BÀI tập vật lý 1 BẰNG TIẾNG ANH đh kỹ THUẬT CÔNG NGHIỆP THÁI NGUYÊN

If the disk is released from rest in the position shown by the blue circle, what is the speed of its center of mass when the disk reaches the position indicated by the dashed circle.. De

PHẦN 1: CƠ HỌC

1 A particle moves according to the equation x =10t2 where x is in meters and t is in seconds (a) Find the average velocity for the time interval from 2.00 s to 3.00 s.(b) Find the average

velocity for the time interval from2.00 s to 2.10 s Ans (a) 50m/s (b) 41m/s

2 A 50.0-g Super Ball traveling at 25.0 m/s bounces off abrick wall and rebounds at 22.0 m/s

A high-speed camera records this event If the ball is in contact with the wall for 3.50 ms,

what is the magnitude of the average acceleration of the ball during this time interval? Ans 1.34x10 4 m/s 2

3 A jet plane comes in for a landing with a speed of 100 m/s, and its acceleration can have a

the runway, what is the minimum time interval needed before it can come to rest? (b) Can this plane land on a small tropical island airport where the runway is 0.800 km long? Explain

your answer Ans (a) 20s

4 A particle moves along the x axis Its position is

given by the equation x = 2 + 3t - 4t2, with x in

meters and t in seconds Determine (a) its position

when it changes direction and (b) its velocity when

it returns to the position it had at t = 0 Ans (a)

2.56m (b) -3m/s

5 An electron in a cathode-ray tube accelerates from a speed of 2.00 x104 m/s to 6.00 x106 m/s over 1.50 cm (a) In what time interval does the electron travel this 1.50 cm? (b) What is its

acceleration? Ans (a) 4.98x10 -9 s (b) 1.2x10 15 m/s 2

6 A firefighter, a distance d from a burning building, directs a stream of water from a fire hose

at angle i above the horizontal as shown in Figure 1 If the initial speed of the stream is vi,

at what height h does the water strike the building?

7 In Fig 2, a stone is projected at a cliff of height h with an initial speed of 42.0 m/s directed at

8 A soccer player kicks a rock horizontally off a

m/s Ans 9.91m/s

Figure 1

Figure 2

9 A train slows down as it rounds a sharp horizontal turn,slowing from 90.0 km/h to 50.0 km/h

in the 15.0 s that it takes to round the bend The radius of the curve is 150 m Compute the

acceleration at the moment the train speed reaches 50.0 km/h Assume it continues to slow

down at this time at the same rate Ans 1.48m/s 2

10 A person standing at the top of a hemispherical rock of

radius R kicks a ball (initially at rest on the top of the

rock) to give it horizontal velocity as shown in Figure 3

(a) What must be its minimum initial speed if the ball is

never to hit the rock after it is kicked? (b) With this initial

speed, how far from the base of the rock does the ball hit

the ground? Ans (a)v i  gR (b)( 2 1)R

11 A car travels due east with a speed of 50.0 km/h

Rain-drops are falling at a constant speed vertically with respect to the Earth The traces of the rain

on the side windows of the car make an angle of 60.0° with the vertical Find the velocity of

the rain with respect to (a) the car and (b) the Earth Ans (a) 57.7km/h (b) 28.9km/h

12 A light string can support a stationary hanging load of 25.0 kg before breaking A 3.00-kg

object attached to the string rotates on a horizontal, frictionless table in a circle of radius

0.800 m, and the other end of the string is held fixed What range of

speeds can the object have before the string breaks? Ans v8.08 /m s

13 A 4.00-kg object is attached to a vertical rod by two strings as shown in

Figure 4 The object rotates in a horizontal circle at constant speed 6.00

m/s Find the tension in (a) the upper string and (b) the lower string Ans

T a =108N; T b =56.2N

14 Two balls with masses M and m are connected by a rigid rod of length

L and negligible mass as shown in Figure 5 For an axis perpendicular to

the rod, show that the system has the minimum moment of inertia when

the axis passes through the center of mass Show that this moment of

inertia is I =L2, where  = mM/(m + M)

15 A uniform, thin solid door has height 2.20 m, width 0.870 m, and

mass 23.0 kg Find its moment of inertia for rotation on its hinges Is

16 The combination of an applied force and a friction force produces a

constant total torque of 36.0 Nm on a wheel rotating about a fixed axis The applied force

acts for 6.00 s During this time, the angular speed of the wheel increases from 0 to 10.0

rad/s The applied force is then removed, and the wheel comes to rest in 60.0 s Find (a) the

Figure 3

Figure 4

Figure 5

moment of inertia of the wheel, (b) the magnitude of the frictional torque, and (c) the total

17 A block of mass m1 = 2.00 kg and a block of mass m2 = 6.00

kg are connected by a massless string over a pulley in the

shape of a solid disk having radius R = 0.250 m and mass M

= 10.0 kg These blocks are allowed to move on a fixed

of kinetic friction is 0.360 for both blocks Draw free-body

diagrams of both blocks and of the pulley Determine (a) the

acceleration of the two blocks and (b) the tensions in the

string on both sides of the pulley Ans 0.309m/s 2 , 7.67N, 9.22N

18 Consider the system shown in Figure 7 with m1=20.0 kg, m2 =12.5 kg, R =

resting on the floor, and object m1 is 4.00 m above the floor when it is released

from rest The pulley axis is frictionless The cord is light, does not stretch, and

does not slip on the pulley Calculate the time interval required for m1 to hit the

floor How would your answer change if the pulley were massless? Ans 2.27s,

1.56s

19 (a) A uniform solid disk of radius R and mass M is free to rotate on a

frictionless pivot through a point on its rim (Fig 8) If the disk is released from

rest in the position shown by the blue circle, what is the speed of its

center of mass when the disk reaches the position indicated by the

dashed circle? (b) What is the speed of the lowest point on the disk in

the dashed position? (c) Repeat part (a) using a uniform hoop Ans (a)

2 gR/ 3 (b)4 gR/ 3 (c) gR

20 A cylinder of mass 10.0 kg rolls without slipping on a horizontal

surface At a certain instant its center of mass has a speed of 10.0 m/s

Determine (a) the translational kinetic energy of its center of mass, (b)

the rotational kinetic energy about its center of mass, and (c) its total energy Ans (a) 500J

(b) 250J (c) 750J

21 A solid sphere is released from height h from the top of an incline making an angle  with the

horizontal Calculate the speed of the sphere when it reaches the bottom of the incline (a) in

the case that it rolls without slipping and (b) in the case that it slides frictionlessly without

rolling (c) Compare the time intervals required to reach the bottom in cases (a) and (b) Ans

(a) 10gh/ 7 (b) 2gh

Figure 6

Figure 7

Figure 8

22 A uniform solid disk and a uniform hoop are placed side by side at the top of an incline of

heighth If they are released from rest at the same time and roll without slip-ping, which

object reaches the bottom first? Verify your answer by calculating their

speeds when they reach the bottom in terms of h Ans the disk reaches the

bottom first

23 The reel shown in Figure 9 has radius R and moment of inertia I One end

of the block of mass m is connected to a spring of force constant k, and the

other end is fastened to a cord wrapped around the reel The reel axle and

the incline are frictionless The reel iswound counterclockwise so that the

spring stretches a dis-tance d from its unstretched position and the reel is then released from

rest (a) Find the angular speed of the reel when the spring is again

unstretched (b) Evaluate the angular speed numerically at this point,

taking I=1.00 kg m2, R = 0.300 m, k = 50.0 N/m, m = 0.500 kg, d =

24 A uniform solid sphere of radius r is placed on the inside surface of a

hemispherical bowl with much larger radius R The sphere is released

from rest at an angle  to the vertical and rolls without slipping (Fig 10)

Determine the angular speed of the sphere when it reaches the bottom

25 A solid sphere of mass m and radius r rolls without slipping along the

track shown in Figure 11 It starts from rest with the lowest point of the

sphere at height h above the bottom of the loop of radius R, much

larger than r (a) What is the minimum value of h (in terms of R) such

that the sphere completes the loop? (b) What are the components of the net

force on the sphere at the point P if h = 3R ? Ans (a) h min =2.7R, (b) F x

=-20mg/7; F y = -5mg/7

26 The system shown in Figure 12 consists of a light, inextensible cord; light,

frictionless pulleys; and blocks of equal mass It is initially held at rest so that

the blocks are at the same height above the ground The blocks are then

released Find the speed of block A at the moment when the vertical

15

A

gh

v 

27 A pendulum, comprising a light string of length L and a small sphere, swings

in the vertical plane The string hits a peg located a distance d below the

point of suspension (Fig 13) (a) Show that if the sphere is released from a

Figure 10

Figure 11

Figure 12

Figure 13 Figure 9

height below that of the peg, it will return to this height after the string strikes the peg (b)

in a complete circle centered on the peg, the minimum value of d must be 3L/5

28 Two blocks are free to slide along the frictionless

wooden track ABC shown in Figure 14 The block of

its front end is the north pole of a strong magnet, which

is repelling the north pole of an identical magnet

kg, initially at rest The two blocks never touch

29 A 5.00g bullet moving with an initial speed of 400 m/s is fired into and

passes through a 1.00-kg block as shown in Figure 15 The block,

initially at rest on a frictionless, horizontal surface, is connected to a

spring with force constant 900 N/m The block moves 5.00 cm to the

right after impact Find (a) the speed at which the bullet emerges from the

block and (b) the mechanical energy converted into internal energy in the

collision Ans (a) 100m/s (b) 374J

30 A particle of mass m is shot with an initial velocity vi

making an angle  with the horizontal

The particle moves in the gravitational field of the Earth Find the angular momentum of the

particle about the origin when the particle is (a) at the origin, (b) at thebhighest point of its

trajectory, and (c) just before it hits the ground (d) What torque causes its angular

momen-tum to change?

31 A conical pendulum consists of a bob of mass m in motion in a circular path in a

horizontal plane as shown in Figure 16 During the motion, the supporting wire of

length l maintains the constant angle  with the vertical Show that the magnitude

of the angular momentum of the bob about the circle’s center is:

2 3 4

sincos

m gl





32 A uniform solid sphere of radius 0.500 m and mass 15.0 kg turns

counterclockwise about a vertical axis through its center Find its vector angular

33 A uniform solid disk of mass 3.00 kg and radius 0.200 m rotates about a fixed axis

perpendicular to its face with angular frequency 6.00 rad/s Calculate the angular mo-mentum

of the disk when the axis of rotation (a) passes through its center of mass and (b) passes

Figure 14

Figure 15

Figure 16

through a point midway between the center and the rim Ans (a) 0.36 kgm 2 /s (b) 0.54

kgm 2 /s

34 A particle of mass 0.400 kg is attached to the 100-cm mark of a meterstick of mass 0.100 kg

The meterstick rotates on a horizontal, frictionless table with an angular speed of 4.00 rad/s

Calculate the angular momentum of the system when the meterstick is pivoted about an axis

(a) perpendicular to the table through the 50.0-cm mark and (b) perpendicular to the table

through the 0-cm mark Ans (a) 0.433 kgm 2 /s (b) 1.73 kgm 2 /s

35 A uniform cylindrical turntable of radius 1.90 m and mass 30.0 kg rotates counterclockwise

in a horizontal plane with an initial angular speed of 4 rad/s The fixed turntable bearing is

frictionless A lump of clay of mass 2.25 kg and negligible size is dropped onto the turntable

from a small distance above it and immediately sticks to the turntable at a point 1.80 m to the

east of the axis (a) Find the final angular speed of the clay and turntable (b) Is mechanical

energy of the turntable-clay system conserved in this process? Explain and use numerical

results to verify your answer (c) Is momentum of the system conserved in this process?

Explain your answer

36 A uniform rod of mass 300 g and length 50.0 cm rotates in a horizontal plane about a fixed,

vertical, frictionless pin through its center Two small, dense beads, each of mass m, are

mounted on the rod so that they can slide without friction along its length Initially, the beads

are held by catches at positions 10.0 cm on each side of the center, and the system is rotating

at an angular speed of 36.0 rad/s The catches are released simultaneously, and the beads slide

outward along the rod Find the angular speed of the system at the instant the beads slide off

the ends of the rod as it depends on m Ans 9.2rad/s

37 A wad of sticky clay with mass m and velocity is fired at a solid cylinder

of mass M and radius R (Figure 17) The cylinder is initially at rest and

is mounted on a fixed horizontal axle that runs through its center of mass

The line of motion of the projectile is perpendicu-lar to the axle and at a

distance d< R from the center (a) Find the angular speed of the system

just after the clay strikes and sticks to the surface of the cylinder (b) Is

mechanical energy of the clay-cylinder system conserved in this process? Explain your

answer (c) Is momentum of the clay-cylinder system conserved in this process? Explain your

38 A wooden block of mass M resting on a frictionless, horizontal surface

is attached to a rigid rod of length l and of negligible mass (Fig 18)

The rod is pivoted at the other end A bullet of mass m traveling

parallel to the horizontal surface and perpendicular to the rod with

speed v hits the block and becomes embedded in it (a) What is the

Figure 17

Figure 18

angular momentum of the bullet–block system? (b) What fraction of the original kinetic

energy is converted into internal energy in the collision? Ans (a)

mvl (b) M/(M+m)

39 A projectile of mass m moves to the right with a speed vi (Fig

P19a) The projectile strikes and sticks to the end of a stationary

rod of mass M, length d, pivoted about a frictionless axle through

its center (Fig P19b) (a) Find the angular speed of the system right

after the collision (b) Determine the fractional loss in mechanical

40 A 2.0-kg disk traveling at 3.0 m/s strikes a 1.0-kg stick of length 4.0

m that is lying flat on nearly frictionless ice as shown in the overhead

view of Figure 20a Assume the collision is elastic and the disk does

not deviate from its original line of motion Find the translational

speed of the disk, the translational speed of the stick, and the angular

speed of the stick after the collision The moment of inertia of the

stick about its center of mass is 1.33 kgm2 Ans v d = 2.3 m/s, v s = 1.3

m/s, and =2.0 rad/s

41 Two objects are connected by a light string passing over a light,

frictionless pulley as shown in Figure P8.7 The object of mass 5.00 kg

is released from rest Using the iso-lated system model, (a) determine

the speed of the 3.00-kg object just as the 5.00-kg object hits the

ground (b) Find the maximum height to which the 3.00-kg object rises

Ans: 4.43m/s; 5m

42 The coefficient of friction between the 3.00-kg block and the surface

in Figure P8.19 is 0.400 The system starts from rest What is the speed

of the 5.00-kg ball when it has fallen 1.50 m? Ans 3.74m/s

43 A 5.00-kg block is set into motion up an inclined plane with an initial

speed of 8.00 m/s (Fig P8.21) The block comes to rest after traveling

3.00 m along the plane, which is inclined at an angle of 30.0° to the

horizontal For this motion, determine (a) the change in the block’s

kinetic energy, (b) the change in the potential energy of the block–Earth

system, and (c) the friction force exerted on the block (assumed to be constant) (d) What is

the coefficient of kinetic friction? Ans (a) -160J (b) 73.5J (c) 28.8N (d) 0.679

Figure 20 Figure 19

44 An 80.0-kg skydiver jumps out of a balloon at an alti-tude of

1 000 m and opens the parachute at an altitude of 200 m (a)

Assuming the total retarding force on the diver is constant at

50.0 N with the parachute closed and con-stant at 3 600 N

with the parachute open, find the sky-diver’s speed when he lands on the ground (b) Do you think the skydiver will be injured? Explain (c) At what height should the parachute be opened so that the final speed of the skydiver when he hits the ground is 5.00 m/s? (d) How

realistic is the assumption that the total retarding force is constant? Explain Ans (a) 24.5m/s (c) 206m

45 A toy cannon uses a spring to project a 5.30-g soft rubber ball The spring is originally

compressed by 5.00 cm and has a force constant of 8.00 N/m When the cannon is fired, the ball moves 15.0 cm through the horizontal bar-rel of the cannon and the barrel exerts a constant friction force of 0.032 0 N on the ball (a) With what speed does the projectile leave the barrel of the cannon? (b) At what point does the ball have maximum

speed? (c) What is this maximum speed? Ans (a) 1.4m/s (b) 4.6 cm

from the start (c) 1.79m/s

46 A 3.00-kg steel ball strikes a wall with a speed of 10.0 m/s at an angle of

60.0° with the surface It bounces off with the same speed and angle (Fig

P9.9) If the ball is in contact with the wall for 0.200 s, what is the

average force exerted by the wall on the ball? Ans -260N

47 A bullet of mass m is fired into a block of mass M initially at rest at

the edge of a frictionless table of height h (Fig.P9.57) The bullet

remains in the block, and after impact the block lands a distance d

from the bottom of the table Determine the initial speed of the

bullet Ans

2 d 2

48 A small block of mass m1 = 0.500 kg is released from rest

= 3.00 kg, which sits on a frictionless horizontal surface as

shown in Figure P9.58a When the block leaves the wedge,

its velocity is measured to be 4.00 m/s to the right as

shown in the figure (a) What is the velocity of the wedge after the block reaches the

horizontal surface? (b) What is the height h of the wedge? Ans (a) - 0.667m/s (b) 0.952m

49 Rigid rods of negligible mass lying along the y axis connect three

particles (Fig P10.22) The system rotates about the x axis with an

angular speed of 2.00 rad/s Find (a) the moment of inertia about the x

axis and the total rotational kinetic energy evaluated from and (b) the

tangential speed of each particle and the total kinetic energy evaluated

from (c) Compare the answers for kinetic energy in parts (a) and (b)

Ans (a) 92kg.m 2 ; 184J (b) v 1 = 6m/s; v 2 = 4m/s; v 3 = 8m/s; 184J

50 A uniform, thin solid door has height 2.20 m, width 0.870 m, and mass

23.0 kg Find its moment of inertia for rotation on its hinges Is any

51 Find the net torque on the wheel in Figure P10.33 about the axle

through O, taking a = 10.0 cm and b =25.0 cm Ans 3.55 Nm

52 A potter’s wheel—a thick stone disk of radius 0.500 m and mass 100

kg—is freely rotating at 50.0 rev/min The pot-ter can stop the wheel

in 6.00 s by pressing a wet rag against the rim and exerting a radially

inward force of 70.0 N Find the effective coefficient of kinetic

friction between wheel and rag Ans 0.312

53 Big Ben, the Parliament tower clock in London, has an hour hand

2.70 m long with a mass of 60.0 kg and a minute hand 4.50 m long

with a mass of 100 kg Calculate the total rotational kinetic energy of the two hands about the

axis of rotation (You may model the hands as long, thin rods.) Ans

1.04x10 -3 J

54 The top in Figure P10.43 has a moment of inertia equal to 4.00x10-4 kg.m2

and is initially at rest It is free to rotate about the stationary axis AA’ A

string, wrapped around a peg along the axis of the top, is pulled in such a

manner as to maintain a constant tension of 5.57 N If the string does not

slip while it is unwound from the peg, what is the angular speed of the top

after 80.0 cm of string has been pulled off the peg? Ans 149 rad/s

55 In Figure P10.45, the sliding block has a mass of 0.850 kg, the

counterweight has a mass of 0.420 kg, and the pulley is a hollow

cylinder with a mass of 0.350 kg, an inner radius of 0.020 0 m,

and an outer radius of 0.030 0 m The coefficient of kinetic

friction between the block and the horizontal surface is 0.250

The pulley turns without friction on its axle The light cord does

not stretch and does not slip on the pulley The block has a

velocity of 0.820 m/s toward the pulley when it passes through a

photogate (a) Use energy methods to predict its speed after it has moved to a second

photogate, 0.700 m away (b) Find the angular speed of the pulley at the same moment Ans (a) 1.59m/s (b) 53.1 rad/s

56 A cylindrical rod 24.0 cm long with mass 1.20 kg and radius 1.50 cm has a ball of diameter

8.00 cm and mass 2.00 kg attached to one end The arrangement is origi-nally vertical and stationary, with the ball at the top The system is free to pivot about the bottom end of the rod after being given a slight nudge (a) After the rod rotates through 90°, what is its rotational kinetic energy? (b) What is the angular speed of the rod and ball? (c) What is the linear speed

of the ball? (d) How does this speed compare with the speed if the ball had fallen freely

through the same distance of 28 cm? Ans (a) 6.9J (b) 8.73rad/s (c) 2.44m/s (d) 1.043 times

57 This problem describes one experimental method for determining

the moment of inertia of an irregularly shaped object such as the

payload for a satellite Figure P10.49 shows a counterweight of

mass m suspended by a cord wound around a spool of radius r,

forming part of a turntable supporting the object The turntable can

rotate without friction When the counterweight is released from

rest, it descends through a distance h, acquiring a speed v Show

that the moment of inertia I of the rotating appa-ratus (including

the turntable) is mr2(2gh/v2-1)

58 A uniform, hollow, cylindrical spool has inside radius R/2, outside

radius R, and mass M (Fig P10.69) It is mounte so that it rotates

on a fixed, horizontal axle A counter-weight of mass m is

connected to the end of a string wound around the spool The

counterweight falls from rest at t = 0 to a position y at time t

Show that the torque due to the friction forces between spool and

axle is:

59 A uniform solid sphere of radius 0.500 m and mass 15.0 kg turns counterclockwise about a

vertical axis through its center Find its vector angular momentum when its angular speed is

3.00 rad/s Ans 4.5 kgm 2 /s

60 A uniform solid disk of mass 3.00 kg and radius 0.200 m rotates about a fixed axis

perpendicular to its face with angular frequency 6.00 rad/s Calculate the angular mo-mentum

of the disk when the axis of rotation (a) passes through its center of mass and (b) passes

PHẦN 2:NHIỆT HỌC

61 Gas is contained in an 8.00-L vessel at a temperature of 20.0°C and a pressure of 9.00 atm

(a) Determine the number of moles of gas in the vessel (b) How many mol-ecules are in the

62 An auditorium has dimensions 10.0 m x20.0 m x30.0 m How many molecules of air fill the

63 A cook puts 9.00 g of water in a 2.00-L pressure cooker and warms it to 500°C What is the pressure inside the container? Ans 15.9 atm

64 The mass of a hot-air balloon and its cargo (not including the air inside) is 200 kg The air

To what temperature must the air in the balloon be warmed before the balloon will lift off? (Air density at 10.0°C

65 A cube 10.0 cm on each edge contains air (with equivalent molar mass 28.9 g/mol) at

atmospheric pressure and temperature 300K Find (a) the mass of the gas, (b) the gravitational force exerted on it, and (c) the force it exerts on each face of the cube (d)

Comment on the physical reason such a small sample can exert such a great force Ans (a) 1.17x10 -3 kg (b) 11.5mN (c) 1.01kN

66 The pressure gauge on a tank registers the gauge pres-sure, which is the difference between

the interior and exterior pressure When the tank is full of oxygen (O2) it contains 12.0 kg of the gas at a gauge pressure of 40.0 atm Determine the mass of oxygen that has been withdrawn from the tank when the pressure reading is 25.0 atm Assume the temperature of

the tank remains constant Ans 4.39kg

67 In state-of-the-art vacuum systems, pressures as low as 10-9 Pa are being attained Calculate the number of molecules in a 1.00m3 vessel at this pressure and a tempera-ture of 27.0°C Ans 2.41x10 11 molecules

68 A cylinder is closed by a piston connected to a spring of constant

2.00x103 N/m (Fig P19.38) With the spring relaxed, the cylinder is

filled with 5.00 L of gas at a pres-sure of 1.00 atm and a

temperature of 20.0°C (a) If the piston has a cross-sectional area of

0.01 m2 and negligi-ble mass, how high will it rise when the

temperature is raised to 250°C? (b) What is the pressure of the gas

69 A vertical cylinder of cross-sectional area A is fitted with a tight-fitting, frictionless piston of

mass m (a) If n moles of an ideal gas are in the cylinder at a tem-perature of T, what is the height h at which the piston is in equilibrium under its own weight? (b) What is the value for

(b) 0.661m

70 A cylinder that has a 40.0cm radius and is 50.0

cm deep is filled with air at 20.0°C and 1.00 atm

(Fig P19.54a) A 20kg piston is now lowered

into the cylinder, compressing the air trapped

inside as it takes equilibrium height hi (Fig

P19.54b) Finally, a 75.0-kg dog stands on the

piston,further compressing the air, which remains at 20°C (Fig.P19.54c) (a) How far down (

be warmed to raise the piston and dog back to hi? Ans (a)

h=0.52cm (b) 276K

71 A sample of ideal gas is expanded to twice its original volume of

1.00 m3 in a quasi-static process for which P =V2 with  =5.00

the expanding gas? Ans -1.18MJ

72 An ideal gas is enclosed in a cylinder with a movable piston on top of it The piston has a

of the gas constant How much work is done on the gas as the temperature of 0.200 mol of

the gas is raised from 20.0°C to 300°C? Ans -466J

73 An ideal gas is enclosed in a cylinder that has a movable piston on top The piston has a

mass m and an area A and is free to slide up and down, keeping the pressure of the gas constant How much work is done on the gas as the temperature of n mol of the gas is raised from T1 to T2? Ans –nR(T 2 – T 1 )

74 One mole of an ideal gas is warmed slowly so that it goes from the PVstate ( Pi, Vi) to( 3Pi, 3Vi), in such a way that the pressure of the gas is directly proportional to the volume (a) How much work is done on the gas in the process? (b) How is the temperature of the gas related to

its volume during this process? Ans (a) -4P i V i , (b)

75 A gas is taken through the cyclic process described in Figure

P20.26 (a) Find the net energy transferred to the sys-tem by

heat during one complete cycle (b) What If? If the cycle is

reversed—that is, the process follows the path ACBA—what is the net energy input per cycle

by heat? Ans12.2kJ

76 A thermodynamic system undergoes a process in which its internal energy decreases by 500

J Over the same time interval, 220 J of work is done on the system Find the energy

transferred to or from it by heat Ans -720J

77 A sample of an ideal gas goes through the process

shown in Figure P20.28 From A to B, the process is

adiabatic; from B to C, it is isobaric with 100 kJ of

energy entering the system by heat From C to D, the

process is isothermal; from D to A, it is isobaric with

150 kJ of energy leaving the system by heat Determine

the difference in internal energy Eint, B - Eint, A Ans

42.9kJ

78 One mole of an ideal gas does 3 000 J of work on its sur-roundings as it expands isothermally

to a final pressure of 1.00 atm and volume of 25.0 L Determine (a) the initial volume and (b)

the temperature of the gas Ans (a) 7.65 L (b) 305K

79 An ideal gas initially at 300 K undergoes an isobaric expansion at 2.50 kPa If the volume

increases from 1.00 m3 to 3.00 m3 and 12.5 kJ is transferred to the gas by heat, what are (a)

the change in its internal energy and (b) its final temperature? Ans (a) 7.5kJ (b) 900K

80 An ideal gas initially at Pi, Vi is taken through a cycle as shown in

Figure P20.34 (a) Find the net work done on the gas per cycle (b)

What is the net energy added by heat to the system per cycle? (c)

Obtain a numerical value for the net work done per cycle for 1.00 mol

of gas initially at 0°C Ans (a) -4P i V i (b) 4P i V i (c) 9.08kJ

81 A 2.00-mol sample of helium gas initially at 300 K and 0.400 atm is

compressed isothermally to 1.20 atm Noting that the helium behaves

as an ideal gas, find (a) the final volume of the gas, (b) the work done

on the gas, and (c) the energy transferred by heat Ans (a) 0.041m 3 (b) 5.48kJ(c) -5.48kJ

82 A cylinder contains 3.00 mol of helium gas at a temperature of 300 K (A) If the gas is heated

at constant volume, how much energy must be transferred by heat to the gas for its ture to increase to 500 K? (B) How much energy must be transferred by heat to the gas at

83 A 1.00-mol sample of hydrogen gas is heated at con-stant pressure from 300 K to 420 K

Calculate (a) the energy transferred to the gas by heat, (b) the increase in its internal energy,

84 A house has well-insulated walls It contains a volume of 100 m3 of air at 300 K (a) Calculate the energy required to increase the temperature of this diatomic ideal gas by

1.00°C (b) What If? If this energy could be used to lift an object of mass m through a height

of 2.00 m, what is the value of m? Ans (a) 118kJ (b) 6.03x10 3 kg

85 A vertical cylinder with a heavy piston contains air at 300 K The initial pressure is 200 kPa,

and the initial volume is 0.350 m3 Take the molar mass of air as 28.9 g/mol and assume Cv = 5R/2 (a) Find the specific heat of air at constant volume in units of J/kg °C (b) Calculate the mass of the air in the cylinder (c) Suppose the piston is held fixed Find the energy input required to raise the temperature of the air to 700K (d) What If? Assume again the conditions of the initial state and assume the heavy piston is free to move Find the energy

input required to raise the temperature to 700K Ans (a) 719J/kg.K (b) 0.811kg (c) 233kJ (d) 327kJ

86 During the compression stroke of a certain gasoline engine, the pressure increases from 1.00

atm to 20.0 atm If the process is adiabatic and the fuel–air mixture behaves as a diatomic ideal gas, (a) by what factor does the volume change and (b) by what factor does the tem-perature change? (c) Assuming the compression starts with 0.016 0 mol of gas at 27.0°C, find the values of Q, A, and U that characterize the process Ans (a) 0.118 (b) 2.35 (c) Q=0, A=U=135J

87 A 2.00-mol sample of a diatomic ideal gas expands slowly and adiabatically from a pressure

of 5.00 atm and a vol-ume of 12.0 L to a final volume of 30.0 L (a) What is the final pressure

of the gas? (b) What are the initial and final temperatures? (c) Find Q , A, and U Ans (a) 1.39atm (b) 365K ;253K (c) Q=0, A=U= - 4.66kJ

88 Air (a diatomic ideal gas) at 27.0°C and atmospheric pressure is drawn into a bicycle pump

that has a cylinder with an inner diameter of 2.50 cm and length 50.0 cm The downstroke adiabatically compresses the air, which reaches a gauge pressure of 800 kPa before entering the tire Determine (a) the volume of the com-pressed air and (b) the temperature of the compressed air (c) What If? The pump is made of steel and has an inner wall that is 2.00

mm thick Assume 4.00 cm of the cylinder’s length is allowed to come to thermal

equi-librium with the air What will be the increase in wall temperature? Ans (a) 5.15x10 -5 m 3 (b) 560K (c) 2.24K

89 A 4.00-L sample of a diatomic ideal gas with specific heat ratio 1.40, confined to a cylinder,

is carried through a closed cycle The gas is initially at 1.00 atm and at 300 K

First, its pressure is tripled under constant volume Then, it expands adiabatically to its original pressure Finally, the gas is compressed isobarically to its original volume (a) Draw a PV diagram of this cycle (b) Determine the volume of the gas at the end of the adiabatic expansion (c) Find the temperature of the gas at the start of the adi-abatic expansion (d) Find the temperature at the end of the cycle (e) What was the net work done on the gas for this

cycle? ans (b) 877L (c) 900K (d) 300K (e) -336J

90 How much work is required to compress 5.00 mol of air at 20.0°C and 1.00 atm to one-tenth

of the original vol-ume (a) by an isothermal process? (b) What If? How much work is required to produce the same compression in an adiabatic process? (c) What is the final

pressure in each of these two cases? Ans (a) 28kJ (b) 46kJ (c) 10atm; 25.1atm

91 The dimensions of a classroom are 4.20m x 3.00m x 2.50m (a) Find the number of molecules

of air in it at atmospheric pressure and 20.0°C (b) Find the mass of this air, assuming the air consists of diatomic molecules with molar mass 28.9 g/mol (c) Find the average kinetic energy of one molecule (d) Find the root-mean-square molecular speed (e) On the

/2 Find the internal energy in the air (f) What If? Find the internal energy of the air in the room at 25.0°C Explain how it compares with the result at 20.0°C and how it happens that

92 As a 1.00-mol sample of a monatomic ideal gas expands adiabatically, the work done on it is

–2 500 J The initial temperature and pressure of the gas are 500 K and 3.60 atm Calculate

(a) the final temperature and (b) the final pressure Ans (a) 300K (b)1atm

93 A Carnot engine has a power output of 150 kW The engine operates between two reservoirs

at 20.0°C and 500°C (a) How much energy does it take in per hour? (b) How much energy is

lost per hour in its exhaust? Ans (a) 869MJ (b) 330MJ

94 An engine operates in a cycle, taking in energy by heat at 180°C and putting out exhaust at

100°C In each cycle, the exhaust energy is 2.00x 104J and the engine does 1.50 x103J of work Explain how the actual efficiency of the engine compares with the efficiency of a

reversible engine operating between the same temperatures Ans (a) 7% (b) 18%

95 A Carnot heat engine operates between temperatures Th and Tc (a) If Th =500 K and Tc= 350

K, what is the efficiency of the engine? (b) What is the change in its efficiency for each degree of increase in Th above 500 K? (c) What is the change in its efficiency for each

96 An ideal gas is taken through a Carnot cycle The isothermal expansion occurs at 250°C, and

the isothermal compression takes place at 50.0°C The gas takes in 1 200 J of energy from the hot reservoir during the isothermal expansion Find (a) the energy expelled to the cold reser-

voir in each cycle and (b) the net work done by the gas in each cycle Ans (a) 741J (b) 459J

97 In a cylinder of an automobile engine, immediately after combustion, the gas is confined to a

volume of 50.0 cm3 and has an initial pressure of 3.00 x106 Pa The piston moves outward to

a final volume of 300 cm3 and the gas expands without energy loss by heat (a) If = 1.40 for the gas, what is the final pressure? (b) How much work is

done by the gas in expanding? Ans (a) 244kPa (b) 192J

98 A 1.00-mol sample of H2 gas is contained in the left side of the container shown in Figure P22.39, which has equal volumes left and right The right side is evacuated When the valve

is opened, the gas streams into the right side What is the final entropy change of the gas? Does the temperature of the gas change? Assume the container is so large that the hydrogen

behaves as an ideal gas Ans 5.76 J/K

99 A 2.00-L container has a center partition that divides it into

two equal parts as shown in Figure P22.40 The left side

contains H2 gas, and the right side contains O2 gas Both

gases are at room temperature and at atmospheric pres-sure

The partition is removed and the gases are allowed to mix

What is the entropy increase of the system? Ans 0.507 J/K

100 In 1816, Robert Stirling, a Scottish clergyman, patented

the Stirling engine , which has found a wide variety of

appli-cations ever since Fuel is burned externally to warm one of the

engine’s two cylinders A fixed quantity of inert gas moves

cyclically between the cylinders, expanding in the hot one and

contracting in the cold one Figure P22.49 represents a model

for its thermodynamic cycle Consider n mol of an ideal

monatomic gas being taken once through the cycle, consisting

of two isothermal processes at temperatures 3Ti and Ti and two constant-volume processes Determine in terms of n, R, and Ti (a) the net energy transferred by heat to the gas and (b) the efficiency of the engine A Stirling engine is easier to manufacture than an internal combustion engine or a turbine It can run on burning garbage It

can run on the energy of sunlight and produce no material

101 A 1.00-mol sample of an ideal monatomic gas is taken

a reversible isothermal expansion Calculate (a) the net work done

by the gas, (b) the energy added to the gas by heat, (c) the energy

exhausted from the gas by heat, and (d) the efficiency of the cycle

(e) Explain how the efficiency compares with that of a Carnot

engine operating between the same temperature extremes Ans (a)

4.11kJ (b) 14.2 kJ (c) – 10.1kJ (d) 28.9%

102 A 1.00-mol sample of a monatomic ideal gas is taken through

the cycle shown in Figure P22.55 At point A, the pressure, volume,

and temperature are Pi, Vi, Ti In terms of R and Ti, find (a) the

total energy entering the system by heat per cycle, (b) the total energy leaving the system by heat per cycle, and (c) the efficiency of an engine operating in this cycle (d) Explain how the efficiency compares with that of an engine operating in a Carnot cycle between the same

temperature extremes Ans (a) 10.5nRT i (b) 8.5nRT i (c) 0.19 (d) 0.833

103 An idealized diesel engine operates in a cycle known as the

air-standard diesel cycle shown in Figure P22.59 Fuel is sprayed

into the cylinder at the point of maximum compression, B

modeled as an isobaric process Show that the efficiency of an

engine operating in this idealized diesel cycle is

PHẦN 3 ĐIỆN TRƯỜNG

104 (a) Two protons in a molecule are 3.80 x 10-10 m apart Find the electrical force exerted

by one proton on the other (b) State how the magnitude of this force com-pares with the magnitude of the gravitational force exerted by one proton on the other (c) What If? What must be a particle’s charge-to-mass ratio if the magnitude of the gravitational force between

two of these particles is equal to the magnitude of electrical force between them? Ans (a) 1.59x10 -9 N (b) 1.29x10 -45 N (c) 8.61x10 -11 C/kg

105 Two small silver spheres, each with a mass of 10.0 g, are separated by 1.00 m Calculate the fraction of the elec-trons in one sphere that must be transferred to the other to produce an attractive force of 1.00 x 104 N (about 1 ton)

between the spheres (The number of electrons per atom of

silver is 47, and the number of atoms per gram is Avogadro’s

number divided by the molar mass of silver, 107.87 g/mol.)

Ans 2.51 charges in every billion

106 Three charged particles are located at the corners of an

equilateral triangle as shown in Figure P23.7 Calculate the

total electric force on the 7.00 C charge Ans 0.872N

107 Two small beads having positive charges 3q and q are fixed at the opposite ends of a horizontal insulating rod, extending from the origin to the point

x= d As shown in Figure P23.8, a third small charged bead is

free to slide on the rod At what position is the third bead in

equilibrium? Explain whether it can be in stable equilibrium Ans

1 1/ 3

d

x



108 Four charged particles are at the corners of a square of side a as

shown in Figure P23.17 (a) Determine the mag-nitude and direction of

the electric field at the location of charge q (b) What is the total electric

111 A uniformly charged insulating rod of length 14.0 cm is bent into the shape of a semicircle as shown in Figure P23.27 The rod has a total charge of -7.50 C Find the

magnitude and direction of the electric field at O, the center of the semicircle Ans 21.6 MV/m

112 A thin rod of length l and uniform charge per unit length  lies

along the x axis as shown in Figure P23.29.(a) Show that the electric

field at P, a distance y from the rod along its perpendicular bisector,

has no x component and is given by E= 2ksin0 /y (b) What If?

Using your result to part (a), show that the field of a rod of infinite

length is E= 2k/y

113 Three solid plastic cylinders all have radius 2.50 cm and length

6.00 cm One (a) carries charge with uniform density 15.0 nC/m2

everywhere on its surface Another (b) carries charge with the same uniform density on its curved lateral surface only The third (c) carries charge with uniform density 500 nC/m3

throughout the plastic Find the charge of each cylinder Ans (a) 2x10 -10 C (b) 1.41x10 -10 C (c) 5.89x10 -11 C

114 Two known charges, -12.0 C and 45.0 C, and a third unknown charge are located on the x axis The charge -12.0 C is at the origin, and the charge 45.0 C is at x =15.0 cm The unknown charge is to be placed so that each charge is in equilibrium under the action of the electric forces exerted by the other two charges Is this situation possible? Is it possible in more than one way? Explain Find the required location, magnitude, and sign of the unknown

116 A charged cork ball of mass 1.00 g is suspended on a

light string in the presence of a uniform electric field as shown

E  ( 3i 5 ) 1 0j V m the ball is in /equilibrium at  =37.0° Find (a) the charge on the ball and (b)

the tension in the string Ans (a) 10.9nC (b) 5.44x10 -3 N

a x



Find an expression for the electric potential at a point P located on the perpendicular central axis of a uniformly charged ring of radius a and total charge Q

A rod of length l located along the x axis has a total

charge Q and a uniform linear charge density = Q/l

Find the electric potential at a point P located on the y

axis a distance a from the origin

140 Ans E=a/20

145 Ans (a) x=-4.83m (b) x=-2m ; x=2/3m 151 Ans (a) E=0 ; V=1.67MV (b) E=5.84MV/m ;

155 Ans

235

kQ W

R



156 Ans



166 Ans

164 Ans 0 1 1

I r



168 Ans 13 T

170 Ans 20 T

169 Ans 13 T

171 Ans 1m/s 174 Ans (a) I

1 = 3.5A ; I 2 =1.4A (b) 34.3W (c) 4.29N

179 Ans -0.421A/s

180 Ans (a) 360mV (b) 180mV (c) 3s

177 Ans 100V

178 Ans 6cos(120t) V

5

6

7

8

9

10

11

12

13

14

15

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18

19

20

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