University Physics with Modern Physics

Roger A. Freedman, Hugh D. Young

Chapter 4

Newton's Laws of Motion - all with Video Answers

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Chapter Questions

Problem 1

Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is $60.0^{\circ} .$ If Rover exerts a force of $270 \mathrm{~N}$ and Fido exerts a force of $300 \mathrm{~N}$, find the magnitude of the resultant force and the angle it makes with Rover's rope.

Ankur S

Ankur S

Numerade Educator

Problem 2

To extricate an SUV stuck in the mud, workmen use three horizontal ropes, producing the force vectors shown in Fig. E4.2. (a) Find the $x$ - and $y$ -components of each of the three pulls. (b) Use the components to find the magnitude and direction of the resultant of the three pulls.

Supratim Pal

Numerade Educator

Problem 3

Due to a jaw injury, a patient must wear a strap (Fig. E4.3) that produces a net upward force of $5.00 \mathrm{~N}$ on his chin. The tension is the same throughout the strap. To what tension must the strap be adjusted to provide the necessary upward force?

Kevin Hayakawa

University of California - Los Angeles

Problem 4

A man is dragging a trunk up the loading ramp of a mover's truck. The ramp has a slope angle of $20.0^{\circ},$ and the man pulls upward with a force $\vec{F}$ whose direction makes an angle of $30.0^{\circ}$ with the ramp (Fig. E4.4). (a) How large a force $\vec{F}$ is necessary for the component $F_{x}$ parallel to the ramp to be $90.0 \mathrm{~N} ?$ (b) How large will the component $F_{y}$ perpendicular to the ramp be then?

Vishal Gupta

Numerade Educator

Problem 5

Forces $\vec{F}_{1}$ and $\vec{F}_{2}$ act at a point. The magnitude of $\vec{F}_{1}$ is $9.00 \mathrm{~N}$, and its direction is $60.0^{\circ}$ above the $x$ -axis in the second quadrant. The magnitude of $\vec{F}_{2}$ is $6.00 \mathrm{~N},$ and its direction is $53.1^{\circ}$ below the $x$ -axis in the third quadrant. (a) What are the $x$ - and $y$ -components of the resultant force?
(b) What is the magnitude of the resultant force?

Kevin Hayakawa

University of California - Los Angeles

Problem 6

An electron $\left(\right.$ mass $\left.=9.11 \times 10^{-31} \mathrm{~kg}\right)$ leaves one end of a TV picture tube with zero initial speed and travels in a straight line to the accelerating grid, which is $1.80 \mathrm{~cm}$ away. It reaches the grid with a speed of $3.00 \times 10^{6} \mathrm{~m} / \mathrm{s}$. If the accelerating force is constant, compute
(a) the acceleration; (b) the time to reach the grid; and (c) the net force, in newtons. Ignore the gravitational force on the electron.

Kevin Hayakawa

University of California - Los Angeles

Problem 7

A $68.5 \mathrm{~kg}$ skater moving initially at $2.40 \mathrm{~m} / \mathrm{s}$ on rough horizontal ice comes to rest uniformly in $3.52 \mathrm{~s}$ due to friction from the ice. What force does friction exert on the skater?

Kevin Hayakawa

University of California - Los Angeles

Problem 8

You walk into an elevator, step onto a scale, and push the "up" button. You recall that your normal weight is $625 \mathrm{~N}$. Draw a free-body diagram. (a) When the elevator has an upward acceleration of magnitude $2.50 \mathrm{~m} / \mathrm{s}^{2},$ what does the scale read? (b) If you hold a $3.85 \mathrm{~kg}$ package by a light vertical string, what will be the tension in this string when the elevator accelerates as in part (a)?

Kevin Hayakawa

University of California - Los Angeles

Problem 9

A box rests on a frozen pond, which serves as a frictionless horizontal surface. If a fisherman applies a horizontal force with magnitude $48.0 \mathrm{~N}$ to the box and produces an acceleration of magnitude $2.20 \mathrm{~m} / \mathrm{s}^{2}$, what is the mass of the box?

Kevin Hayakawa

University of California - Los Angeles

Problem 10

A dockworker applies a constant horizontal force of $80.0 \mathrm{~N}$ to a block of ice on a smooth horizontal floor. The frictional force is negligible. The block starts from rest and moves $11.0 \mathrm{~m}$ in $5.00 \mathrm{~s}$. (a) What is the mass of the block of ice? (b) If the worker stops pushing at the end of $5.00 \mathrm{~s},$ how far does the block move in the next $5.00 \mathrm{~s} ?$

Vishal Gupta

Numerade Educator

Problem 11

A hockey puck with mass $0.160 \mathrm{~kg}$ is at rest at the origin $(x=0)$ on the horizontal, frictionless surface of the rink. At time $t=0$ a player applies a force of $0.250 \mathrm{~N}$ to the puck, parallel to the $x$ -axis; she continues to apply this force until $t=2.00 \mathrm{~s}$. (a) What are the position and speed of the puck at $t=2.00 \mathrm{~s} ?$ (b) If the same force is again applied at $t=5.00 \mathrm{~s},$ what are the position and speed of the puck at $t=7.00 \mathrm{~s} ?$

Kevin Hayakawa

University of California - Los Angeles

Problem 12

A crate with mass $32.5 \mathrm{~kg}$ initially at rest on a warehouse floor is acted on by a net horizontal force of $14.0 \mathrm{~N}$. (a) What acceleration is produced? (b) How far does the crate travel in $10.0 \mathrm{~s} ?$ (c) What is its speed at the end of $10.0 \mathrm{~s} ?$

Kevin Hayakawa

University of California - Los Angeles

Problem 13

A $4.50 \mathrm{~kg}$ experimental cart undergoes an acceleration in a straight line (the $x$ -axis). The graph in Fig. E4.13 shows this acceleration as a function of time.
(a) Find the maximum net force on this cart. When does this maximum force occur? (b) During what times is the net force on the cart a constant? (c) When is the net force equal to zero?

Kevin Hayakawa

University of California - Los Angeles

Problem 14

A $2.75 \mathrm{~kg}$ cat moves in a straight line (the $x$ -axis). Figure E4.14 shows a graph of the $x$ component of this cat's velocity as a function of time. (a) Find the maximum net force on this cat. When does this force occur? (b) When is the net force on the cat equal to zero? (c) What is the net force at time $8.5 \mathrm{~s} ?$

Kevin Hayakawa

University of California - Los Angeles

Problem 15

A small $8.00 \mathrm{~kg}$ rocket burn fuel that exerts a time-varying upward force on the rocket (assume constant mass) as the rocket moves upward from the launch pad. This force obeys the equation $F=A+B t^{2} .$ Measurements show that at $t=0,$ the force is $100.0 \mathrm{~N}$ and at the end of the first $2.00 \mathrm{~s}$, it is $150.0 \mathrm{~N}$. (a) Find the constants $A$ and $B$, including their SI units. (b) Find the net force on this rocket and its acceleration (i) the instant after the fuel ignites and (ii) $3.00 \mathrm{~s}$ after the fuel ignites. (c) Suppose that you were using this rocket in outer space, far from all gravity. What would its acceleration be 3.00 s after fuel ignition?

Bianca Gualtieri

Bianca Gualtieri

Numerade Educator

Problem 16

An astronaut's pack weighs $17.5 \mathrm{~N}$ when she is on the earth but only $3.24 \mathrm{~N}$ when she is at the surface of a moon. (a) What is the acceleration due to gravity on this moon? (b) What is the mass of the pack on this moon?

Kevin Hayakawa

University of California - Los Angeles

Problem 17

Superman throws a $2400 \mathrm{~N}$ boulder at an adversary. What horizontal force must Superman apply to the boulder to give it a horizontal acceleration of $12.0 \mathrm{~m} / \mathrm{s}^{2} ?$

Kevin Hayakawa

University of California - Los Angeles

Problem 18

(a) An ordinary flea has a mass of $210 \mu \mathrm{g}$. How many newtons does it weigh? (b) The mass of a typical froghopper is 12.3 mg. How many newtons does it weigh? (c) A house cat typically weighs $45 \mathrm{~N}$. How many pounds does it weigh, and what is its mass in kilograms?

Kevin Hayakawa

University of California - Los Angeles

Problem 19

At the surface of Jupiter's moon Io, the acceleration due to gravity is $g=1.81 \mathrm{~m} / \mathrm{s}^{2}$. A watermelon weighs $44.0 \mathrm{~N}$ at the surface of the earth. (a) What is the watermelon's mass on the earth's surface?
(b) What would be its mass and weight on the surface of Io?

Kevin Hayakawa

University of California - Los Angeles

Problem 20

Estimate the mass in kilograms and the weight in pounds of a typical sumo wrestler. How do your estimates for the wrestler compare to your estimates of the average mass and weight of the students in your physics class? Do a web search if necessary to help make the estimates. In your solution list what values you assume for the quantities you use in making your estimates.

Bianca Gualtieri

Bianca Gualtieri

Numerade Educator

Problem 21

World-class sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has magnitude $15 \mathrm{~m} / \mathrm{s}^{2} .$ How much horizontal force must a $55 \mathrm{~kg}$ sprinter exert on the starting blocks to produce this acceleration? Which object exerts the force that propels the sprinter: the blocks or the sprinter herself?

Bianca Gualtieri

Bianca Gualtieri

Numerade Educator

Problem 22

A small car of mass $380 \mathrm{~kg}$ is pushing a large truck of mass 900 $\mathrm{kg}$ due east on a level road. The car exerts a horizontal force of $1600 \mathrm{~N}$ on the truck. What is the magnitude of the force that the truck exerts on the car?

Kevin Hayakawa

University of California - Los Angeles

Problem 23

Boxes $A$ and $B$ are in contact on a horizontal, frictionless surface (Fig. E4.23). Box A has mass $20.0 \mathrm{~kg}$ and box $B$ has mass $5.0 \mathrm{~kg} .$ A horizontal force of $250 \mathrm{~N}$ is exerted on box $A$. What is the
magnitude of the force that box $A$ exerts on box $B ?$

Kevin Hayakawa

University of California - Los Angeles

Problem 24

The upward normal force exerted by the floor is $620 \mathrm{~N}$ on an elevator passenger who weighs $650 \mathrm{~N}$. What are the reaction forces to these two forces? Is the passenger accelerating? If so, what are the magnitude and direction of the acceleration?

Aaron Shoolroy

Numerade Educator

Problem 25

A student of mass $45 \mathrm{~kg}$ jumps off a high diving board. What is the acceleration of the earth toward her as she accelerates toward the earth with an acceleration of $9.8 \mathrm{~m} / \mathrm{s}^{2}$ ? Use $6.0 \times 10^{24} \mathrm{~kg}$ for the mass of the earth, and assume that the net force on the earth is the force of gravity she exerts on it.

Kevin Hayakawa

University of California - Los Angeles

Problem 26

You pull horizontally on block $B$ in Fig. E4.26, causing both blocks to move together as a unit. For this moving system, make a carefully labeled freebody diagram of block $A$ if (a) the table is frictionless and (b) there is friction between block $B$ and the table and the pull is equal in magnitude to the friction force on block $B$ due to the table.

Bianca Gualtieri

Bianca Gualtieri

Numerade Educator

Problem 27

Crates $A$ and $B$ sit at rest side by side on a frictionless horizontal surface. They have masses $m_{A}$ and $m_{B},$ respectively. When a horizontal force $\overrightarrow{\boldsymbol{F}}$ is applied to crate $A$, the two crates move off to the right.
(a) Draw clearly labeled free-body diagrams for crate $A$ and for crate
B. Indicate which pairs of forces, if any, are third-law action-reaction pairs. (b) If the magnitude of $\vec{F}$ is less than the total weight of the two crates, will it cause the crates to move? Explain.

Bianca Gualtieri

Bianca Gualtieri

Numerade Educator

Problem 28

A .22 caliber rifle bullet traveling at $350 \mathrm{~m} / \mathrm{s}$ strikes a large tree and penetrates it to a depth of $0.130 \mathrm{~m}$. The mass of the bullet is $1.80 \mathrm{~g}$. Assume a constant retarding force. (a) How much time is required for the bullet to stop? (b) What force, in newtons, does the tree exert on the bullet?

Kevin Hayakawa

University of California - Los Angeles

Problem 29

A ball is hanging from a long string that is tied to the ceiling of a train car traveling eastward on horizontal tracks. An observer inside the train car sees the ball hang motionless. Draw a clearly labeled free-body diagram for the ball if (a) the train has a uniform velocity and
(b) the train is speeding up uniformly. Is the net force on the ball zero in either case? Explain.

Farhanul Hasan

Numerade Educator

Problem 30

A chair of mass $12.0 \mathrm{~kg}$ is sitting on the horizontal floor; the floor is not frictionless. You push on the chair with a force $F=40.0 \mathrm{~N}$ that is directed at an angle of $37.0^{\circ}$ below the horizontal, and the chair slides along the floor. (a) Draw a clearly labeled free-body diagram for the chair. (b) Use your diagram and Newton's laws to calculate the normal force that the floor exerts on the chair.

Aaron Shoolroy

Numerade Educator

Problem 31

Estimate the average force that a major-league pitcher exerts on the baseball when he throws a fastball. Express your answer in pounds. In your solution, list the quantities for which you estimate values and any assumptions you make. Do a web search to help determine the values you use in making your estimates.

Bianca Gualtieri

Bianca Gualtieri

Numerade Educator

Problem 32

You have just landed on Planet X. You release a $100 \mathrm{~g}$ ball from rest from a height of $10.0 \mathrm{~m}$ and measure that it takes $3.40 \mathrm{~s}$ to reach the ground. Ignore any force on the ball from the atmosphere of the planet. How much does the $100 \mathrm{~g}$ ball weigh on the surface of Planet X?

Mukesh Devi

Numerade Educator

Problem 33

A $5.60 \mathrm{~kg}$ bucket of water is accelerated upward by a cord of negligible mass whose breaking strength is $75.0 \mathrm{~N}$. If the bucket starts from rest, what is the minimum time required to raise the bucket a vertical distance of $12.0 \mathrm{~m}$ without breaking the cord?

Kevin Hayakawa

University of California - Los Angeles

Problem 34

Block $A$ rests on top of block $B$ as shown in Fig. E4.26. The table is frictionless but there is friction (a horizontal force) between blocks $A$ and $B .$ Block $B$ has mass $6.00 \mathrm{~kg}$ and block $A$ has mass $2.00 \mathrm{~kg} .$ If the horizontal pull applied to block $B$ equals $12.0 \mathrm{~N},$ then block $B$ has an acceleration of $1.80 \mathrm{~m} / \mathrm{s}^{2}$. What is the acceleration of blosk $A$ ?

Bianca Gualtieri

Bianca Gualtieri

Numerade Educator

Problem 35

Two adults and a child want to push a wheeled cart in the direction marked $x$ in Fig. $\mathbf{P} 4.35$ (next page). The two adults push with horizontal forces $\vec{F}_{1}$ and $\vec{F}_{2}$ as shown. (a) Find the magnitude and direction of the smallest force that the child should exert. Ignore the effects of friction.
(b) If the child exerts the minimum force found in part (a), the cart accelerates at $2.0 \mathrm{~m} / \mathrm{s}^{2}$ in the $+x$ -direction. What is the weight of the cart?

Vishal Gupta

Numerade Educator

Problem 36

An advertisement claims that a particular automobile can "stop on a dime." What net force would be necessary to stop a $850 \mathrm{~kg}$ automobile traveling initially at $45.0 \mathrm{~km} / \mathrm{h}$ in a distance equal to the diameter of a dime, $1.8 \mathrm{~cm} ?$

Kevin Hayakawa

University of California - Los Angeles

Problem 37

Two crates, one with mass $4.00 \mathrm{~kg}$ and the other with mass $6.00 \mathrm{~kg},$ sit on the frictionless surface of a frozen pond, connected by a light rope (Fig. P4.37). A woman wearing golf shoes (for traction) pulls horizontally on the $6.00 \mathrm{~kg}$ crate with a force $F$ that gives the crate an acceleration of $2.50 \mathrm{~m} / \mathrm{s}^{2}$. (a) What is the acceleration of the $4.00 \mathrm{~kg}$ crate? (b) Draw a free-body diagram for the $4.00 \mathrm{~kg}$ crate. Use that diagram and Newton's second law to find the tension $T$ in the rope that connects the two crates.
(c) Draw a free-body diagram for the $6.00 \mathrm{~kg}$ crate. What is the direction of the net force on the $6.00 \mathrm{~kg}$ crate? Which is larger in magnitude, $T$ or $F ?$
(d) Use part (c) and Newton's second law to calculate the magnitude of $F$.

Vishal Gupta

Numerade Educator

Problem 38

Two blocks connected by a light horizontal rope sit at rest on a horizontal, frictionless surface. Block $A$ has mass $15.0 \mathrm{~kg},$ and block $B$ has mass $m$. A constant horizontal force $F=60.0 \mathrm{~N}$ is applied to block $A$ (Fig. $\mathbf{P 4 . 3 8}$ ). In the first 5.00 s after the force is applied, block $A$ moves $18.0 \mathrm{~m}$ to the right. (a) While the blocks are moving, what is the tension $T$ in the rope that connects the two blocks? (b) What is the mass of block $B ?$

Vishal Gupta

Numerade Educator

Problem 39

To study damage to aircraft that collide with large birds, you design a test gun that will accelerate chicken-sized objects so that their displacement along the gun barrel is given by $x=\left(9.0 \times 10^{3} \mathrm{~m} / \mathrm{s}^{2}\right) t^{2}-\left(8.0 \times 10^{4} \mathrm{~m} / \mathrm{s}^{3}\right) t^{3} .$ The object leaves
the end of the barrel at $t=0.025 \mathrm{~s}$. (a) How long must the gun barrel be? (b) What will be the speed of the objects as they leave the end of the barrel? (c) What net force must be exerted on a $1.50 \mathrm{~kg}$ object at
(i) $t=0$ and (ii) $t=0.025 \mathrm{~s} ?$

Prashant Bana

Numerade Educator

Problem 40

On a test flight a rocket with mass $400 \mathrm{~kg}$ blasts off from the surface of the earth. The rocket engines apply a constant upward force $F$ until the rocket reaches a height of $100 \mathrm{~m}$ and then they shut off. If the rocket is to reach a maximum height of $400 \mathrm{~m}$ above the surface of the earth, what value of $F$ is required? Assume the change in the rocket's mass is negligible.

Vishal Gupta

Numerade Educator

Problem 41

After an annual checkup, you leave your physician's office, where you weighed $683 \mathrm{~N}$. You then get into an elevator that, conveniently, has a scale. Find the magnitude and direction of the elevator's acceleration if the scale reads (a) $725 \mathrm{~N}$ and (b) $595 \mathrm{~N}$.

Kevin Hayakawa

University of California - Los Angeles

Problem 42

A loaded elevator with very worn cables has a total mass of $2200 \mathrm{~kg},$ and the cables can withstand a maximum tension of $28,000 \mathrm{~N}$
(a) Draw the free-body force diagram for the elevator. In terms of the forces on your diagram, what is the net force on the elevator? Apply Newton's second law to the elevator and find the maximum upward acceleration for the elevator if the cables are not to break. (b) What would be the answer to part
(a) if the elevator were on the moon, where $g=1.62 \mathrm{~m} / \mathrm{s}^{2} ?$

Kevin Hayakawa

University of California - Los Angeles

Problem 43

A batter swings at a baseball (mass $0.145 \mathrm{~kg}$ ) that is moving horizontally toward him at a speed of $40.0 \mathrm{~m} / \mathrm{s} .$ He hits a line drive with the ball moving away from him horizontally at $50.0 \mathrm{~m} / \mathrm{s}$ just after it leaves the bat. If the bat and ball are in contact for $8.00 \mathrm{~ms}$, what is the average force that the bat applies to the ball?

Bianca Gualtieri

Bianca Gualtieri

Numerade Educator

Problem 44

An object with mass $m$ is moving along the $x$ -axis according to the equation $x(t)=\alpha t^{2}-2 \beta t,$ where $\alpha$ and $\beta$ are positive constants. What is the magnitude of the net force on the object at time $t=0 ?$

Bianca Gualtieri

Bianca Gualtieri

Numerade Educator

Problem 45

Boxes $A$ and $B$ are connected to each end of a light vertical rope (Fig. $\mathbf{P 4 . 4 5}$ ). A constant upward force $F=80.0 \mathrm{~N}$ is applied to box $A .$ Starting from rest, box $B$ descends $12.0 \mathrm{~m}$ in $4.00 \mathrm{~s}$. The tension in the rope connecting the two boxes is $36.0 \mathrm{~N}$. What are the masses of $(\mathrm{a})$ box $B,(\mathrm{~b})$ box $A ?$

Vishal Gupta

Numerade Educator

Problem 46

The two blocks in Fig. $\mathbf{P} 4.46$ are connected by a heavy uniform rope with a mass of $4.00 \mathrm{~kg}$. An upward force of $200 \mathrm{~N}$ is applied as shown. (a) Draw three free-body diagrams: one for the $6.00 \mathrm{~kg}$ block, one for the $4.00 \mathrm{~kg}$ rope, and another one for the $5.00 \mathrm{~kg}$ block. For each force, indicate what object exerts that force. (b) What is the acceleration of the system? (c) What is the tension at the top of the heavy rope?
(d) What is the tension at the midpoint of the rope?

Vishal Gupta

Numerade Educator

Problem 47

A small rocket with mass $20.0 \mathrm{~kg}$ is moving in free fall toward the earth. Air resistance can be neglected. When the rocket is $80.0 \mathrm{~m}$ above the surface of the earth, it is moving downward with a speed of $30.0 \mathrm{~m} / \mathrm{s}$. At that instant the rocket engines start to fire and produce a constant upward force $F$ on the rocket. Assume the change in the rocket's mass is negligible. What is the value of $F$ if the rocket's speed becomes zero just as it reaches the surface of the earth, for a soft landing? (Hint: The net force on the rocket is the combination of the upward force $F$ from the engines and the downward weight of the rocket.)

Vishal Gupta

Numerade Educator

Problem 48

You have landed on an unknown planet, Newtonia, and want to know what objects weigh there. When you push a certain tool, starting from rest, on a frictionless horizontal surface with a $12.0 \mathrm{~N}$ force, the tool moves $16.0 \mathrm{~m}$ in the first $2.00 \mathrm{~s}$. You next observe that if you release this tool from rest at $10.0 \mathrm{~m}$ above the ground, it takes $2.58 \mathrm{~s}$ to reach the ground. What does the tool weigh on Newtonia, and what does it weigh on earth?

Kevin Hayakawa

University of California - Los Angeles

Problem 49

A mysterious rocket-propelled object of mass $45.0 \mathrm{~kg}$ is initially at rest in the middle of the horizontal, frictionless surface of an ice-covered lake. Then a force directed east and with magnitude $F(t)=(16.8 \mathrm{~N} / \mathrm{s}) t$ is applied. How far does the object travel in the first $5.00 \mathrm{~s}$ after the force is applied?

Kevin Hayakawa

University of California - Los Angeles

Problem 50

Starting at time $t=0$, net force $F_{1}$ is applied to an object that is initially at rest. (a) If the force remains constant with magnitude $F_{1}$ while the object moves a distance $d$, the final speed of the object is $v_{1} .$ What is the final speed $v_{2}$ (in terms of $v_{1}$ ) if the net force is $F_{2}=2 F_{1}$ and the object moves the same distance $d$ while the force is being applied? (b) If the force $F_{1}$ remains constant while it is applied for a time $T,$ the final speed of the object is $v_{1} .$ What is the final speed $v_{2}$ (in terms of $v_{1}$ ) if the applied force is $F_{2}=2 F_{1}$ and is constant while it is applied for the same time $T ?$ In a later chapter we'll call force times distance work and force times time impulse and associate work and impulse with the change in speed.)

Vishal Gupta

Numerade Educator

Problem 51

The table $^{*}$ gives automobile performance data for a few types of cars: (a) During an acceleration of 0 to $60 \mathrm{mph},$ which car has the largest average net force acting on it? The smallest? (b) During this acceleration, for which car would the average net force on a $72.0 \mathrm{~kg}$ passenger be the largest? The smallest? (c) When the Ferrari F430 accelerates from 0 to $100 \mathrm{mph}$ in $8.6 \mathrm{~s},$ what is the average net force acting on it? How does this net force compare with the average net force during the acceleration from 0 to $60 \mathrm{mph} ?$ Explain why these average net forces might differ. (d) Discuss why a car has a top speed. What is the net force on the Ferrari $\mathrm{F} 430$ when it is traveling at its top speed, $196 \mathrm{mph} ?$

Paul A.

California State Polytechnic University, Pomona

Problem 52

The position of a training helicopter (weight $2.75 \times 10^{5} \mathrm{~N}$ ) in a test is given by $\hat{r}=\left(0.020 \mathrm{~m} / \mathrm{s}^{3}\right) t^{3} \hat{\imath}+$
$(2.2 \mathrm{~m} / \mathrm{s}) t \hat{\jmath}-\left(0.060 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2} \hat{k} .$ Find the net force on the helicopter at
$t=5.0 \mathrm{~s}$

Kevin Hayakawa

University of California - Los Angeles

Problem 53

You are a Starfleet captain going boldly where no one has gone before. You land on a distant planet and visit an engineering testing lab. In one experiment a short, light rope is attached to the top of a block and a constant upward force $F$ is applied to the free end of the rope. The block has mass $m$ and is initially at rest. As $F$ is varied, the time for the block to move upward $8.00 \mathrm{~m}$ is measured. The values that you collected are given in the table:
(a) Plot $F$ versus the acceleration $a$ of the block. (b) Use your graph to determine the mass $m$ of the block and the acceleration of gravity $g$ at the surface of the planet. Note that even on that planet, measured values contain some experimental error.

David González Cornejo

David González Cornejo

Numerade Educator

Problem 54

An $8.00 \mathrm{~kg}$ box sits on a level floor. You give the box a sharp push and find that it travels $8.22 \mathrm{~m}$ in $2.8 \mathrm{~s}$ before coming to rest again. (a) You measure that with a different push the box traveled $4.20 \mathrm{~m}$ in $2.0 \mathrm{~s}$. Do you think the box has a constant acceleration as it slows down? Explain your reasoning. (b) You add books to the box to increase its mass. Repeating the experiment, you give the box a push and measure how long it takes the box to come to rest and how far the box travels. The results, including the initial experiment with no added mass, are given in the table:
In each case, did your push give the box the same initial speed? What is the ratio between the greatest initial speed and the smallest initial speed for these four cases? (c) Is the average horizontal force $f$ exerted on the box by the floor the same in each case? Graph the magnitude of force $f$ versus the total mass $m$ of the box plus its contents, and use your graph to determine an equation for $f$ as a function of $m$

Kevin Hayakawa

University of California - Los Angeles

Problem 55

A block of mass $2.00 \mathrm{~kg}$ is initially at rest at $x=0$ on a slippery horizontal surface for which there is no friction. Starting at time $t=0,$ a horizontal force $F_{x}(t)=\beta-\alpha t$ is applied to the block, where $\alpha=6.00 \mathrm{~N} / \mathrm{s}$ anwd $\beta=4.00 \mathrm{~N}$. (a) What is the largest positive value of $x$ reached by the block? How long does it take the block to reach this point, starting from $t=0,$ and what is the magnitude of the force when the block is at this value of $x ?$ (b) How long from $t=0$ does it take the block to return to $x=0,$ and what is its speed at this point?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 56

An object of mass $m$ is at rest in equilibrium at the origin. At $t=0$ a new force $F(t)$ is applied that has components
$$
F_{x}(t)=k_{1}+k_{2} y \quad F_{y}(t)=k_{3} t
$$
where $k_{1}, k_{2},$ and $k_{3}$ are constants. Calculate the position $\vec{r}(t)$ and velocity $\overrightarrow{\boldsymbol{v}}(t)$ vectors as functions of time.

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 57

What is the magnitude of the average force that her neck exerts on her head during the landing?
(a) $0 \mathrm{~N} ;$ (b) $60 \mathrm{~N} ;$ (c) $120 \mathrm{~N} ;$ (d) $180 \mathrm{~N}$.

Kevin Hayakawa

University of California - Los Angeles

Problem 58

Compared with the force her neck exerts on her head during the landing, the force her head exerts on her neck is (a) the same;
(b) greater; (c) smaller; (d) greater during the first half of the landing and smaller during the second half of the landing.

Kevin Hayakawa

University of California - Los Angeles

Problem 59

While the dancer is in the air and holding a fixed pose, what is the magnitude of the force her neck exerts on her head?
(a) $0 \mathrm{~N} ;$ (b) $60 \mathrm{~N}$
(c) $120 \mathrm{~N} ;$ (d) $180 \mathrm{~N}$.

Kevin Hayakawa

University of California - Los Angeles

Problem 60

The forces on a dancer can be measured directly when a dancer performs a jump on a force plate that measures the force between her feet and the ground. A graph of force versus time throughout a vertical jump performed on a force plate is shown in Fig. $\mathbf{P 4 . 6 0 .}$. What is happening at $0.4 \mathrm{~s}$ ? The dancer is (a) bending her legs so that her body is accelerating downward; (b) pushing her body up with her legs and is almost ready to leave the ground; (c) in the air and at the top of her jump;
(d) landing and her feet have just touched the ground.

Vishal Gupta

Numerade Educator