the circular path.
c. The maximum tension the string can have without breaking is Tmax Derive an expression for
vmax, the maximum speed the ball can have at point Q without breaking the string.
d. Suppose that the string breaks at the instant the ball is at point P. Describe the motion of the ball immediately after the string breaks.
CW: VERTICAL CIRCLES
17. To study circular motion, two students use the hand-held device shown above, which consists of a rod on which a spring scale is attached. A polished glass tube attached at the top serves as a guide for a light cord attached the spring scale. A ball of mass 0.200 kg is attached to the other end of the cord. One student swings the teal around at constant speed in a horizontal circle with a radius of 0.500 m. Assume friction and air resistance al negligible.
a. Explain how the students, by using a timer and the information given above, can determine the speed of the ball as it is revolving.
b. The speed of the ball is determined to be 3.7 m/s. Assuming that the cord is horizontal as it swings, calculate the expected tension in the cord. (5.5 N)
d. The students find that, despite their best efforts, they cannot swing the ball so that the cord remains exactly horizontal.
i. On the picture of the ball below, draw vectors to represent the forces acting on the ball and identify the force that each vector represents.
ii. Explain why it is not possible for the ball to swing so that the cord remains exactly horizontal.
iii. Calculate the angle that the cord makes with the horizontal. (20°)
KEPLER'S LAWS
KEPLER’S FIRST LAW
KEPLER’S THIRD LAW
"If T is the period and r is the length of the semi-major axis of a planet’s orbit, then the ratio T2/r3 is the same for all planets."
4.17 The mean distance from the Earth to the Sun is 1.496x108 km and the period of its motion about the Sun is one year. The period of Jupiter’s motion around the Sun is 11.86 years. Determine the mean distance from the Sun to Jupiter.
NEWTON’S LAW OF UNIVERSAL GRAVITATION
Newton eventually proved that Kepler’s first two laws imply a law of gravitation: Any two objects in the Universe exert an attractive force on each other -called the gravitational force- whose strength is directly proportional to the product of the objects’ masses and inversely proportional to the square of the distance between them. If we let G be the universal
KEPLER’S SECOND LAW
Units: Newton (N)
Cavendish determined the first reasonably accurate numerical value for G more than one hundred years after Newton’s Law was published. To three decimal places, the currently accepted value is: G = 6.67 x10-11 N.m2/kg
4.18 Derive Kepler’s Third Law from Newton’s Law of Gravitation.
4.19 What is the force of gravity acting on a 2000 kg spacecraft when it orbits two Earth radii from the Earth’s center above the Earth’s surface? (RE = 6380 km, ME=5.98 x1024 kg.)
4.21 a. Derive the expression for g from the Law of Universal Gravitation.
SATELLITES
A satellite is put into orbit by accelerating it to a sufficiently high tangential speed with the use of a rocket. Satellites are usually put into circular (or nearly circular) orbits because they require the least takeoff speed. At the very high speed a satellite has, it would quickly fly out into space if it weren’t for the gravitational force of the Earth pulling it into orbit. In fact, a satellite is falling (accelerating toward the Earth), but its high tangential speed keeps it from hitting Earth. The force that gives a satellite its acceleration is the force of gravity.
Artificial satellites are launched at different speeds in order to obtain different orbits: for a circular orbit: 27,000 km/h; 30,000 km/h for an elliptical orbit and 40,000 km/h for an escape velocity.
4.20 A geosynchronous satellite is one that stays above the same point on the equator of the Earth. Determine:
a. The height above the Earth’s surface such a satellite must orbit and
CW: GRAVITATION
18. The radius of a planet is 3400 km. If an object weighs 550 N at the surface of the planet, what is its weight 210 km above the surface? (490 N)
19. Determine the speed of the Hubble space telescope orbiting at a height of 596 km above the Earth's surface (RE = 6.38 x106 m, ME = 5.98 x1024 kg) (7.56 x103 m/s)
21. Find the net force on the Moon (mM= 7.35 x1022 kg) due to the gravitational attraction of both the Earth and the Sun (mS= 1.99 x103 kg), assuming they are at right angles to each other.
UNIT 5
CONSERVATION LAWS WORK
In order for work to be done, three things are necessary:
o There must be an applied force.
o The force must act through a certain distance, called the displacement.
o The force must have a component along the displacement.
Work is a scalar quantity equal to the product of the magnitudes of the displacement and the component of the force in the direction of the displacement.
W = F . x W = F cos x
The units of work are the units of N.m, this unit is called a Joule (J)
If the force acting on an object varies in magnitude and/or direction during the object’s displacement, graphical analysis can be used to determine the work done. F is plotted on the y-axis and the distance through which the object moves is plotted on the x-axis. The work done is represented by the area under the curve.
5.1 A push of 200 N moves a 100 N block up a 30inclined plane. The coefficient of kinetic friction is 0.25 and the length of the plane is 12 meters.
b. Show that the net work done by these forces is the same as the work of the resultant force.
ENERGY
Energy is that which can be converted into work. When something has energy, it is able to perform work or, in a general sense, to change some aspect of the physical world. In mechanics we are concerned with two kinds of energy:
KINETIC ENERGY: K, energy possessed by a body by virtue of its motion. Units: Joules (J)
POTENTIAL ENERGY: PE, energy possessed by a system by virtue of position or condition. PE = m g h Units: Joules (J)
WORK-ENERGY THEOREM: The work of a resultant external force on a body is equal to the change in kinetic energy of the body.
W = K Units: Joules (J)
CONSERVATIVE AND NON-CONSERVATIVE FORCES
The work done by a conservative force depends only on the initial and final position of the object acted upon. An example of a conservative force is gravity. The work done equals the change in potential energy and depends only on the initial and final positions above the ground and NOT on the path taken.
Friction is a conservative force and the work done in moving an object against a non-conservative force depends on the path. For example, the work done in sliding a box of books against friction from one end of a room to the other depends on the path taken.
For mechanical systems involving conservative forces, the total mechanical energy equals the sum of the kinetic and potential energies of the objects that make up the system and is always conserved.
ΔPE = ΔK
In real life applications, some of the mechanical energy is lost due to friction. The work due to non-conservative forces has to be taken into account:
WNC = ΔPE + ΔK or WNC = Ef - Eo
5.3 A ballistic pendulum apparatus has a 40-g ball that is caught by a 500-g suspended mass. After impact, the two masses rise a vertical distance of 45 mm. Find the velocity of the combined masses just after impact.
b. Find the work due to non-conservative forces during the descent if the actual velocity at the bottom is 41 m/s.
CW: WORK AND ENERGY
1. A 10-kg sled traveling initially at 2 m/s is given a steady push P for a distance of 12 m. What must be the value of P in order to increase the speed to 4 m/s in this distance?
Calculate first from work-energy considerations and then using Newton's Second Law. (5 N)
2. Consider a 64 N object at a height of 10 m above a concrete floor. From energy
3. a. Find the work required to move at constant speed a 4-kg box up an incline 20 m long and 16 m high if the k = 0.3 (768 J)
b. Calculate the tension in the horizontal string. (10.18 N)
c. The horizontal string is now cut close to the bob, and the pendulum swings down. Calculate the speed of the bob at its lowest position. (2.5 m/s)
4. A simple pendulum consists of a bob of mass 1.8 kg attached to a string of length
2.3 m. The pendulum is held at an angle of 30° from the vertical by a light horizontal string attached to a wall, as shown.
5. A roller coaster ride at an amusement park lifts a car of mass 700 kg to point A at a height of 90 m above the lowest point on the track, as shown above. The car starts from rest at point A, rolls with negligible friction down the incline and follows the track around a loop of radius 20 m. Point B, the highest point on the loop, is at a height of 50 m above the lowest point on the track. a. i. Indicate on the figure the point P at which the maximum speed of the car is attained.
ii. Calculate the value vmax. of this maximum speed. (42 m/s)
b. Calculate the speed vB of the car at point B. (28 m/s)
c.i. On the figure of the car below, draw and label vectors to represent the forces acting on the car when it is upside down at point B.
POWER
Power is the rate at which work is performed. P = work/time P = W = F x = F v Units: J/s: watt (W) t t
5.6 A 1100-kg car accelerates from rest in a time of 5.0 s. The magnitude of the acceleration is 4.6 m/s2. What power must the motor produce to cause this acceleration?
CW: POWER
7. The Sojourner rover vehicle shown in the sketch above was used to explore the surface of Mars as part of the Pathfinder mission in 1997. Use the data in the tables below to answer the questions that follow.
Mars Data Sojourner Data
Radius: 0.53 x Earth's radius Mass of Sojourner vehicle: 11.5 kg
Mass: 0.11 x Earth's mass Wheel diameter: 0.13 m
Stored energy available: 5.4 x 105 J Power required for driving
under average conditions: 10 W
Land speed: 6.7 x 10-3 m/s a. Determine the maximum distance that Sojourner can travel on a horizontal Martian surface using its stored energy. (362 m)
b. Suppose that 0.010% of the power for driving is expended against atmospheric drag as Sojourner travels on the Martian surface. Calculate the magnitude of the drag force. (0.15 N)
ELASTIC POTENTIAL ENERGY
The force Fpapplied to a spring to stretch it or to compress it an amount x is directly proportional to x. That is:
Fs = - k x Units: Newtons (N)
Where k is a constant called the spring constant and is a measure of the stiffness of the particular spring. The spring itself exerts a force in the opposite direction:
