MP02: Motion Diagrams
Velocity and Acceleration of a Power Ball
Learning Goal: To understand the distinction between velocity and acceleration with the use of motion diagrams. In common usage, velocity and acceleration both can imply having considerable speed. In physics, they are sharply defined concepts that are not at all synonymous. Distinguishing clearly between them is a prerequisite to
understanding motion. Moreover, an easy way to study motion is to draw a motion diagram, in which the position of the object in motion is sketched at several equally spaced instants of time, and these sketches (or snapshots) are combined into one single picture.
In this problem, we make use of these concepts to study the motion of a power ball. This discussion assumes that we have already agreed on a coordinate system from which to measure the position (also called the position vector) of objects as a function of time. Let and be velocity and acceleration, respectively.
Harvaran Ghai Consider the motion of a power ball that is dropped on the floor and bounces back. In the following questions, you will describe its motion at various points in its fall in terms of its velocity and acceleration.
Part A
You drop a power ball on the floor. The motion diagram of the ball is sketched in the figure . Indicate whether the magnitude of the velocity of the ball is increasing, decreasing, or not changing.
increasing decreasing not changing Correct
While the ball is in free fall, the magnitude of its velocity is increasing, so the ball is accelerating. Part B
Since the length of is directly proportional to the length of , the vector connecting each dot to the next could represent velocity vectors as well as position vectors, as shown in the figure here . Indicate whether the velocity and acceleration of the ball are, respectively, positive (upward), negative, or zero.
Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and acceleration with a comma.
N,N Correct
Part C
Now, consider the motion of the power ball once it bounces upward. Its motion diagram is shown in the figure here . Indicate whether the magnitude of the velocity of the ball is increasing, decreasing, or not changing.
increasing decreasing not changing Correct
Since the magnitude of the velocity of the ball is decreasing, the ball must be accelerating (specifically, slowing down).
Part D
The next figure shows the velocity vectors corresponding to the upward motion of the power ball. Indicate whether its velocity and acceleration, respectively, are positive (upward), negative, or zero.
Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and acceleration with a comma.
P,N Correct
Part E
The power ball has now reached its highest point above the ground and starts to descend again. The motion diagram representing the velocity vectors is the same as that after the initial release, as shown in the figure of Part B. Indicate whether the velocity and acceleration of the ball at its highest point are positive (upward), negative, or zero.
Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and acceleration with a comma.
Z,N Correct
These examples should show you that the velocity and acceleration can have opposite or similar signs or that one of them can be zero while the other has either sign. Try hard to think carefully about them as distinct physical
Motion of Two Rockets Learning Goal: To learn to use images of an object in motion to determine velocity and acceleration. Two toy rockets are traveling in the same direction (taken to be the x axis). A diagram is shown of a timeexposure image where a stroboscope has illuminated the rockets at the uniform time intervals indicated. Harvaran Ghai Part A At what time(s) do the rockets have the same velocity? at time only at time only at times and at some instant in time between and at no time shown in the figure Correct Part B At what time(s) do the rockets have the same x position? at time only at time only at times and at some instant in time between and
at no time shown in the figure Correct Part C At what time(s) do the two rockets have the same acceleration? at time only at time only at times and at some instant in time between and at no time shown in the figure Correct Part D The motion of the rocket labeled A is an example of motion with uniform (i.e., constant) __________. and nonzero acceleration velocity displacement time Correct Part E The motion of the rocket labeled B is an example of motion with uniform (i.e., constant) __________. and nonzero acceleration velocity displacement time Correct Part F At what time(s) is rocket A ahead of rocket B? before only after only before and after
between and at no time(s) shown in the figure
Correct
PSS 1.1: (Almost) a Dozen Diagrams
Learning Goal: To practice ProblemSolving Strategy 1.1 for constructing motion diagrams. A car is traveling with constant velocity along a highway. The driver notices he is late for work so he stomps down on the gas pedal and the car begins to accelerate. The car has just achieved double its initial velocity when the driver spots a policeman behind him and applies the brakes. The car then decelerates, coming to rest at a stoplight ahead. In this problem, you will be asked several questions related to construction of a motion diagram for this situation and a few others. Harvaran Ghai MODEL:Represent the moving object as a particle. Make simplifying assumptions when interpreting the problem statement. VISUALIZE:A complete motion diagram consists of: The position of the object in each frame of the film, shown as a dot. Use five or six dots to make the motion clear but without overcrowding the picture. More complex motions may need more dots.
The average velocity vectors, found by connecting each dot in the motion diagram to the next with a vector arrow. There is one velocity vector linking each set of two position dots. Label the row of velocity vectors . The average acceleration vectors, found using Tactics Box 1.3. There is one acceleration vector linking each set of two velocity vectors. Each acceleration vector is drawn at the dot between the two velocity vectors it links. Use to indicate a point at which the acceleration is zero. Label the row of acceleration vectors . Model It is appropriate to use the particle model for the car. You should also make some simplifying assumptions. Part A Which of the following simplifying assumptions is it reasonable to make in this problem? A. During each of the three different stages of its motion, the car is moving with constant (possibly zero) acceleration. B. During each of the three different stages of its motion, the car is moving with constant (possibly zero) velocity. C. The highway is straight (i.e., there are no curves). D. The highway is level (i.e., there are no hills or valleys). Enter the letters of all the correct answers in alphabetical order. Do not use commas. For example, if you think that assumptions C and D are reasonable, enter CD. ACD Correct Visualize Now draw a motion diagram, including all the elements listed in the problemsolving strategy. Use your diagram to answer the following questions. In interpreting the diagrams that follow, assume that the car is moving in a straight line to the right. Refer to this set of motion diagrams in answering the following. Part B Which of the diagrams best describes the position and the velocity of the car before the driver notices he is late? A B C Correct
Part C Which of the diagrams best describes the position and the velocity of the car after the driver hits the gas, but before he notices the policeman? A B C Correct Part D Which of the diagrams best describes the position and the velocity of the car after the driver notices the policeman? A B C Correct Part E Which of these diagrams most accurately depicts the acceleration of the car during the events described in the problem introduction? Assume that the car is initially moving to the right. A B C Correct Now let's use our results for the car and apply them to some other problems. Consider these three situations: A train has its brakes released and pulls out of the station, slowly picking up speed. A sled is given a quick push along a horizontal surface; the sled comes to a stop after covering some distance. A motorcycle is moving along a straight highway at 105 km/h (the legal speed limit in many states). Part F Of the three situations described, which object corresponds to the position and velocity diagram shown here? the train the sled
the motorcycle Correct Note that the diagram shown is not a complete motion diagram; it lacks the vector representing the acceleration of the object. Part G Of the three situations described, which object corresponds to the motion diagram shown here? the train the sled the motorcycle Correct Let us now consider another scenario. A car and a truck are moving at the same velocity along a straight highway. Both drivers apply the brakes at the same moment. The car and truck both come to a stop. The car takes less time to stop than the truck. Refer to the motion diagrams shown here in answering the following. Assume that both cars are moving to the right. Part H Which of the three diagrams shown best describes the motion of the car and truck after the brakes have been applied? A B C Correct Part I Diagram (B) is incorrect because, according to it: The car and truck move in different directions. The car is moving at constant velocity. The truck is speeding up. The car and the truck have the same acceleration. Correct
Part J Diagram (C) is incorrect because, according to it: The car and truck move in different directions. The car is moving at constant velocity. The truck is speeding up. The car and the truck have the same acceleration. Correct
MP04: Using Motion Diagrams
Curved Motion Diagram
The motion diagram shown here represents a pendulum released from rest at an angle of 45 from the vertical. The dots in the motion diagram represent the positions of the pendulum bob at eleven moments separated by equal time intervals. The green arrows represent the average velocity between adjacent dots. Also given is a "compass rose" of directions in which the different directions are labeled with the letters of the alphabet. Harvaran Ghai Part A What is the direction of the acceleration of the pendulum bob at moment 5? Enter the letter of the arrow with this direction from the compass rose in the figure. Type Z if the acceleration vector has zero length. A Correct
Part B What is the direction of the acceleration of the pendulum bob at moments 0 and 10? Enter the letters of the arrows with these directions from the compass rose in the figure, separated by commas. Type Z if the acceleration vector has zero length. directions at moment 0, moment 10 =D,F Correct Part C In which of the following other scenarios could the motion reasonably be represented by the motion diagram in the introduction? A. A weight placed on the rim of a bicycle wheel that is being held off the ground so it can rotate freely B. An airplane pulling out of a dive C. A race car rounding a turn D. A marble released part way up the inside surface of a smoothly rounded bowl Enter the letters of all possible correct scenarios in alphabetical order. Do not use commas. AD Correct Part D Assume that the diagram in the problem introduction represents the motion of a ball tied on the end of a stringthat is, a pendulum. Also assume that the interval between each time step in the diagram is 0.10 s. The total time it takes for this pendulum to swing back and forth, also called the period of the pendulum, is then approximately 2 s. An interesting fact about the pendulum is that its period is essentially independent of the weight of the ball (or whatever other object is used). It depends only on the length of the string (with longer period for longer strings) and the strength of the force of gravity (which is essentially constant over the surface of the earth). Based on observation or comparison with other reallife pendula, estimate the length of the string needed for the pendulum to have a period of 2 s. Express in meters. A factor of 3 error is allowed in either direction. =1.0 Correct Physics can often seem to be a science of very precise answers. However, having a solid grasp of the fundamental concepts allows a physicist to make reasonably accurate estimates like this one very quickly. Even if you were ultimately looking for a more precise answer, an initial estimate gives you a way of checking whether the result of a long calculation is reasonable.
Average Velocity from a Position vs. Time Graph
Learning Goal: To learn to read and interpolate on a graph of position versus time and to change units.
In this problem you must find the average velocity from a graph of . We will use the notation to indicate the average velocity over the time interval from to . Thus is the average velocity over the time interval from 1 to 3 s. Harvaran Ghai Part A Find the average velocity over the time interval from 0 to 1 second. Answer to the nearest integer. =0 Correct Part B Find the average velocity over the time interval from 1 to 3 seconds. Answer to the nearest integer. =20 Correct Part C
Give your answer to three significant figures.
=13.3
Correct
Note that is not equal to the simple arithmetic average of and , because they are averages for time intervals of different length. You would have to double the weight given to because it is for an interval twice as long. Part D Find the average velocity over the time interval from 1 to 5 seconds. You will need to interpolate to find the position at time . Do not simply eyeball the position or you will likely not be able to obtain the solution to the desired accuracy. Round your answer to two significant figures. =6.7 Correct Part E Obtaining this answer required some interpolation on the graph. Now see if you can express this result in terms of kilometers per hour. Express your answer to the nearest integer. =24 Correct Part F Find the average velocity over the time interval from 2.5 to 6.0 seconds. Express your answer to two significant figures. =8.6 Correct
Running and Walking
Tim and Rick both can run at speed and walk at speed , with . They set off together on a journey of distance . Rick walks half of the distance and runs the other half. Tim walks half of the time and runs the other half.
Harvaran Ghai Part A
How long does it take Rick to cover the distance ? Express the time taken by Rick in terms of , , and . = Correct Part B Find Rick's average speed for covering the distance . Express Rick's average speed in terms of , , and . = Correct Part C How long does it take Tim to cover the distance? Express the time taken by Tim in terms of , , and . = Correct Part D Who covers the distance more quickly? Think logically, but without using the detailed answers in the previous parts. Rick Tim Neither. They cover the distance in the same amount of time. Correct Part E In terms of given quantities, by what amount of time, , does Tim beat Rick? It will help you check your answer if you simplify it algebraically and check the special case . Express the difference in time, in terms of , , and .
=
Correct
In the special case that , what would be Tim's margin of victory ?
0
Correct
Graph of v(t) for a Sports Car
The graph shows the velocity of a sports car as a function of time . Use the graph to answer the following questions. Harvaran Ghai Part A Find the maximum velocity of the car. Express your answer in meters per second to the nearest integer. =55 Correct Part B During which time interval is the acceleration positive? Indicate the most complete answer. to to to to to Correct
Part C Find the maximum acceleration of the car. Express your answer in meters per second squared to the nearest integer. =30 Correct Part D Find the minimum magnitude of the acceleration of the car. Express your answer in meters per second squared to the nearest integer. =0 Correct Part E Find the distance traveled by the car between 0 and 2 s. Express your answer in meters to the nearest integer. =55 Correct
Rearending Drag Racer To demonstrate the tremendous acceleration of a top fuel drag racer, you attempt to run your car into the back of a dragster that is "burning out" at the red light before the start of a race. (Burning out means spinning the tires at high speed to heat the tread and make the rubber sticky.) You drive at a constant speed of toward the stopped dragster, not slowing down in the face of the imminent collision. The dragster driver sees you coming but waits until the last instant to put down the hammer, accelerating from the starting line at constant acceleration, . Let the time at which the dragster starts to accelerate be .
Harvaran Ghai Part A What is , the longest time after the dragster begins to accelerate that you can possibly run into the back of the dragster if you continue at your initial velocity? = Correct Part B Assuming that the dragster has started at the last instant possible (so your front bumper almost hits the rear of the dragster at ), find your distance from the dragster when he started. If you calculate positions on the way to this solution, choose coordinates so that the position of the drag car is 0 at . Remember that you are solving for a distance (which is a magnitude, and can never be negative), not a position (which can be negative). = Correct Part C
Find numerical values for and in seconds and meters for the (reasonable) values (26.8 m/s)
and .
Separate your two numerical answers by commas, and give your answer to two significant figures. , =0.54,7.2 s, m
The blue curve shows how the car, initially at , continues at constant velocity (blue) and just barely touches the accelerating drag car (red) at .
Motion of a Shadow
A small source of light is located at a distance from a vertical wall. An opaque object with a height of moves toward the wall with constant velocity of magnitude . At time , the object is located at the source .
Harvaran Ghai Part A
Find an expression for , the magnitude of the velocity of the top of the object's shadow, at time . Express the speed of the top of the object's shadow in terms of , , , and .
= Correct Rocket Height A rocket, initially at rest on the ground, accelerates straight upward from rest with constant net acceleration , until time , when the fuel is exhausted. Harvaran Ghai Part A Find the maximum height that the rocket reaches (neglecting air resistance).
Express the maximum height in terms of , , and/or . Note that in this problem, is a positive number equal to the magnitude of the acceleration due to gravity.
= a*t1*((a*t1)/g)(.5g((a*t1)/g)2)+(.5(t1)2*a) Correct
Part B
If the rocket's net acceleration is for , what is the maximum height the rocket will reach? Express your answer numerically in meters, using .
=1470 m Correct
A Flower Pot Falling Past a Window
As you look out of your dorm window, a flower pot suddenly falls past. The pot is visible for a time , and the vertical length of your window is . Take down to be the positive direction, so that downward velocities are positive and the acceleration due to gravity is the positive quantity . Assume that the flower pot was dropped by someone on the floor above you (rather than thrown downward). Harvaran Ghai Part A From what height above the bottom of your window was the flower pot dropped?
Express your answer in terms of , , and .
=
Correct
Part B
If the bottom of your window is a height above the ground, what is the velocity of the pot as it hits the ground? You may introduce the new variable , the speed at the bottom of the window, defined by
.
Express your answer in terms of some or all of the variables , , , , and .
=
Correct
Resolving Vector Components with Trigonometry
Often a vector is specified by a magnitude and a direction; for example, a rope with tension exerts a force of magnitude in a direction 35 degrees north of east. This is a good way to think of vectors; however, to calculate results with vectors, it is best to select a coordinate system and manipulate the components of the vectors in that coordinate system.
Harvaran Ghai Part A
Find the components of the vector with length and angle with respect to the x axis as shown, named . Don't forget that when multiplying two factors, you must include a multiplication symbol; also, the cos and sin functions must have parentheses around their arguments. For example, a vector might take the form p*sin(Q),m*cos(N). Write the components in the form x,y. = Correct Part B
Find the components of the vector with length and angle with respect to the x axis as shown, named . Write the components in the form x,y.
=
Correct
Notice that vectors and have the same form despite their placement with respect to the y axis on the drawing. Part C
Find the components of the vector with length and angle as shown, named . Express your answer in terms of and . Write the components in the form x,y.
=
Tracking a Plane
A radar station, located at the origin of xz plane, as shown in the figure , detects an airplane coming straight at the station from the east. At first observation (point A), the position of the airplane relative to the origin is . The position vector has a magnitude of 360 and is located at exactly 40 above the horizon. The airplane is tracked for another 123 in the vertical eastwest plane for 5.0 , until it has passed directly over the station and reached point B. The position of point B relative to the origin is (the magnitude of is 880 ). The contact points are shown in the diagram, where the x axis represents the ground and the positive z direction is upward. Harvaran Ghai Part A Define the displacement of the airplane while the radar was tracking it: . What are the components of ? Express in meters as an ordered pair, separating the x and z components with a comma, to two significant figures. =1100,26 Correct
Two Forces Acting at a Point
Two forces, and , act at a point. has a magnitude of 9.80 and is directed at an angle of 60.0 above the negative x axis in the second quadrant. has a magnitude of 6.40 and is directed at an angle of 53.2 below the negative x axis in the third quadrant. Harvaran Ghai Part A What is the x component of the resultant force? Express your answer in newtons. 8.73 Correct Part B What is the y component of the resultant force? Express your answer in newtons. 3.36 Correct Part C What is the magnitude of the resultant force? Express your answer in newtons. 9.36 Correct A Push or a Pull? Learning Goal: To understand the concept of force as a push or a pull and to become familiar with everyday forces. A force can be simply defined as a push or a pull exerted by one object upon another. Although such a definition may not sound too scientific, it does capture three essential properties of forces: Each force is created by some object. Each force acts upon some other object.
The action of a force can be visualized as a push or a pull. Since each force is created by one object and acts upon another, forces must be described as interactions. The proper words describing the force interaction between objects A and B may be any of the following: "Object A acts upon object B with force ." "Object A exerts force upon object B." "Force is applied to object B by object A." "Force due to object A is acting upon object B." One of the biggest mistakes you may make is to think of a force as "something an object has." In fact, at least two objects are always required for a force to exist. Each force has a direction: Forces are vectors. The main result of such interactions is that the objects involved change their velocities: Forces cause acceleration. However, in this problem, we will not concern ourselves with accelerationnot yet. Harvaran Ghai Some common types of forces that you will be dealing with include the gravitational force (weight), the force of tension, the force of friction, and the normal force. It is sometimes convenient to classify forces as either contact forces between two objects that are touching or as longrange forces between two objects that are some distance apart. Contact forces include tension, friction, and the normal force. Longrange forces include gravity and electromagnetic forces. Note that such a distinction is useful but not really fundamental: For instance, on a microscopic scale the force of friction is really an electromagnetic force. In this problem, you will identify the types of forces acting on objects in various situations. First, consider a book resting on a horizontal table. Part A Which object exerts a downward force on the book? the book itself the earth the surface of the table Correct
Part B The downward force acting on the book is __________. a contact force a longrange force Correct Part C What is the downward force acting on the book called? tension normal force weight friction Correct Part D Which object exerts an upward force on the book? the book itself the earth the surface of the table Correct Part E The upward force acting on the book is __________. a contact force a longrange force Correct Part F What is the upward force acting on the book called? tension normal force weight friction
Correct Now consider a different situation. A string is attached to a heavy block. The string is used to pull the block to the right along a rough horizontal table. Part G Which object exerts a force on the block that is directed toward the right? the block itself the earth the surface of the table the string Correct Part H The force acting on the block and directed to the right is __________. a contact force a longrange force Correct To exert a tension force, the string must be connected to (i.e., touching) the block. Part I What is the force acting on the block and directed to the right called? tension normal force weight friction Correct Part J Which object exerts a force on the block that is directed toward the left? the block itself the earth the surface of the table the string Correct
Part K The force acting on the block and directed to the left is __________. a contact force a longrange force Correct Part L What is the force acting on the block and directed to the left called? tension normal force weight friction Correct Now consider a slightly different situation. The same block is placed on the same rough table. However, this time, the string is disconnected and the block is given a quick push to the right. The block slides to the right and eventually stops. The following questions refer to the motion of the block after it is pushed but before it stops. Part M How many forces are acting on the block in the horizontal direction? 0 1 2 3 Correct Once the push has commenced, there is no force acting to the right: The block is moving to the right because it was given a velocity in this direction by some force that is no longer applied to the block (probably, the normal force exerted by a student's hand or some spring launcher). Once the contact with the launching object has been lost, the only horizontal force acting on the block is directed to the leftwhich is why the block eventually stops. Part N What is the force acting on the block that is directed to the left called? tension normal force
weight friction Correct
The force of friction does not disappear as long as the block is moving. Once the block stops, fricion becomes zero (assuming the table is perfectly horizontal).
Free-Body Diagrams: Introduction
Learning Goal: To learn to draw freebody diagrams for various reallife situations. Imagine that you are given a description of a reallife situation and are asked to analyze the motion of the objects involved. Frequently, that analysis would involve finding the acceleration of the objects. That, in turn, requires that you find the net force. To find the net force, you must first identify all of the forces involved and then add them as vectors. Such a procedure is not always trivial. It is helpful to replace the sketch of the situation by the drawing of the object (represented as a particle) and all the forces applied to it. Such a drawing is called a freebody diagram. This problem will walk you through several examples of freebody diagrams and will demonstrate some of the possible pitfalls. Here is the general strategy for drawing freebody diagrams: Identify the object of interest. This may not always be easy: A sketch of the situation may contain many objects, each of which has a different set of forces acting on it. Including forces acting on different objects in the same diagram will lead to confusion and a wrong solution. Draw the object as a dot. Draw and clearly label all the forces acting on the object of interest. The forces should be shown as vectors originating from the dot representing the object of interest. There are two possible difficulties here: omitting some forces and drawing the forces that either don't exist at all or are applied to other objects. To avoid these two pitfalls, remember that every force must be applied to the object of interest by some other objector, as some like to say, "every force must have a source." Once all of the forces are drawn, draw the coordinate system. The origin should coincide with the dot representing the object of interest and the axes should be chosen so that the subsequent calculations of vector components of the forces will be relatively simple. That is, as many forces as possible must be either parallel or perpendicular to one of the axes.
Harvaran Ghai It should come as good news that, even though real life can present us with a wide variety of situations, we will be mostly dealing with a very small number of forces. Here are the principal ones of interest: Weight, or the force due to gravity. Weight acts on every object and is directed straight down unless we are considering a problem involving the nonflat earth (e.g., satellites). Normal force. The normal force exists between two surfaces that are pressed against each other; it is always perpendicular to the surfaces. Force of tension. Tension exists in strings, springs, and other objects of finite length. It is directed along the string or a spring. Keep in mind that a spring can be either compressed or stretched whereas a string can only be stretched. Force of friction. A friction force exists between two surfaces that either move or have a tendency to move relative to each other. Sometimes, the force of air drag, similar in some ways to the force of friction, may come into play. These forces are directed so that they resist the relative motion of the surfaces. Keep in mind that to simplify problems you often assume friction is negligible on smooth surfaces. In addition, the word friction commonly refers to resistive forces other than air drag that are caused by contact between surfaces so you can ignore air drag in problems unless you are told to consider its effects. The following examples should help you learn to draw freebody diagrams. We will start with relatively simple situations in which the object of interest is either explicitly suggested or fairly obvious. Part A A hockey puck slides along a horizontal, smooth icy surface at a constant velocity as shown. Draw a freebody diagram for the puck. Which of the following forces are acting on the puck? A. weight B. friction C. force of velocity D. force of push E. normal force
F. air drag G. acceleration Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For instance, if you think that only answers C and D are correct, type CD. AE Correct There is no such thing as "the force of velocity." If the puck is not being pushed, there are no horizontal forces acting on it. Of course, some horizontal force must have acted on it before, to impart the velocityhowever, in the situation described, no such "force of push" exists. Also, the air drag in such cases is assumed to be negligible. Finally, the word "smooth" usually implies negligible surface friction. Your freebody diagram should look like the one shown here. Part B Consider a block pulled by a horizontal rope along a horizontal surface at a constant velocity as shown. The tension in the rope is nonzero. Draw a freebody diagram for the block. Which of the following forces are acting on the block? A. weight B. friction C. force of velocity D. force of tension E. normal force F. air drag G. acceleration Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For instance, if you think that only answers C and D are correct, type CD.
ABDE Correct Because the velocity is constant, there must be a force of friction opposing the force of tension. Since the block is moving, it is kinetic friction. Your freebody diagram should look like that shown here. Consider the following situation in parts C F. A block is resting on a slope as shown. Part C Which of the following forces are acting on the block? A. weight B. kinetic friction C. static friction D. force of push E. normal force Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For instance, if you think that only answers C and D are correct, type CD. ACE Correct Part D What is the direction of the force due to gravity acting on the block? vertically upward vertically downward perpendicular to the slope
upward along the slope downward along the slope Correct Part E What is the direction of the normal force acting on the block? vertically upward vertically downward perpendicular to the slope upward along the slope downward along the slope Correct Part F Draw the freebody diagram for the block. What is the direction of the force of friction acting on the block? vertically upward vertically downward perpendicular to the slope upward along the slope downward along the slope Correct Without friction, the block would slide down the slope; so the force of static friction must oppose such a motion and be directed upward along the slope. Your freebody diagram should look like that shown here.
Now consider a block sliding up a rough slope after having been given a quick push as shown. Part G Which of the following forces are acting on the block? A. weight B. kinetic friction C. static friction D. force of push E. normal force F. the force of velocity Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For instance, if you think that only answers C and D are correct, type CD. ABE Correct The word "rough" implies the presence of friction. Since the block is in motion, it is kinetic friction. Once again, there is no such thing as "the force of velocity." However, it seems a tempting choice to some students since the block is going up. Part H Draw the freebody diagram for the block. What is the direction of the force of friction acting on the block? vertically upward vertically downward perpendicular to the slope upward along the slope downward along the slope Correct
The force of kinetic friction opposes the upward motion of the block. Your freebody diagram should look like the one shown here. Part I Now consider a block being pushed up a smooth slope. The force pushing the block is parallel to the slope. Which of the following forces are acting on the block? A. weight B. kinetic friction C. static friction D. force of push E. normal force Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For instance, if you think that only answers C and D are correct, type CD. ADE Correct
Your freebody diagram should look like the one shown here.
The force of push is the normal force exerted, possibly, by the palm of the hand of the person pushing the block.
In all the previous situations just described, the object of interest was explicitly given. Let us consider a situation where choosing the objects for which to draw the freebody diagrams is up to you.
Two blocks of masses and are connected by a light string that goes over a light frictionless pulley. The block of mass is sliding to the right on a rough horizontal surface of a lab table. Part J To solve for the acceleration of the blocks, you will have to draw the freebody diagrams for which objects? A. the block of mass B. the block of mass C. the connecting string D. the pulley E. the table F. the earth Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For instance, if you think that only answers C and D are correct, type CD. AB Correct Part K Draw the freebody diagram for the block of mass . How many forces are exerted on this block? none one two
three four Correct Your freebody diagram should look like that shown here. Part L Draw the freebody diagram for the block of mass . How many forces are exerted on this block? none one two three four Correct
Your freebody diagram should look like that shown here.
Understanding Newton's Laws
Harvaran Ghai Part A An object cannot remain at rest unless which of the following holds? The net force acting on it is zero. The net force acting on it is constant and nonzero. There are no forces at all acting on it. There is only one force acting on it. Correct If there is a net force acting on a body, regardless of whether it is a constant force, the body accelerates. If the body is at rest and the net force acting on it is zero, then it will remain at rest. The net force could be zero either because there are no forces acting on the body at all or because several forces are acting on the body but they all cancel out. Part B If a block is moving to the left at a constant velocity, what can one conclude? There is exactly one force applied to the block. The net force applied to the block is directed to the left. The net force applied to the block is zero. There must be no forces at all applied to the block. Correct
If there is a net force acting on a body, regardless of whether the body is already moving, the body accelerates. If a body is moving with constant velocity, then it is not accelerating and the net force acting on it is zero. The net force could be zero either because there are no forces acting on the body at all or because several forces are acting on the body but they all cancel out.
Part C
A block of mass is acted upon by two forces: (directed to the left) and (directed to the right). What can you say about the block's motion? It must be moving to the left. It must be moving to the right. It must be at rest. It could be moving to the left, moving to the right, or be instantaneously at rest. Correct The acceleration of an object tells you nothing about its velocitythe direction and speed at which it is moving. In this case, the net force on (and therefore the acceleration of) the block is to the right, but the block could be moving left, right, or in any other direction. Part D A massive block is being pulled along a horizontal frictionless surface by a constant horizontal force. The block must be __________. continuously changing direction moving at constant velocity moving with a constant nonzero acceleration moving with continuously increasing acceleration Correct Since there is a net force acting, the body does not move at a constant velocity, but it accelerates instead. However, the force acting on the body is constant. Hence, according to Newton's 2nd law of motion, the acceleration of the body is also constant. Part E
Two forces, of magnitude and , are applied to an object. The relative direction of the forces is unknown. The net force acting on the object __________.
A. cannot be equal to B. cannot be equal to
D. must be greater than
Enter the letters of all the correct answers in alphabetical order. Do not use commas. For example, if you think only the last option is correct, enter D.
A Correct
Conceptual Questions on Newton's 1st and 2nd Laws
Learning Goal: To understand the meaning and the basic applications of Newton's 1st and 2nd laws. In this problem, you are given a diagram representing the motion of an objecta motion diagram. The dots represent the object's position at moments separated by equal intervals of time. The dots are connected by arrows representing the object's average velocity during the corresponding time interval. Your goal is to use this motion diagram to determine the direction of the net force acting on the object. You will then determine which force diagrams and which situations may correspond to such a motion. Harvaran Ghai Part A What is the direction of the net force acting on the object at position A? upward downward to the left to the right The net force is zero.
Correct The velocity vectors connecting position A to the adjacent positions appear to have the same magnitude and direction. Therefore, the acceleration is zeroand so is the net force. Part B What is the direction of the net force acting on the object at position B? upward downward to the left to the right The net force is zero. Correct The velocity is directed to the right; however, it is decreasing. Therefore, the acceleration is directed to the leftand so is the net force. Part C What is the direction of the net force acting on the object at position C? upward downward to the left to the right The net force is zero. Correct The horizontal component of the velocity does not change. The vertical component of the velocity increases. Therefore, the accelerationand the net forceare directed straight downward. The next four questions are related to the force diagrams numbered 1 to 6. These diagrams represent the forces acting on a moving object. The number next to each arrow represents the magnitude of the force in newtons. Part D Which of these diagrams may possibly correspond to the situation at point A on the motion diagram? Type, in increasing order, the numbers corresponding to the correct diagrams. Do not use commas. For instance, if you think that only diagrams 3 and 4 are correct, type 34. 6 Correct Part E
Which of these diagrams may possibly correspond to the situation at point B on the motion diagram? Type, in increasing order, the numbers corresponding to the correct diagrams. Do not use commas. For instance, if you think that only diagrams 3 and 4 are correct, type 34. 35 Correct Part F Which of these diagrams may possibly correspond to the situation at point C on the motion diagram? Type, in increasing order, the numbers corresponding to the correct diagrams. Do not use commas. For instance, if you think that only diagrams 3 and 4 are correct, type 34. 24 Correct Part G Which of these diagrams correspond to a situation where the moving object (not necessarily the one shown in the motion diagram) is changing its velocity? Type, in increasing order, the numbers corresponding to the correct diagrams. Do not use commas. For instance, if you think that only diagrams 3 and 4 are correct, type 34. 12345 Correct Consider the following situations: A. A car is moving along a straight road at a constant speed. B. A car is moving along a straight road while slowing down. C. A car is moving along a straight road while speeding up. D. A hockey puck slides along a smooth (i.e., frictionless) icy surface. E. A hockey puck slides along a rough concrete surface. F. A cockroach is speeding up from rest. G. A rock is thrown horizontally; air resistance is negligible. H. A rock is thrown horizontally; air resistance is substantial. I. A rock is dropped vertically; air resistance is negligible. J. A rock is dropped vertically; air resistance is substantial. Part H Which of these situations describe the motion shown in the motion diagram at point A? Type the letters corresponding to all the right answers in alphabetical order. Do not use commas. For instance, if you think that only situations C and D are correct, type CD. AD Correct
Part I Which of these situations describe the motion shown in the motion diagram at point B? Type the letters corresponding to all the right answersin alphabetical order. Do not use commas. For instance, if you think that only situations C and D are correct, type CD. BE Correct Part J Which of these situations describe the motion shown in the motion diagram at point C? Type the letters corresponding to all the right answers in alphabetical order. Do not use commas. For instance, if you think that only situations C and D are correct, type CD. G Correct A World-Class Sprinter Worldclass sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has magnitude . Harvaran Ghai Part A
How much horizontal force must a sprinter of mass 54.0 exert on the starting blocks to produce this acceleration? Express your answer in newtons. =810 Correct Part B Which body exerts the force that propels the sprinter, the blocks or the sprinter? the blocks the sprinter Correct To start moving forward, sprinters push backward on the starting blocks with their feet. As a reaction, the blocks push forward on their feet with a force of the same magnitude. This external force accelerates the sprinter forward.
Block on an Incline
A block lies on a plane raised an angle from the horizontal. Three forces act upon the block: , the force of gravity; , the normal force; and , the force of friction. The coefficient of friction is large enough to prevent the block from sliding . Harvaran Ghai Part A Consider coordinate system a, with the x axis along the plane. Which forces lie along the axes? only only only and and and and and Correct Part B Which forces lie along the axes of the coordinate system b, in which the y axis is vertical?
only only only and and and and and Correct Now you are going to ignore the general rule (actually, a strong suggestion) that you should pick the coordinate system with the most vectors, especially unknown ones, along the coordinate axes. You will find the normal force, , using vertical coordinate system b. In these coordinates you will find the magnitude appearing in both the x and y equations, each multiplied by a trigonometric function. Part C Because the block is not moving, the sum of the y components of the forces acting on the block must be zero. Find an expression for the sum of the y components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables , , , and . Correct Part D Because the block is not moving, the sum of the x components of the forces acting on the block must be zero. Find an expression for the sum of the x components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables , , , and . Correct Part E
To find the magnitude of the normal force, you must express in terms of since is an unknown. Using the equations you found in the two previous parts, find an expression for involving and but not .
= Correct
Congratulations on working this through. Now realize that in coordinate system a, which is aligned with the plane, the ycoordinate equation is , which leads immediately to the result obtained here for . CONCLUSION: A thoughtful examination of which coordinate system to choose can save a lot of algebra. Hanging Chandelier A chandelier with mass is attached to the ceiling of a large concert hall by two cables. Because the ceiling is covered with intricate architectural decorations (not indicated in the figure, which uses a humbler depiction), the workers who hung the chandelier couldn't attach the cables to the ceiling directly above the chandelier. Instead, they attached the cables to the ceiling near the walls. Cable 1 has tension and makes an angle of with the ceiling. Cable 2 has tension and makes an angle of with the ceiling.
Harvaran Ghai Part A
Find an expression for , the tension in cable 1, that does not depend on .
Express your answer in terms of some or all of the variables , , and , as well as the magnitude of the acceleration due to gravity .
=
Problem 5.11 Harvaran Ghai Part A An astronaut's weight on earth is 805 . What is his weight on Mars, where 309 N Correct Problem 5.12 A woman has a mass of . Harvaran Ghai Part A What is her weight on earth? 539 N Correct Part B What is her mass on the moon, where 55.0 kg Correct Part C What is her weight on the moon? 89.1 N Correct Problem 5.14 The figure shows the velocity graph of a passenger in an elevator.
Harvaran Ghai Part A What is the passenger's apparent weight at ? 1040 N Correct Part B At t = ? 735 N Correct Part C At t = ? 585 N Correct
Pushing a Chair along the Floor
A chair of weight 120 lies atop a horizontal floor; the floor is not frictionless. You push on the chair with a force of = 35.0 directed at an angle of 41.0 below the horizontal and the chair slides along the floor.
Harvaran Ghai Part A Using Newton's laws, calculate , the magnitude of the normal force that the floor exerts on the chair. Express your answer in newtons. =143 Correct
Board Pulled Out from under a Box
A small box of mass is sitting on a board of mass and length . The board rests on a frictionless horizontal surface. The coefficient of static friction between the board and the box is . The coefficient of kinetic friction between the board and the box is, as usual, less than .
Throughout the problem, use for the magnitude of the acceleration due to gravity. In the hints, use for the magnitude of the friction force between the board and the box.
Harvaran Ghai Part A
Find , the constant force with the least magnitude that must be applied to the board in order to pull the board out from under the the box (which will then fall off of the opposite end of the board).
Express your answer in terms of some or all of the variables , , , , and . Do not include in your answer.
=
Correct
Friction Force on a Dancer on a Drawbridge
A dancer is standing on one leg on a drawbridge that is about to open. The coefficients of static and kinetic friction between the drawbridge and the dancer's foot are and , respectively. represents the normal force exerted on
the dancer by the bridge, and represents the gravitational force exerted on the dancer, as shown in the drawing . For all the questions, we can assume that the bridge is a perfectly flat surface and lacks the curvature characteristic of most bridges. Harvaran Ghai Part A Before the drawbridge starts to open, it is perfectly level with the ground. The dancer is standing still on one leg. What is the x component of the friction force, ? Express your answer in terms of some or all of the variables , , and/or . =0 Correct This shows a very important point. When you are not told that an object is slipping or on the verge of slipping, then the friction force is determined using Newton's laws of motion in conjunction with the observed motion and the other forces on the object. Under these circumstances the friction force is limited by or but is otherwise not
necessarily related to or . Part B
The drawbridge then starts to rise and the dancer continues to stand on one leg. The drawbridge stops just at the point where the dancer is on the verge of slipping. What is the magnitude of the frictional force now?
Express your answer in terms of some or all of the variables , , and/or . The angle should not appear in your answer.
=
Part C
Then, because the bridge is old and poorly designed, it falls a little bit and then jerks. This causes the person to start to slide down the bridge at a constant speed. What is the magnitude of the frictional force now?
Express your answer in terms of some or all of the variables , , and/or . The angle should not appear in your answer. = Correct Part D The bridge starts to come back down again. The dancer stops sliding. However, again because of the age and design of the bridge it never makes it all the way down; rather it stops half a meter short. This half a meter corresponds to an angle degree (see the diagram, which has the angle exaggerated). What is the force of friction now? Express your answer in terms of some or all of the variables , , , , and/or . = Correct Skydiving A sky diver of mass 80.0 (including parachute) jumps off a plane and begins her descent. Throughout this problem use 9.80 for the magnitude of the acceleration due to gravity. Harvaran Ghai Part A At the beginning of her fall, does the sky diver have an acceleration? No; the sky diver falls at constant speed. Yes and her acceleration is directed upward. Yes and her acceleration is directed downward. Correct
This Error! Hyperlink reference not valid. shows the sky diver (not to scale) with her position, speed, and acceleration graphed as functions of time. You can see how her acceleration drops to zero over time, giving constant speed after a long time.
Part B At some point during her free fall, the sky diver reaches her terminal speed. What is the magnitude of the drag force due to air resistance that acts on the sky diver when she has reached terminal speed? Express your answer in newtons. =784 Correct Part C For an object falling through air at a high speed , the drag force acting on it due to air resistance can be expressed as , where the coefficient depends on the shape and size of the falling object and on the density of air. For a human body, the numerical value for is about 0.250 .
Using this value for , what is the terminal speed of the sky diver? Express you answer in meters per second. =56.0 Correct Recreational sky divers can control their terminal speed to some extent by changing their body posture. When oriented in a headfirst dive, a sky diver can reach speeds of about 54 meters per second (120 miles per hour). For maximum drag and stability, sky divers often will orient themselves "bellyfirst." In this position, their terminal speed is typically around 45 meters per second (100 miles per hour). Part D When the sky diver descends to a certain height from the ground, she deploys her parachute to ensure a safe landing. (Usually the parachute is deployed when the sky diver reaches an altitude of about 900 3000 .) Immediately after deploying the parachute, does the skydiver have a nonzero acceleration? No; the sky diver keeps falling at constant speed. Yes and her acceleration is directed downward. Yes and her acceleration is directed upward. Correct Part E
When the parachute is fully open, the effective drag coefficient of the sky diver plus parachute increases to 60.0 . What is the drag force acting on the sky diver immediately after she has opened the parachute? Express your answer in newtons. =1.88×105 Correct Part F What is the terminal speed of the sky diver when the parachute is opened? Express your answer in meters per second. =3.61 Correct A typical "student" parachute for recreational skydiving has a drag coefficient that gives a terminal speed for landing of about 2 meters per second (5 miles per hour). If this seems slow based on video or reallife sky divers you have seen, that may be because the sky divers you saw were using highperformance parachutes; these offer the sky divers more maneuverability in the air but increase the terminal speed up to 4 meters per second (10 miles per hour).
An Object Accelerating on a Ramp
Learning Goal: Understand that the acceleration vector is in the direction of the change of the velocity vector. In one dimensional (straight line) motion, acceleration is accompanied by a change in speed, and the acceleration is always parallel (or antiparallel) to the velocity. When motion can occur in two dimensions (e.g. is confined to a tabletop but can lie anywhere in the xy plane), the definition of acceleration is in the limit . In picturing this vector derivative you can think of the derivative of a vector as an instantaneous quantity by thinking of the velocity of the tip of the arrow as the vector changes in time. Alternatively, you can (for small ) approximate the acceleration as .
Obviously the difference between and is another vector that can lie in any direction. If it is longer but in the same direction, will be parallel to . On the other hand, if has the same magnitude as
but is in a slightly different direction, then will be perpendicular to . In general, can differ from in both magnitude and direction, hence can have any direction relative to . This problem contains several examples of this.Consider an object sliding on a frictionless ramp as depicted here. The object is already moving along the ramp toward position 2 when it is at position 1. The following questions concern the direction of the object's acceleration vector, . In this problem, you should find the direction of the acceleration vector by drawing the velocity vector at two points near to the position you are asked about. Note that since the object moves along the track, its velocity vector at a point will be tangent to the track at that point. The acceleration vector will point in the same direction as the vector difference of the two velocities. (This is a result of the equation given above.) Harvaran Ghai Part A Which direction best approximates the direction of when the object is at position 1? straight up downward to the left downward to the right straight down Correct Part B Which direction best approximates the direction of when the object is at position 2? straight up upward to the right
straight down downward to the left Correct Even though the acceleration is directed straight up, this does not mean that the object is moving straight up. Part C Which direction best approximates the direction of when the object is at position 3? upward to the right to the right straight down downward to the right Correct Problem 6.3
A particle's trajectory is described by and , where is in s.
Harvaran Ghai Part A What is the particle's speed at t 2.00 m/s Correct Part B What is the particle's speed at = 4.50 ? 12.6 m/s Correct Part C What is the particle's direction of motion, measured from the xaxis, at 0 ? 270 counterclockwise from the +x axis Correct
Part D
What is the particle's direction of motion, measured from the xaxis, at = 4.50 ? 11.4 counterclockwise from the +x axis
Correct
Projectile Motion Tutorial
Learning Goal: Understand how to apply the equations for 1dimensional motion to the y and x directions separately in order to derive standard formulae for the range and height of a projectile.
A projectile is fired from ground level at time , at an angle with respect to the horizontal. It has an initial speed . In this problem we are assuming that the ground is level.
Harvaran Ghai Part A
Find the time it takes the projectile to reach its maximum height.
Express in terms of , , and (the magnitude of the acceleration due to gravity).
=
Part B Find , the time at which the projectile hits the ground. Express the time in terms of , , and . = Correct Part C Find , the maximum height attained by the projectile. Express the maximum height in terms of , , and . = Correct Part D Find the total distance (often called the range) traveled in the x direction; in other words, find where the projectile lands. Express the range in terms of , , and . = Correct The actual formula for is less important than how it is obtained: 1. Consider the x and y motion separately. 2. Find the time of flight from the ymotion 3. Find the xposition at the end of the flight this is the range. If you remember these steps, you can deal with many variants of the basic problem, such as: a cannon on a hill that fires horizontally (i.e. the second half of the trajectory), a projectile that lands on a hill, or a projectile that must hit a moving target.
Horizontal Cannon on a Cliff
A cannonball is fired horizontally from the top of a cliff. The cannon is at height above ground level, and the ball is fired with initial horizontal speed .