Question 1

You are whirling a stone on the end of a string in a horizontal circle of radius R = 0.65 m with a frequency of 4 rev/s when the string breaks. Just after the string breaks, the velocity of the stone is
A) straight down.
B) 32 m/s along a tangent to the circle.
C) 16 m/s along the radius away from the center.
D) 1.0 m/s along the radius toward the center.
E) none of these.

Accepted Answer

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Question 2

A wheel rotates with a constant nonzero angular acceleration. Which of the following quantities remains constant in magnitude? A) v ⃗, tangential velocity B) a ⃗_r, radial acceleration C) a_t ⃗, tangential acceleration D) ω ⃗, angular velocity E) All of these are correct.

Accepted Answer

The tangential acceleration remains constant in magnitude because it is directly related to the constant angular acceleration.

Question 3

A wheel slows from 20 rad/s to 12 rad/s in 5 s under the influence of a constant frictional torque. In these 5 s, the wheel turns through an angle of
A) 2.4 rad.
B) 43 rad.
C) 60 rad.
D) 80 rad.
E) 100 rad.

Accepted Answer

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Question 4

The Crab Pulsar rotates 30 times per second. However, its period is increasing by about s per year. How many years will it take before its period of rotation has doubled? A) 3.0  10⁶ yr B) 6.7  10³ yr C) 3.3  10³ yr D) 1.05  10¹⁰ yr E) 2.10  10¹⁰ yr

Accepted Answer

Let T be the initial period of rotation, then after one year, the period will be T + T/τ, where τ is the characteristic age of the pulsar.

We can use the formula for the characteristic age:

τ = P/2Ṗ = T/2(T/τ) = τ(T/2)²

So, we have (T + T/τ) = 2T after some number of years:

T + T/τ = 2T
T/τ = 1
τ = T

Substituting T = 1/30 s and using the formula for τ, we get:

τ = 3.3 x 10¹³ s ≈ 10¹³ s

So, the time it takes for the period to double is:

2T = T + T/τ + nT/τ
T = nT/τ
n = τ = 10¹³ s

Therefore, the answer is 3.3 x 10¹³ s / (365.25 days/year) ≈ 3.3 x 10¹³ s / (3.156 x 10⁷ s/year) ≈ 1.05 x 10¹¹ years, which is closest to choice (C).

Question 5

Torque is defined as
A) a force tending to cause rotation.
B) the cross product of force and displacement.
C) the product of the force and the angular displacement.
D) the product of the force and the angular velocity.
E) the rotational work done.

Accepted Answer

The answer of Torque is defined as
A) a force tending...

Question 6

The torque exerted on a perfectly spherical satellite by the gravitational pull of the sun is
A) zero.
B) directed along Earth's axis to the north pole.
C) directed along Earth's axis to the south pole.
D) in the direction of Earth's orbit.
E) directed toward the sun.

Accepted Answer

A perfectly spherical satellite experiences no torque because the gravitational force of the sun acts through the center of mass of the satellite, resulting in no net torque.

Question 7

The data used to construct the graph were taken from the tachometer of an airplane a tachometer measures the angular speed of an engine). The angular acceleration during the 10 s interval was A) "3.0 rad/s²." B) "6.0 rad/s²." C) "8.0 rad/s²." D) "20 rad/s²." E) "38 rad/s²."

Accepted Answer

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Question 8

A turntable rotating at 8.0 rad/s slows to a stop in 10 s. If the acceleration is constant, the angle through which the turntable rotates in the 10 s is
A) 0.80 rad.
B) 0.40 rad.
C) 40 rad.
D) 80 rad.
E) 16 rad.

Accepted Answer

We can use the formula of angular acceleration to find the initial angular velocity of the turntable:

ω = ω0 + αt

where ω0 is the initial angular velocity (which is 8.0 rad/s), α is the angular acceleration (which is constant), and t is the time (which is 10 s).

Rearranging the formula, we get:

α = (ω - ω0)/t

α = (0 - 8.0)/10

α = -0.8 rad/s^2

Now we can use the formula of angular displacement to find the angle through which the turntable rotates in the 10 s:

θ = ω0t + (1/2)αt^2

θ = 8.0 x 10 + (1/2)(-0.8)(10)^2

θ = 40 - 40

θ = 0.40 rad

Therefore, the answer is B) 0.40 rad.

Question 9

Two blocks m₁ = 0.4 kg) and m₂ = 0.6 kg) are initially positioned at x = 0.6 and x = 1.1m. One cord is used to couple the blocks together and another to attach m₁ to a vertical pole at the origin. The blocks are made to rotate in a horizontal circle on a frictionless surface. If the period of rotation is 0.4 s, then calculate the ratio of the tension for the inner cord divided by the tension in the outer cord. A) 0.64 B) 1.4 C) 1.6 D) 0.74 E) 1.8

Accepted Answer

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Question 10

What physical quantity is represented by the slope of the curve shown on the graph?
A) displacement
B) angular acceleration
C) tangential acceleration
D) velocity
E) None of these is correct.

Accepted Answer

The answer of  What physical quantity is represented by...

Question 11

A wheel starting from rest has a constant angular acceleration. After 3.0 s the angular velocity of the wheel is 7.5 rad/s. This same angular acceleration continues for a further 7.0 s after which it drops suddenly to zero. In the first 20.0 s how many revolutions does the wheel make?
A) 20
B) 40
C) 60
D) 80
E) 100

Accepted Answer

First we can find the angular acceleration using the formula:
angular acceleration (alpha) = change in angular velocity / time
alpha = (7.5 rad/s - 0 rad/s) / 3.0 s
alpha = 2.5 rad/s^2

We can now find the angular displacement of the wheel during the first 7.0 s of constant angular acceleration:
θ = (1/2) * alpha * t^2
θ = (1/2) * 2.5 rad/s^2 * (7.0 s)^2
θ = 61.25 rad

During the next 7.0 s with constant angular acceleration, the angular displacement is:
θ = alpha * t^2
θ = 2.5 rad/s^2 * (7.0 s)^2
θ = 122.5 rad

Finally, during the period with zero angular acceleration (the last 6.0 s), the angular displacement is:
θ = omega * t
θ = 7.5 rad/s * 6.0 s
θ = 45 rad

The total angular displacement of the wheel in the first 20.0 s is:
total θ = 61.25 rad + 122.5 rad + 45 rad
total θ = 228.75 rad

To convert from radians to revolutions, we divide by 2π:
total revolutions = total θ / (2π)
total revolutions = 228.75 rad / (2π)
total revolutions ≈ 36.45

Therefore, the closest answer choice is C) 60 revolutions.

Question 12

Water is drawn from a well in a bucket tied to the end of a rope whose other end wraps around a cylinder of mass 50 kg and diameter 25 cm. As you turn this cylinder with a crank, the rope raises the bucket. If the mass of a bucket of water is 20 kg, what torque must you apply to the crank to raise the bucket of water at a constant speed?
A) 24 N · m
B) 2.5 N · m
C) 80 N · m
D) 2.4  103 N · m
E) 49 N · m

Accepted Answer

To raise the bucket at a constant speed, the torque applied must balance the gravitational force on the bucket. The force due to gravity is 20 kg * 9.8 m/s² = 196 N. The radius of the cylinder is 0.125 m (half of the diameter). Torque = Force * Radius = 196 N * 0.125 m = 24.5 N·m, which is closest to 24 N·m.

Question 13

When an object is moving in a circle at constant speed, its acceleration is
A) constantly increasing.
B) constant in direction.
C) zero.
D) constant in magnitude.
E) constant in both magnitude and direction.

Accepted Answer

The acceleration of an object moving in a circle at constant speed is centripetal acceleration, which is constant in magnitude but always directed towards the center of the circle, thus changing direction as the object moves.

Question 14

A uniform stick 1 m long is placed horizontally on the ground along an east-west axis. A force of 1.0 N is applied to the center of the stick in a direction 30^o west of north. The torque exerted by the force relative to the east end of the stick is A) zero. B) 0.25 m, clockwise. C) 0.43 m, clockwise. D) 0.25 m, counterclockwise. E) 0.43 m, counterclockwise.

Accepted Answer

To find the torque, we need to find the force's moment arm, which is the perpendicular distance from the line of action of the force to the pivot point (in this case, the east end of the stick). We can draw a diagram to help us visualize the problem:

30°
         |-------|
         |       |
      N  |   *   |  S
         |       |
         |-------|

The force is applied at an angle of 30° west of north, which means it has both northward and westward components. We can use trigonometry to find the magnitudes of these components:

cos 30° = adjacent/hypotenuse
adjacent = hypotenuse * cos 30°
adjacent = 1.0 N * cos 30°
adjacent = 0.87 N

sin 30° = opposite/hypotenuse
opposite = hypotenuse * sin 30°
opposite = 1.0 N * sin 30°
opposite = 0.50 N

The westward component is the one that creates the torque, so we need to find the moment arm corresponding to that component. Since the force is applied at the center of the stick, the moment arm is half the length of the stick:

moment arm = 1 m / 2
moment arm = 0.50 m

The torque is then given by:

torque = force * moment arm * sin θ (where θ is the angle between the force and moment arm)

torque = 0.87 N * 0.50 m * sin 30°
torque = 0.43 Nm (clockwise)

So the answer is (C) 0.43 m, clockwise.

Question 15

The Empire's space station is a long way from any star. It is circular and has a radius of 5.10 km. The angular velocity that is needed to give the station an artificial gravity of 9.80 m/s² at its circumference is A) 4.4  10^-² rad/s. B) 7.0  10^-³ rad/s. C) 0.28 rad/s. D) - 0.22 rad/s. E) 1.3  103 rad/s.

Accepted Answer

The answer of The Empire's space station is a long...

Question 16

A point P is at a distance R from the axis of rotation of a rigid body. The linear speed, centripetal acceleration, and tangential acceleration of the point can be expressed 
A)A
B)B
C)C
D)D
E)E

Accepted Answer

The answer of A point P is at a distance...

Question 17

You are pedaling a bicycle at 9.8 m/s. The radius of the wheels of the bicycle is 51.9 cm. The angular velocity of rotation of the wheels is
A) 19 rad/s.
B) 2.5 rad/s.
C) 4.5 rad/s.
D) 3.0 rad/s.
E) 6.3 rad/s.

Accepted Answer

The answer of You are pedaling a bicycle at 9.8...

Question 18

The information on a compact disk is scanned by a laser initially at a radius of 2.4 cm and then out to a maximum of 6.0 cm. Because the dimensions of the pit information remain constant with radius, the disk motor adjusts so that the tangential velocity remains constant. What is the ratio of the inner to outer rotational frequencies?
A) 0.16
B) 6.3
C) 1.0
D) 2.5
E) 0.40

Accepted Answer

Since the tangential velocity must be constant, the angular velocity must increase as the radius increases. From the formula for tangential velocity, $v=\omega r$, we have $v=	ext{constant} \Rightarrow \omega \propto r^{-1}$. Therefore, the ratio of the inner to outer rotational frequencies is $\frac{\omega_{	ext{inner}}}{\omega_{	ext{outer}}}=\frac{r_{	ext{outer}}}{r_{	ext{inner}}}=\frac{6.0	ext{ cm}}{2.4	ext{ cm}}=\boxed{2.5	ext{ (D)}}$.

Question 19

A serious audiophile prefers to listen to music on his vinyl record turntable. The 12-inch diameter disk rotates at a constant 33 rpm. How fast is the record passing by the pickup cartridge tip at a radius of 4 inches 1 inch = 2.54 cm)? A) 5.6 cm/s B) 35 cm/s C) 3.3  10² cm/s D) 21 m/s E) 18 cm/s

Accepted Answer

First, we need to convert the radius from inches to cm:

4 inches x 2.54 cm/inch = 10.16 cm

Next, we can use the formula for linear velocity:

v = ωr

where v is the linear velocity, ω is the angular velocity (in radians/second), and r is the radius. We know that the record rotates at 33 rpm, which is equivalent to 3.47 radians/second (since 1 revolution = 2π radians and there are 60 seconds in a minute):

ω = 33 rpm x 2π/60 = 3.47 rad/s

Plugging in the values, we get:

v = (3.47 rad/s) x (10.16 cm) = 35.3 cm/s

Rounding to the nearest tenth, the answer is 35 cm/s (choice B).

Question 20

A 7.00-kg mass and a 4.00-kg mass are mounted on a spindle that is free to turn about the x axis as shown. Assume the mass of the arms and the spindle to be negligible. The magnitude of the resultant torque is approximately
A) 82.2 N · m.
B) 157 N · m.
C) 225 N · m.
D) 392 N · m.
E) 461 N · m.

Accepted Answer

The answer of  A 7.00-kg mass and a 4.00-kg...

Quiz 8: Rotational Motion

You are whirling a stone on the end of a string in a horizontal circle of radius R = 0.65 m with a frequency of 4 rev/s when the string breaks. Just after the string breaks, the velocity of the stone is

A wheel rotates with a constant nonzero angular acceleration. Which of the following quantities remains constant in magnitude?

A wheel slows from 20 rad/s to 12 rad/s in 5 s under the influence of a constant frictional torque. In these 5 s, the wheel turns through an angle of

The Crab Pulsar rotates 30 times per second. However, its period is increasing by about s per year. How many years will it take before its period of rotation has doubled?

Torque is defined as

The torque exerted on a perfectly spherical satellite by the gravitational pull of the sun is

The data used to construct the graph were taken from the tachometer of an airplane a tachometer measures the angular speed of an engine) . The angular acceleration during the 10 s interval was

A turntable rotating at 8.0 rad/s slows to a stop in 10 s. If the acceleration is constant, the angle through which the turntable rotates in the 10 s is

What physical quantity is represented by the slope of the curve shown on the graph?

A wheel starting from rest has a constant angular acceleration. After 3.0 s the angular velocity of the wheel is 7.5 rad/s. This same angular acceleration continues for a further 7.0 s after which it drops suddenly to zero. In the first 20.0 s how many revolutions does the wheel make?

When an object is moving in a circle at constant speed, its acceleration is

A uniform stick 1 m long is placed horizontally on the ground along an east-west axis. A force of 1.0 N is applied to the center of the stick in a direction 30^o west of north. The torque exerted by the force relative to the east end of the stick is

The Empire's space station is a long way from any star. It is circular and has a radius of 5.10 km. The angular velocity that is needed to give the station an artificial gravity of 9.80 m/s² at its circumference is

A point P is at a distance R from the axis of rotation of a rigid body. The linear speed, centripetal acceleration, and tangential acceleration of the point can be expressed

You are pedaling a bicycle at 9.8 m/s. The radius of the wheels of the bicycle is 51.9 cm. The angular velocity of rotation of the wheels is

A serious audiophile prefers to listen to music on his vinyl record turntable. The 12-inch diameter disk rotates at a constant 33 rpm. How fast is the record passing by the pickup cartridge tip at a radius of 4 inches 1 inch = 2.54 cm) ?

A 7.00-kg mass and a 4.00-kg mass are mounted on a spindle that is free to turn about the x axis as shown. Assume the mass of the arms and the spindle to be negligible. The magnitude of the resultant torque is approximately