Physics Principles with Applications

Douglas C. Giancoli

Chapter 6

Work and Energy - all with Video Answers

Educators

+ 3 more educators

Chapter Questions

Problem 1

(I) How much work is done by the gravitational force when a 265 -kg pile driver falls 2.80 $\mathrm{m} ?$

Dading Chen

Dading Chen

Numerade Educator

Problem 2

(I) A 65.0 -kg firefighter climbs a flight of stairs 20.0 $\mathrm{m}$ high. How much work is required?

Vishal Gupta

Numerade Educator

Problem 3

(I) A $1300-\mathrm{N}$ crate rests on the floor. How much work is required to move it at constant speed (a) 4.0 $\mathrm{m}$ along the floor against a friction force of $230 \mathrm{N},$ and $(b) 4.0 \mathrm{m}$ vertically?

Dading Chen

Numerade Educator

Problem 4

(I) How much work did the movers do (horizontally) pushing a $160-\mathrm{kg}$ crate 10.3 $\mathrm{m}$ across a rough floor without acceleration, if the effective coefficient of friction was 0.50$?$

Dading Chen

Numerade Educator

Problem 5

(II) A box of mass 5.0 $\mathrm{kg}$ is accelerated from rest across a floor at a rate of 2.0 $\mathrm{m} / \mathrm{s}^{2}$ for 7.0 $\mathrm{s}$ . Find the net work done on the box.

Dading Chen

Numerade Educator

Problem 6

(II) Eight books, each 4.3 $\mathrm{cm}$ thick with mass 1.7 $\mathrm{kg}$ , lie flat on a table. How much work is required to stack them one on top of another?

Dading Chen

Numerade Educator

Problem 7

(II) A lever such as that shown in Fig. $6-35$ can be used to lift objects we might not otherwise be able to lift. Show that the ratio of output force, $F_{\mathrm{O}},$ to input force, $F_{1},$ is related to the lengths $l_{1}$ and $l_{0}$ from the pivot point by $F_{\mathrm{O}} / F_{1}=l_{1} / l_{\mathrm{O}}$ (ignoring friction and the mass of the lever), given that the work output equals work input.

Keshav Singh

Numerade Educator

Problem 8

(II) A 330 -kg piano slides 3.6 $\mathrm{m}$ down a $28^{\circ}$ incline and is kept from accelerating by a man who is pushing back on it parallel to the incline (Fig. $6-36$ ). The effective coefficient of kinetic friction is $0.40 .$ Calculate: $(a)$ the force exerted by the man, $(b)$ the work done by the man on the piano, $(c)$ the work done by the friction force, (d) the work done by the force of gravity, and $(e)$ the net work done on the piano.

Vishal Gupta

Numerade Educator

Problem 9

(II) $(a)$ Find the force required to give a helicopter of mass $M$ an acceleration of 0.10 $\mathrm{g}$ upward. $(b)$ Find the work done by this force as the helicopter moves a distance $h$ upward.

Susan Lu

Numerade Educator

Problem 10

(II) What is the minimum work needed to push a $950-\mathrm{kg}$ car 810 $\mathrm{m}$ up along a $9.0^{\circ}$ incline?
(a) Ignore friction.
(b) Assume the effective coefficient of friction retarding the car is 0.25 .

Dading Chen

Numerade Educator

Problem 11

(II) In Fig. $6-6 a$ , assume the distance axis is linear and that $d_{A}=10.0 \mathrm{m}$ and $d_{B}=35.0 \mathrm{m}$ . Estimate the work done by force $F$ in moving a $2.80-\mathrm{kg}$ object from $d_{\mathrm{A}}$ to $d_{\mathrm{B}}$

Dading Chen

Numerade Educator

Problem 12

(II) The force on an object, acting along the $x$ axis, varies as shown in Fig. $6-37 .$ Determine the work done by this force to move the object $(a)$ from $x=0.0$ to $x=10.0 \mathrm{m},$ and $(b)$ from $x=0.0$ to $x=15.0 \mathrm{m}$

Dading Chen

Numerade Educator

Problem 13

(II) A spring has $k=88 \mathrm{N} / \mathrm{m}$ . Use a graph to determine the work needed to stretch it from $x=3.8 \mathrm{cm}$ to $x=5.8 \mathrm{cm},$ where $x$ is the displacement from its unstretched length.

Dading Chen

Numerade Educator

Problem 14

(II) The net force exerted on a particle acts in the $+x$ direction. Its magnitude increases linearly from zero at $x=0$ , to 24.0 $\mathrm{N}$ at $x=3.0 \mathrm{m}$ . It remains constant at 24.0 $\mathrm{N}$ from $x=3.0 \mathrm{m}$ to $x=8.0 \mathrm{m}$ , and then decreases linearly to zero at $x=13.0 \mathrm{m}$ . Determine the work done to move the particle from $x=0$ to $x=13.0 \mathrm{m}$ graphically by determining the area under the $F_{x}$ vs. $x$ graph.

Dading Chen

Numerade Educator

Problem 15

(I) At room temperature, an oxygen molecule, with mass of $5.31 \times 10^{-26} \mathrm{kg}$ , typically has a $\mathrm{KE}$ of about $6.21 \times 10^{-21} \mathrm{J}$ . How fast is the molecule moving?

Dading Chen

Numerade Educator

Problem 16

(I) $(a)$ If the KE of an arrow is doubled, by what factor has its speed increased? $(b)$ If its speed is doubled, by what factor does its $\mathrm{KE}$ increase?

Dading Chen

Numerade Educator

Problem 17

(I) How much work is required to stop an electron $m=9.11 \times 10^{-31} \mathrm{kg}$ , which is moving with a speed of $1.90 \times 10^{6} \mathrm{m} / \mathrm{s}$ ?

Dading Chen

Numerade Educator

Problem 18

(I) How much work must be done to stop a 1250 -kg car traveling at 105 $\mathrm{km} / \mathrm{h}$ ?

Dading Chen

Numerade Educator

Problem 19

(II) An $88-g$ arrow is fired from a bow whose string exerts an average force of 110 $\mathrm{N}$ on the arrow over a distance of $78 \mathrm{cm} .$ What is the speed of the arrow as it leaves the bow?

Dading Chen

Numerade Educator

Problem 20

(II) A baseball $(m=140 \mathrm{g})$ traveling 32 $\mathrm{m} / \mathrm{s}$ moves a fielder's glove backward 25 $\mathrm{cm}$ when the ball is caught. What was the average force exerted by the ball on the glove?

Dading Chen

Numerade Educator

Problem 21

(II) If the speed of a car is increased by 50$\%$ , by what factor will its minimum braking distance be increased, assuming all else is the same? Ignore the driver's reaction time.

Averell Hause

Carnegie Mellon University

Problem 22

(II) At an accident scene on a level road, investigators measure a car's skid mark to be 88 $\mathrm{m}$ long. The accident occurred on a rainy day, and the coefficient of kinetic friction was estimated to be $0.42 .$ Use these data to determine the speed of the car when the driver slammed on (and locked) the brakes. (Why does the car's mass not matter?)

Nishant Kumar

Numerade Educator

Problem 23

(II) A softball having a mass of 0.25 $\mathrm{kg}$ is pitched at 95 $\mathrm{km} / \mathrm{h}$ . By the time it reaches the plate, it may have slowed by $10 \% .$ Neglecting gravity, estimate the average force of air resistance during a pitch, if the distance between the plate and the pitcher is about $15 \mathrm{m} .$

Dading Chen

Numerade Educator

Problem 24

(II) How high will a 1.85 -kg rock go if thrown straight up by someone who does 80.0 $\mathrm{J}$ of work on it? Neglect air resistance.

Dading Chen

Numerade Educator

Problem 25

(III) A $285-\mathrm{kg}$ load is lifted 22.0 $\mathrm{m}$ vertically with an acceleration $a=0.160 \mathrm{g}$ by a single cable. Determine
(a) the tension in the cable,
(b) the net work done on the load,
(c) the work done by the cable on the load, $(d)$ the work done by gravity on the load, and $(e)$ the final speed of the load assuming it started from rest.

Dading Chen

Numerade Educator

Problem 26

(I) A spring has a spring stiffness constant, $k,$ of 440 $\mathrm{N} / \mathrm{m}$ . How much must this spring be stretched to store 25 $\mathrm{J}$ of potential energy?

Dading Chen

Numerade Educator

Problem 27

(I) A 7.0 -kg monkey swings from one branch to another 1.2 $\mathrm{m}$ higher. What is the change in potential energy?

Dading Chen

Numerade Educator

Problem 28

(I) By how much does the gravitational potential energy of a $64-\mathrm{kg}$ pole vaulter change if his center of mass rises about 4.0 $\mathrm{m}$ during the jump?

Dading Chen

Numerade Educator

Problem 29

(II) A 1200 -kg car rolling on a horizontal surface has speed $v=65 \mathrm{km} / \mathrm{h}$ when it strikes a horizontal coiled spring and is brought to rest in a distance of $2.2 \mathrm{m} .$ What is the spring stiffness constant of the spring?

Dading Chen

Numerade Educator

Problem 30

(II) A $1.60-\mathrm{m}$ tall person lifts a $2.10-\mathrm{kg}$ book from the ground so it is 2.20 $\mathrm{m}$ above the ground. What is the potential energy of the book relative to $(a)$ the ground, and (b) the top of the person's head? (c) How is the work done by the person related to the answers in parts (a) and (b)?

Dading Chen

Numerade Educator

Problem 31

(II) A $55-\mathrm{kg}$ hiker starts at an elevation of 1600 $\mathrm{m}$ and climbs to the top of a 3300$\cdot \mathrm{m}$ peak. ( $a$ ) What is the hiker's change in potential energy? (b) What is the minimum work required of the hiker? $(c)$ Can the actual work done be more than this? Explain why.

Dading Chen

Numerade Educator

Problem 32

(II) A spring with $k=53 \mathrm{N} / \mathrm{m}$ hangs vertically next to a ruler. The end of the spring is next to the $15-\mathrm{cm}$ mark on the ruler. If a $2.5-\mathrm{kg}$ mass is now attached to the end of the spring, where will the end of the spring line up with the ruler marks?

Dading Chen

Numerade Educator

Problem 33

(I) Jane, looking for Tarzan, is running at top speed $(5.3 \mathrm{m} / \mathrm{s})$ and grabs a vine hanging vertically from a tall tree in the jungle. How high can she swing upward? Does the length of the vine affect your answer?

Dading Chen

Numerade Educator

Problem 34

(I) A novice skier, starting from rest, slides down a frictionless $35.0^{\circ}$ incline whose vertical height is 185 $\mathrm{m}$ . How fast is she going when she reaches the bottom?

Dading Chen

Numerade Educator

Problem 35

(I) A sled is initially given a shove up a frictionless $28.0^{\circ}$ incline. It reaches a maximum vertical height 1.35 $\mathrm{m}$ higher than where it started. What was its initial speed?

Dading Chen

Numerade Educator

Problem 36

(II) In the high jump. Fran's kinetic energy is transformed into gravitational potential energy without the aid of a pole. With what minimum speed must Fran leave the ground in order to lift her center of mass 2.10 $\mathrm{m}$ and cross the bar with a speed of 0.70 $\mathrm{m} / \mathrm{s} ?$

Dading Chen

Numerade Educator

Problem 37

(II) A $65-\mathrm{kg}$ trampoline artist jumps vertically upward
from the top of a platform with a speed of 5.0 $\mathrm{m} / \mathrm{s}$ .
(a) How fast is he going as he lands on the trampoline, 3.0 $\mathrm{m}$ below (Fig. $6-38 )$ ?
(b) If the trampoline behaves like a spring with spring stiffness constant $6.2 \times 10^{4} \mathrm{N} / \mathrm{m}$ , how far does he depress it?
figure can't copy

Dading Chen

Numerade Educator

Problem 38

(II) A projectile is fired at an upward angle of $45.0^{\circ}$ from the top of a $265-\mathrm{m}$ cliff with a speed of 185 $\mathrm{m} / \mathrm{s}$ . What will be its speed when it strikes the ground below? (Use conservation of energy.)

Prabhu Ramji

Numerade Educator

Problem 39

(II) A vertical spring (ignore its mass), whose spring stiffness constant is $950 \mathrm{N} / \mathrm{m},$ is attached to a table and is compressed down 0.150 $\mathrm{m}$ . (a) What upward speed can it give to a $0.30-\mathrm{kg}$ ball when released? (b) How high above its original position (spring compressed) will the ball fly?

Dading Chen

Numerade Educator

Problem 40

(II) A block of mass $m$ slides without friction along the looped track shown in Fig. $6-39 .$ If the block is to remain on the track, even at the top of the circle (whose radius is $r )$ , from what minimum height $h$ must it be released?

Dading Chen

Numerade Educator

Problem 41

(II) A block of mass $m$ is attached to the end of a spring (spring stiffness constant $k ),$ Fig. $6-40$ . The block is given an initial displacement $x_{0}$ , after which it oscillates back and forth. Write a formula for the total mechanical energy (ignore friction and the mass of the spring) in terms of $x_{0}$ , position $x$ , and speed $v$ .

Dading Chen

Numerade Educator

Problem 42

(II) A 62 -kg bungee jumper jumps from a bridge. She is tied to a bungee cord whose unstretched length is 12 $\mathrm{m}$ , and falls a total of 31 $\mathrm{m}$ . (a) Calculate the spring stiffness constant $k$ of the bungee cord, assuming Hooke's law applies.
(b) Calculate the maximum acceleration she experiences.

Dading Chen

Numerade Educator

Problem 43

(II) The roller-coaster car shown in Fig. $6-41$ is dragged up to point 1 where it is released from rest. Assuming no friction, calculate the speed at points $2,3,$ and $4 .$

Esmaeil Mahdavi

Esmaeil Mahdavi

Numerade Educator

Problem 44

(II) A $0.40-\mathrm{kg}$ ball is thrown with a speed of 12 $\mathrm{m} / \mathrm{s}$ at an angle of $33^{\circ} .(a)$ What is its speed at its highest point, and
(b) how high does it go? (Use conservation of energy, and ignore air resistance.)

Dading Chen

Numerade Educator

Problem 45

(III) An engineer is designing a spring to be placed at the bottom of an elevator shaft. If the elevator cable should break when the elevator is at a height $h$ above the top of the spring, calculate the value that the spring stiffness constant $k$ should have so that passengers undergo an acceleration of no more than 5.0 $\mathrm{g}$ when brought to rest. Let $M$ be the total mass of the elevator and passengers.

Dading Chen

Numerade Educator

Problem 46

(III) A cyclist intends to cycle up a $7.8^{\circ}$ hill whose vertical height is 150 $\mathrm{m}$ . Assuming the mass of bicycle plus cyclist is $75 \mathrm{kg},(a)$ calculate how much work must be done against gravity. (b) If each complete revolution of the pedals moves the bike 5.1 $\mathrm{m}$ along its path, calculate the average force that must be exerted on the pedals tangent to their circular path. Neglect work done by friction and other losses. The pedals turn in a circle of diameter 36 $\mathrm{cm}$ .

Dading Chen

Numerade Educator

Problem 47

(I) Two railroad cars, each of mass 7650 $\mathrm{kg}$ and traveling 95 $\mathrm{km} / \mathrm{h}$ in opposite directions, collide head-on and come to rest. How much thermal energy is produced in this collision?

Dading Chen

Numerade Educator

Problem 48

(II) A 21.7 -kg child descends a slide 3.5 $\mathrm{m}$ high and reaches the bottom with a speed of 2.2 $\mathrm{m} / \mathrm{s}$ . How much thermal energy due to friction was generated in this process?

Dading Chen

Numerade Educator

Problem 49

(II) A ski starts from rest and slides down a $22^{\circ}$ incline 75 $\mathrm{m}$ long. $(a)$ If the coefficient of friction is $0.090,$ what is the ski's speed at the base of the incline? $(b)$ If the snow is level at the foot of the incline and has the same coefficient of friction, how far will the ski travel along the
level? Use energy methods.

Dading Chen

Numerade Educator

Problem 50

(II) A 145 -g baseball is dropped from a tree 13.0 $\mathrm{m}$ above the ground. (a) With what speed would it hit the ground if air resistance could be ignored? (b) If it actually hits the ground with a speed of 8.00 $\mathrm{m} / \mathrm{s}$ , what is the average force of air resistance exerted on it?

Dading Chen

Numerade Educator

Problem 51

(II) You drop a ball from a height of 2.0 $\mathrm{m}$ , and it bounces back to a height of 1.5 $\mathrm{m}$ . $(a)$ What fraction of its initial energy is lost during the bounce? (b) What is the ball's speed just as it leaves the ground after the bounce? (c) Where did the energy go?

Dading Chen

Numerade Educator

Problem 52

(II) A $110-\mathrm{kg}$ crate, starting from rest, is pulled across a floor with a constant horizontal force of 350 $\mathrm{N}$ . For the first 15 $\mathrm{m}$ the floor is frictionless, and for the next 15 $\mathrm{m}$ the coefficient of friction is $0.30 .$ What is the final speed of the crate?

Dading Chen

Numerade Educator

Problem 53

(II) Suppose the roller coaster in Fig. $6-41$ passes point 1 with a speed of 1.70 $\mathrm{m} / \mathrm{s}$ . If the average force of friction is equal to one-fifth of its weight, with what speed will it
reach point 2 ? The distance traveled is 45.0 $\mathrm{m}$ .

Dading Chen

Numerade Educator

Problem 54

(II) A skier traveling 12.0 $\mathrm{m} / \mathrm{s}$ reaches the foot of a steady upward $18.0^{\circ}$ incline and glides 12.2 $\mathrm{m}$ up along this slope before coming to rest. What was the average coefficient of friction?

Dading Chen

Numerade Educator

Problem 55

(III) A 0.620 -kg wood block is firmly attached to a very light horizontal spring $(k=180 \mathrm{N} / \mathrm{m})$ as shown in Fig. $6-40 .$ It is noted that the block-spring system, when compressed 5.0 $\mathrm{cm}$ and released, stretches out 2.3 $\mathrm{cm}$ beyond the equilibrium position before stopping and turning back. What is the coefficient of kinetic friction between the block and the table?

Brittany Carnathan

Brittany Carnathan

Numerade Educator

Problem 56

(III) A 280 -g wood block is firmly attached to a very light horizontal spring, Fig. $6-40 .$ The block can slide along a table where the coefficient of friction is $0.30 .$ A force of22 $\mathrm{N}$ compresses the spring $18 \mathrm{cm} .$ If the spring is released from this position, how far beyond its equilibrium position will it stretch at its first maximum extension?

Dading Chen

Numerade Educator

Problem 57

(III) Early test flights for the space shuttle used a "glider" (mass of 980 $\mathrm{kg}$ including pilot) that was launched horizontally at 500 $\mathrm{km} / \mathrm{h}$ from a height of 3500 $\mathrm{m}$ . The glider eventually landed at a speed of 200 $\mathrm{km} / \mathrm{h}$ . (a) What would its landing speed have been in the absence of air resistance? (b) What was the average force of air resistance exerted on it if it came in at a constant glide of $10^{\circ}$ to the Earth?

Dading Chen

Numerade Educator

Problem 58

(I) How long will it take a 1750 - W motor to lift a $315-\mathrm{kg}$ piano to a sixth-story window 16.0 $\mathrm{m}$ above?

Dading Chen

Numerade Educator

Problem 59

(I) If a car generates 18 hp when traveling at a steady 88 $\mathrm{km} / \mathrm{h}$ , what must be the average force exerted on the car due to friction and air resistance?

Dading Chen

Numerade Educator

Problem 60

(1) A $1400-\mathrm{kg}$ sports car accelerates from rest to 95 $\mathrm{km} / \mathrm{h}$ in 7.4 $\mathrm{s}$ . What is the average power delivered by the engine?

Dading Chen

Numerade Educator

Problem 61

(I) $(a)$ Show that one British horsepower $(550 \mathrm{ft} \cdot \mathrm{Ib} / \mathrm{s})$ is equal to 746 $\mathrm{W}$ . (b) What is the horsepower rating of a $75-\mathrm{W}$ lightbulb?

Dading Chen

Numerade Educator

Problem 62

(II) Electric energy units are often expressed in the form of "kilowatt-hours" (a) Show that one kilowatt-hour $(\mathrm{kWh})$ is equal to $3.6 \times 10^{6} \mathrm{J} .(b)$ If a typical family of four uses electric energy at an average rate of $520 \mathrm{W},$ how many $\mathrm{k} \mathrm{Wh}$ would their electric bill be for one month, and (c) how many joules would this be? $(d)$ At a cost of $\$ 0.12$ per $\mathrm{kWh}$ , what would their monthly bill be in dollars? Does the monthly bill depend on the rate at which they use the electric energy?

Dading Chen

Numerade Educator

Problem 63

(II) A driver notices that her $1150-\mathrm{kg}$ car slows down from 85 $\mathrm{km} / \mathrm{h}$ to 65 $\mathrm{km} / \mathrm{h}$ in about 6.0 $\mathrm{s}$ on the level when it is in neutral. Approximately what power (watts and hp) is needed to keep the car traveling at a constant 75 $\mathrm{km} / \mathrm{h}$ ?

Dading Chen

Numerade Educator

Problem 64

(II) How much work can a 3.0 -hp motor do in 1.0 $\mathrm{h} ?$

Farhanul Hasan

Numerade Educator

Problem 65

(II) A shot-putter accelerates a 7.3 -kg shot from rest to 14 $\mathrm{m} / \mathrm{s}$ . If this motion takes 1.5 $\mathrm{s}$ , what average power was developed?

Rachel Wellington

Rachel Wellington

University of Georgia

Problem 66

(II) A pump is to lift 18.0 $\mathrm{kg}$ of water per minute through a height of 3.60 $\mathrm{m}$ . What output rating (watts) should the pump motor have?

Dading Chen

Numerade Educator

Problem 67

(II) During a workout, the football players at State U. ran up the stadium stairs in 66 s. The stairs are 140 $\mathrm{m}$ long and inclined at an angle of $32^{\circ} .$ If a typical player has a mass of 95 $\mathrm{kg}$ , estimate the average power output on the way up. Ignore friction and air resistance.

Dading Chen

Numerade Educator

Problem 68

(II) How fast must a cyclist climb a $6.0^{\circ}$ hill to maintain a power output of 0.25 $\mathrm{hp}$ ? Neglect work done by friction, and assume the mass of cyclist plus bicycle is 68 $\mathrm{kg}$ .

Dading Chen

Numerade Educator

Problem 69

(II) A 1200 -kg car has a maximum power output of 120 hp. How steep a hill can it climb at a constant speed of 75 $\mathrm{km} / \mathrm{h}$ if the frictional forces add up to 650 $\mathrm{N}$ ?

Dading Chen

Numerade Educator

Problem 70

(II) What minimum horsepower must a motor have to be able to drag a $310-\mathrm{kg}$ box along a level floor at a speed of 1.20 $\mathrm{m} / \mathrm{s}$ if the coefficient of friction is 0.45$?$

Dading Chen

Numerade Educator

Problem 71

(III) A bicyclist coasts down a $7.0^{\circ}$ hill at a steady speed of 5.0 $\mathrm{m} / \mathrm{s}$ . Assuming a total mass of 75 $\mathrm{kg}$ (bicycle plus rider), what must be the cyclist's power output to climb the same hill at the same speed?

Dading Chen

Numerade Educator

Problem 72

Designers of today's cars have built $\cdots 5 \mathrm{mi} / \mathrm{h}$ (8 $\mathrm{km} / \mathrm{h} )$ bumpers" that are designed to compress and rebound elastically without any physical damage at speeds below 8 $\mathrm{km} / \mathrm{h}$ . If the material of the bumpers permanently deforms after a compression of $1.5 \mathrm{cm},$ but remains like an elastic spring up to that point, what must the effective spring stiffess constant of the bumper be, assuming the car has a mass of 1300 $\mathrm{kg}$ and is tested by ramming into a solid wall?

Dading Chen

Numerade Educator

Problem 73

In a certain library the first shelf is 10.0 $\mathrm{cm}$ off the ground, and the remaining four shelves are each spaced 30.0 $\mathrm{cm}$ above the previous one. If the average book has a mass of 1.5 $\mathrm{kg}$ with a height of $21 \mathrm{cm},$ and an average shelf holds 25 books, how much work is required to fill all the shelves, assuming the books are all laying flat on the floor to start?

Dading Chen

Numerade Educator

Problem 74

A film of Jesse Owens's famous long jump (Fig. $6-42 )$ in the 1936 Olympics shows that his center of mass rose 1.1 $\mathrm{m}$ from launch point to the top of the arc. What minimum speed did he need at launch if he was traveling at 6.5 $\mathrm{m} / \mathrm{s}$ at the top of the arc?

Farhanul Hasan

Numerade Educator

Problem 75

The block of mass $m$ sliding without friction along the looped track shown in Fig. $6-39$ is to remain on the track at all times, even at the very top of the loop of radius $r .$ (a) In terms of the given quantities, determine the minimum release height $h$ (as in Problem $40 ) .$ Next, if the actual release height is $2 h,$ calculate $(b)$ the normal force exerted by the track at the bottom of the loop, $(c)$ the normal force exerted by the track at the top of the loop, and $(d)$ the normal force exerted by the track after the block exits the loop onto the flat section.

Dading Chen

Numerade Educator

Problem 76

An airplane pilot fell 370 $\mathrm{m}$ after jumping from an aircraft without his parachute opening. He landed in a snowbank, creating a crater 1.1 $\mathrm{m}$ deep, but survived with only minor injuries. Assuming the pilot's mass was 78 $\mathrm{kg}$ and his terminal velocity was 35 $\mathrm{m} / \mathrm{s}$ , estimate $(a)$ the work done by the snow in bringing him to rest; $(b)$ the average force
exerted on him by the snow to stop him; and $(c)$ the work done on him by air resistance as he fell.

Dading Chen

Numerade Educator

Problem 77

A ball is attached to a horizontal cord of length $L$ whose other end is fixed (Fig. $6-43 ) .(a)$ If the ball is released, what will be its speed at the lowest point of its path? (b) A peg is located a distance $h$ directly
below the point of attachment of the cord. If $h=0.80 L$ , what will be the speed of the ball when it
reaches the top of its circular path about the peg?
figure can't copy

Dading Chen

Numerade Educator

Problem 78

A $65-\mathrm{kg}$ hiker climbs to the top of a 3700 -m-high mountain. The climb is made in 5.0 $\mathrm{h}$ starting at an elevation of $2300 \mathrm{m} .$ Calculate (a) the work done by the hiker against gravity, (b) the average power output in watts and in horsepower, and $(c)$ assuming the body is 15$\%$ efficient, what rate of energy input was required.

Nishant Kumar

Numerade Educator

Problem 79

An elevator cable breaks when a 920 -kg elevator is 28 $\mathrm{m}$ above a huge spring $\left(k=2.2 \times 10^{5} \mathrm{N} / \mathrm{m}\right)$ at the bottom of the shaft. Calculate $(a)$ the work done by gravity on the elevator before it hits the spring. (b) the speed of the elevator just before striking the spring, and $(c)$ the amount the spring compresses (note that work is done by both the spring and gravity in this part).

Dading Chen

Numerade Educator

Problem 80

Squaw Valley ski area in California claims that its lifts can move $47,000$ people per hour. If the average lift carries people about 200 $\mathrm{m}$ (vertically) higher, estimate the power needed.

Dading Chen

Numerade Educator

Problem 81

Water flows $(v \approx 0)$ over a dam at the rate of 650 $\mathrm{kg} / \mathrm{s}$ and falls vertically 81 $\mathrm{m}$ before striking the turbine blades. Calculate $(a)$ the speed of the water just before striking the turbine blades (neglect air resistance), and (b) the rate at which mechanical energy is transferred to the turbine blades, assuming 58$\%$ efficiency.

Dading Chen

Numerade Educator

Problem 82

Show that on a roller coaster with a circular vertical loop (Fig. $6-44 )$ , the difference in your apparent weight at the top of the circular loop and the bottom of the circular loop is $6 g^{\prime} s-$ that is, six times your weight. Ignore friction. Show also that as long as your speed is above the minimum needed, this answer doesn't depend on the size of the loop or how fast you go through it.
figure can't copy

Dading Chen

Numerade Educator

Problem 83

(a) If the human body could convert a candy bar directly into work, how high could an 82 -kg man climb a ladder if he were fueled by one bar $(=1100 \mathrm{kJ}) ?(b)$ If the man then jumped off the ladder, what will be his speed when he reaches the bottom?

Dading Chen

Numerade Educator

Problem 84

A projectile is fired at an upward angle of $45.0^{\circ}$ from the top of a $165-\mathrm{m}$ cliff with a speed of 175 $\mathrm{m} / \mathrm{s}$ . What will be its speed when it strikes the ground below? (Use conservation of energy and neglect air resistance.)

Dading Chen

Numerade Educator

Problem 85

If you stand on a bathroom scale, the spring inside the scale compresses $0.60 \mathrm{mm},$ and it tells you your weight is 710 $\mathrm{N}$ . Now if you jump on the scale from a height of $1.0 \mathrm{m},$ what does the scale read at its peak?

Dading Chen

Numerade Educator

Problem 86

A $65-\mathrm{kg}$ student runs at 5.0 $\mathrm{m} / \mathrm{s}$ , grabs a rope, and swings out over a lake (Fig. $6-45 ) .$ He releases the rope when his velocity is zero. $(a)$ What is the angle $\theta$ when he releases the rope? $(b)$ What is the tension in the rope just before he releases it? $(c)$ What is the maximum tension in the rope?

Dading Chen

Numerade Educator

Problem 87

In the rope climb, a 72 -kg athlete climbs a vertical distance of 5.0 $\mathrm{m}$ in 9.0 $\mathrm{s}$ . What minimum power output was used to accomplish this feat?

Dading Chen

Numerade Educator

Problem 88

Some electric-power companies use water to store energy. Water is pumped by reversible turbine pumps from a low to a high reservoir. To store the energy produced in 1.0 hour by a $120-\mathrm{MW}\left(120 \times 10^{6} \mathrm{W}\right)$ electric-power plant, how many cubic meters of water will have to be pumped from the lower to the upper reservoir? Assume the upper reservoir is 520 $\mathrm{m}$ above the lower and we can neglect the small change in depths within each. Water has a mass of 1000 $\mathrm{kg}$ for every $1.0 \mathrm{m}^{3} .$

Dading Chen

Numerade Educator

Problem 89

A spring with spring stiffness constant $k$ is cut in half. What is the spring stiffness constant for each of the two resulting springs?

Dading Chen

Numerade Educator

Problem 90

A $6.0-\mathrm{kg}$ block is pushed 8.0 $\mathrm{m}$ up a rough $37^{\circ}$ inclined plane by a horizontal force of 75 $\mathrm{N}$ . If the initial speed of the block is 2.2 $\mathrm{m} / \mathrm{s}$ up the plane and a constant kinetic friction force of 25 $\mathrm{N}$ opposes the motion, calculate $(a)$ the initial kinetic energy of the block; $(b)$ the work done by the $75-\mathrm{N}$ force; $(c)$ the work done by the friction force; $(d)$ the work done by gravity; (e) the work done by the normal force; $(f)$ the final kinetic energy of the block.

Dading Chen

Numerade Educator

Problem 91

If a $1500-\mathrm{kg}$ car can accelerate from 35 $\mathrm{km} / \mathrm{h}$ to 55 $\mathrm{km} / \mathrm{h}$ in 3.2 $\mathrm{s}$ , how long will it take to accelerate from 55 $\mathrm{km} / \mathrm{h}$ to 75 $\mathrm{km} / \mathrm{h}$ ? Assume the power stays the same, and neglect frictional losses.

Vishal Gupta

Numerade Educator

Problem 92

In a common test for cardiac function (the "stress test"), the patient walks on an inclined treadmill (Fig. $6-46$ ). Estimate the power required from a $75-\mathrm{kg}$ patient when the treadmill is sloping at an angle of $15^{\circ}$ and the velocity is 3.3 $\mathrm{km} / \mathrm{h}$ . (How does this power compare to the power rating of a lightbulb?)

Dading Chen

Numerade Educator

Problem 93

(a) If a volcano spews a $500-\mathrm{kg}$ rock vertically upward a distance of 500 $\mathrm{m}$ , what was its velocity when it left the volcano? $(b)$ If the volcano spews the equivalent of 1000 rocks of this size every minute, what is power output?

Dading Chen

Numerade Educator

Problem 94

Water falls onto a water wheel from a height of 2.0 $\mathrm{m}$ at a rate of 95 $\mathrm{kg} / \mathrm{s}$ . (a) If this water wheel is set up to provide electricity output, what is its maximum power output? (b) What is the speed of the water as it hits the wheel?

Dading Chen

Numerade Educator