AP Physics

Chapter 10

Review

€

at t = 0, ωo = 2 radsec at t = 2sec, Δθ = 32rad

1. At t = 0, a wheel rotating about a fixed axis at a constant angular acceleration has an angular velocity of 2.0

€

radsec . Two seconds later it has turned through 32

radians. What is the angular acceleration of this wheel?

2

€

Δθ =ωot + 12α t2

2 Δθ −ωot( )

t 2 =α

2 32rad − 2 radsec( ) 2sec( )( )

2sec( )2 =α

14 radsec2 =α

€

at t = 0, ωo = 2 radsec at t = 2sec, Δθ = 32rad

3

#1

€

θ = 3− 2t 3

ω =dθdt

= −6t 2

α =dωdt

= −12t

at t = 2sec, α = −24 radsec2

4

2. A wheel rotating about a fixed axis has an angular position given by , where θ is measured in radians and t in seconds. What is the angular acceleration of the wheel at t = 2.0 s?

€

θ = 3− 2t 3

€

Δθ =ω ot + 12α t2

Δθ − 12α t 2( )t

=ωo

16rad − 12 2 rad

sec2( ) 2sec( )2[ ]2sec

=ωo

6 radsec =ωo

€

ω =ωo +α t

ω = 6 radsec( ) + 2 rad

sec2( ) 2sec( )ω =10 rad

sec

5

3. A wheel rotating about a fixed axis with a constant angular acceleration of 2.0 turns through 16 radians during a 2.0-sec time interval. What is the angular velocity at the end of this time interval?

€

radsec2

€

θ =ω t

θ =20 + 10( ) rad

sec

2

5sec

θ = 75rad

6

4. A wheel rotates about a fixed axis with an initial angular velocity of 20 . During a 5.0 second interval the angular velocity decreases to 10 . Assume that the angular acceleration is constant during the 5.0 second interval. Through how many radians does the wheel turn during the 5.0 second interval?

€

radsec

€

radsec

5

€

37°

€

53°

€

mg

€

mgcosθ

€

τ = Iα F⊥R = Iα

mgcos37°L2

=13

mL2α

g 45

12

=

13

Lα

25

g =13

Lα

6g5L

=α

7

5. A thin uniform rod of length L and mass m is pivoted about a horizontal, frictionless pin through one end of the rod. ( ) The rod is released when it makes an angle of 37° with the horizontal. What is the angular acceleration of the rod at the instant it is released?

€

I = 13 ML2

€

θo

€

θ2

€

θ3

€

80 rad

€

Δθ3−2 =ω2t + 12α t2

Δθ3−2 −12α t2

t=ω2

80rad − 12 4 rad

sec2( ) 4sec( )2

4 sec=ω2

12 radsec =ω2

€

ω2 =ωo +α tω2

α= t

12 radsec

4 radsec2

= t = 3sec

8

6. A wheel rotating about a fixed axis has a constant angular acceleration of 4.0

€

radsec2 . In a 4.0 second interval the wheel

turns through an angular displacement of 80 radians. Assuming the wheel started from rest, how long had the wheel been in motion at the start of the 4.0 second interval?

time line

€

m1€

m2

7. A mass (m1 = 5.0 kg) is connected by a light cord to a mass (m2 = 4.0 kg) which slides on a smooth surface, as shown in the figure. The pulley (radius = 0.20m) rotates about a frictionless axle. The acceleration of m2 is 3.5 What is the moment of inertia of the pulley?

€

msec2

9

€

FNet = mam1g− T1 + T2 = m1 + m2( )am1g− T1 −T2( ) = m1 + m2( )a

m1g−IaR2

= m1 + m2( )a

m1g− m1 + m2( )a =IaR2

R2 m1g− m1 + m2( )a( )a

= I

0.04m2 49N − 31.5N( )3.5 m

sec2

= I

I = 0.20Kg ⋅m2

10

€

m1€

m2

m1 = 5.0 kgm2 = 4.0 kga = 3.5

€

msec2

8. A wheel (radius = 0.25 m) is mounted on a frictionless, horizontal axis. The moment of inertia of the wheel about the axis is . A light cord wrapped around the wheel supports a 0.50Kg object as shown in the figure. The object is released from rest. What is the magnitude of the acceleration of the 0.50Kg object?

€

0.040Kg ⋅m 2

€

0.5kg

11

€

FNet = mamg−T = ma

mg− IaR2 = ma

mg = m +I

R2

a

mg

m +I

R2

= a

€

0.5Kg 9.8 msec2( )

0.5Kg +0.04Kg ⋅m 2

0.0625m2

= a

4.3 msec2 = a

12

13

9. A wheel rotating about a fixed axis with a constant angular acceleration of 2 starts from rest at t = 0. The wheel has a diameter of 20 cm. What is the magnitude of the total linear acceleration of a point on the outer edge of the wheel at t = 0.60 s?€

radsec2

€

10cm

9

€

at = Rα = 0.1m( ) 2 radsec2( ) = 0.2 m

sec2

ω =α t = 2 radsec2( ) 0.6sec( ) =1.2 rad

sec

aR = Rω2 = 0.1m( ) 1.2 radsec( )2

= 0.144 msec2

a = at2 + aR

2 = 0.2 msec2( )2

+ 0.144 msec2( )2

a = 0.25 msec2

14

10.A cylinder rotating about its axis with a constant angular acceleration of 1.6 starts from rest at t = 0. At the instant when it has turned through 0.40 radian, what is the magnitude of the total linear acceleration of a point on the rim (radius = 13 cm)?€

radsec2

15€

θ = 12α t2

2θα

= t =2 0.4rad( )

1.6 radsec2

t = 0.707sec

€

ω =α t

ω = 1.6 radsec2( ) 0.707sec( )

ω = 1.13 radsec

10

€

at = Rα = 0.13m( ) 1.6 radsec2( ) = 0.208 m

sec2

aR = Rω 2 = 0.13m( ) 1.13 radsec( )2

= 0.166 msec2

a = at2 + aR

2 = 0.208 msec2( )2

+ 0.166 msec2( )2

a = 0.27 msec2

16

What is the magnitude of the total linear acceleration of a point on the rim?

€

aR = at

Rω 2 = Rαα 2t 2 =α

α t 2 = 1

t =1α

=1

2.1 radsec2

t = 0.69sec

17

11. A horizontal disk with a radius of 10 cm rotates about a vertical axis through its center. The disk starts from rest at t = 0 and has a constant angular acceleration of 2.1 . At what value of t will the radial and tangential components of the linear acceleration of a point on the rim of the disk be equal in magnitude?

€

radsec2

18

12. A mass m = 4.0 kg is connected, as shown, by a light cord to a mass M = 6.0 kg, which slides on a smooth horizontal surface. The pulley rotates about a frictionless axle and has a radius R = 0.12 m and a moment of inertia . The cord does not slip on the pulley. What is the magnitude of the acceleration of m?

€

I = 0.090Kg ⋅m2

M

m

12

€

FNet = ma

mg − IaR2 = m + M( )a

mg = m + M +I

R2

a

m

m + M +I

R2

g = a

4Kg

4Kg + 6Kg +0.09Kg ⋅ m2

0.12m( )2

9.8 msec2 = a

2.41 msec2 = a

19

13. Two particles (m1 = 0.20 kg, m2 = 0.30 kg) are positioned at the ends of a 2.0-m long rod of negligible mass. What is the moment of inertia of this rigid body about an axis perpendicular to the rod and through the center of mass?

€

m1

€

m2

€

l = 2m

€

C of G

20

13

€

xCofM =mx∑m∑

=0.2Kg( ) 0( ) + 0.3Kg( ) 2m( )

0.5Kg

xCofM =1.2m from the 0.2Kg massxCofM = 0.8m from the 0.3Kg mass

€

I = mR2 = 0.2Kg 1.2m( )2+ 0.3Kg 0.8m( )2∑

I = 0.288 + 0.192( )Kg ⋅ m2

I = 0.48Kg ⋅ m2

21

14

€

0.5Kg

€

2m

€

2m €

3m

€

3m

€

3.25m

€

I = mR2∑ = 4 0.5Kg( ) 3.25m( )2

= 6.5Kg ⋅m2

14. Four identical particles (mass of each = 0.5 kg) are placed at the vertices of a rectangle (2.0 m by 3.0 m) and held in those positions by four light rods which form the sides of the rectangle. What is the moment of inertia of this rigid body about an axis that passes through the center of mass of the body and is perpendicular to the page?

22

15

€

K = 12 Iω2

2KI

=ω 2

2K

2 M( )L2 + 2 2M( ) L2

2 =ω 2

2K3ML2 =ω 2

1L

2K3M

=ω

23

15. The rigid object shown is rotated about an axis perpendicular to the paper and through point P. The total kinetic energy of the object as it rotates is equal to K. If mass = M and length = L , what is the angular velocity of the object? Neglect the mass of the connecting rods and treat the masses as particles.

€

L

€

L

€

L2

€

L2

€

P

€

M

€

M

€

2M

€

2M

16

€

xCofM =mx∑m∑

=ML + 3M 2L( )

5M=

75

L from the left edge.

I = mR2 = M 7

5L

2

∑ + M 2

5L

2

+ 3M 3

5L

2

=16

5ML2

24

€

M

€

M

€

3M

€

L

€

L

16. If the mass of each connecting rod shown above is negligible, what is the moment of inertia about an axis perpendicular to the paper through the center of mass?

€

τ∑ = Iα

F⊥R∑ = Iα

25

17. A uniform rod of mass M and length L, lying on a frictionless horizontal plane, is free to pivot about a vertical axis through one end, as shown. The moment of inertia of the rod about this axis is given by

€

13 ML2 . If a force F is applied at some

angle θ , and acts as shown, what is the resulting angular acceleration about the pivot point?

€

θ

€

F sinθ

€

F

€

F sinθL =13

ML2α

F sinθ =13

MLα

3 F sinθ = MLα3F sinθ

ML=α

€

θ

€

F sinθ

18

€

60°

€

60°

€

mg€

mgcos60°

€

τ = Iα F⊥R = Iα

mgcos60°L

2=

1

3mL2α

3g2L

12

=α

3g4L

=α

18. A uniform rod of mass m and length L is free to rotate about a frictionless pivot at one end. The rod is released from rest in the horizontal position. What is the angular acceleration of the rod at the instant it is 60° below the horizontal? 26

19. A uniform meter stick is pivoted to rotate about a horizontal axis through the 25-cm mark on the stick. The stick is released from rest in a horizontal position. The moment of inertia of a uniform rod about an axis perpendicular to the rod and through the end of the rod is given by Determine the magnitude of the initial angular acceleration of the stick.

€

13 ML2 .

0 m

28

19

€

I = mR2 =13

14

M

L4

2

∑ +13

34

M

3L4

2

I =1

12M L2

16

+

14

M 9L2

16

=

112

M L2

16

+

2712

M L2

16

I =2812

M L2

16

=

748

ML2

€

I =ML

x 2− L

4

3 L4∫ dx =

ML

13

x 3

− L4

3L4

=ML

27L3

3⋅ 64+

L3

3⋅ 64

I =28

192ML2 =

748

ML2

29

Find the moment of inertia!

19.3

€

mg

€

τ = Iα F⊥R = Iα

Mg L4

=7

48ML2α

g =7

12Lα

12g7L

=α

30

€

30°

€

L4

€

L4

€

UTop = KBottom

mg L4

=12

Iω2

mg L4

=12

13

mL2

ω 2

g 12

=13

Lω2

3g2L

=ω

31

20. A uniform rod of length L and mass M is pivoted about a horizontal frictionless pin through one end. The rod is released from rest at an angle of 30° below the horizontal. What is the angular speed of the rod when it passes through the vertical position?

2 4 6 8 10 12 14 16

5

10

-5

-10

€

ω radsec( )

€

t min( )

21. The figure below shows a graph of angular velocity as a function of time for a car driving around a circular track. Through how many radians does the car travel in the first 10 minutes?

€

20 + 40 + 10( ) rad ⋅minsec

60secmin

= 4200rad

32

€

KNet = K + Kθ

KNet =12

mv2 +12

Iω 2

KNet =12

mv2 +12

25

mR2

v2

R2

KNet =7

10mv2 =

710

10Kg( ) 2 msec( )2

KNet = 28J

33

22. What is the total Kinetic Energy of a 10 Kg solid sphere whose radius 25 cm and is rolling across a horizontal floor at

€

msec

€

12

Iω 2

12

mv 2=

23

mR2 v 2

R2

mv2 =23

34

23. A 2 Kg, hollow sphere is rolling on a horizontal surface with no slippage. What is the ratio of the sphere’s rotational Kinetic energy to its translational kinetic energy?

35

24.The graphs below show angular velocity as a function of time. In which one is the magnitude of the angular acceleration constantly decreasing?

A) B) C) D)

E)

36

€

ω radmin( )

€

t min( )2 4 6 8 10 12 14 16

2π

4π3π

π

-2π-π

25. The figure below shows a graph of angular velocity versus time for a woman bicycling around a circular track. What is her angular displacement (in rad) in the first 8 minutes?

€

Δθ =4π rad

min( ) 8 min( )2

Δθ =16π rad

€

Kθ =12

Iω2

Kθ =12

25

mR2

ω 2

Kθ =15

5Kg( ) 0.5m( )2 6 radsec( )2

Kθ = 9J

37

26. What is the Rotational Kinetic Energy of a solid sphere of mass 5 kg and radius 0.5 m if it is rotating about its diameter at a constant rate of 6 ? rad

sec

€

τ = Iα FR = Iα

FR12 mR2 =α

2FmR

=α

2 50N( )6.25Kg 4m( )

=α

4 radsec2 =α

€

ω =α tω = 4 rad

sec2 1sec( )ω = 4 rad

sec38

27. Two Forces of magnitude 50 N, as shown in the figure below, act on a cylinder of radius 4 m and mass 6.25 kg. The cylinder, which is initially at rest, sits on a frictionless surface. After 1 second, the velocity and angular velocity of the cylinder in and are respectively:

€

msec

€

radsec

€

m1

€

m2

28. Two blocks, m1 = 1 kg and m2 = 2 kg, are connected by a light string as shown in the figure. If the radius of the pulley is 1 m and its moment of inertia is what fraction of g is the acceleration of the system?

€

5Kg ⋅m2,

39

28

€

FNet = mam2g −T2 + T1 −m1g = m1 + m2( )a

m2 −m1( )g − IaR2 = m1 + m2( )a

m2 −m1( )g = m1 + m2 +I

R2

a

m2 −m1( )g

m1 + m2 +I

R2

= a

18

g = a

40

€

m1

€

m2

m1 = 1 kgm2 = 2 kg

€

I = 5Kg ⋅ m2

41

€

α = slope =−4π − 4π( ) rad

sec

10min 60secmin

=−8π600

radsec2 = −

π75

radsec2

29. The figure below shows a graph of angular velocity versus time for a man bicycling around a circular track. What is his average angular acceleration in the first 10 minutes?

€

mvo = −25

mvo +75

mvo

€

v'= pm

=

75

mvo

45

m=

74

vo

40

A mass m is moving with a speed vo as it moves left to right across the page as shown below. At some time t, the mass explodes into two pieces with a piece m moving to the left with a velocity -2vo.

€

15

30. What is the velocity of the second piece?

31. What is the velocity of the center of mass after the explosion?

€

vCofM =mv∑m∑

=−

25

mvo +75

mvo

m= vo

€

τ Net = F⊥R∑ = 20N 0.5m( ) − 7N + 5N( ) 1.5m( )τ Net = −8Nm

a

b

20 N

5 N

7 N

radius:a = 0.5mb = 1.5m

32. The Net Torque on the following system is most nearly.

43

€

4Kg

€

m

€

τ CCW =τCW

mLg 2R( ) = mR gR

2 4Kg( ) = mR

8Kg = mR

44

33. A tiered-solid disk of mass M is free to rotate on a frictionless axis. Two masses are hung over the disk so that it makes an Atwood's machine. Someone tells you the inner radius is half as large as the outer radius, and that the moment of inertia of the disk is

If the left hand mass is 4 Kg, what value of mass m would keep the system in static equilibrium?

€

35Kg ⋅ m 2 .