# Worksheet: Rotational Variables

In this worksheet, we will practice calculating the kinematic properties of objects that are rotating or moving along circular paths.

Q1:

A boy rides his bicycle which has wheels with a radius of 30.0 cm. If the boy on the bicycle accelerates from rest to a speed of 10.0 m/s in 10.0 s, what is the angular acceleration of the tires?

Q2:

A cyclist is riding such that the wheels of the bicycle have a rotation rate of 3.0 rev/s. If the cyclist brakes such that the rotation rate of the wheels decrease at a rate of 0.3 rev/s2, how long does it take for the cyclist to come to a complete stop?

Q3:

Calculate the angular velocity of the rotational motion of the Earth around its axis.

Q4:

Our solar system orbits the center of the Milky Way galaxy. Assuming a circular orbit 30,000 ly in radius and an orbital speed of 250 km/s, how many years does it take for one revolution? Give your answer to three significant figures.

• A years
• B years
• C years
• D years
• E years

Q5:

Each piston in a race car’s eight-cylinder engine makes a sharp sound every alternate revolution of the engine. The car is driving at a speed where the engine turns through revolutions per kilometer. The repeated sharp sounds produce a sound of frequency 0.750 kHz.

What speed is the race car driving at?

At how many revolutions per minute is the engine rotating?

• A rpm
• B rpm
• C rpm
• D rpm
• E rpm

Q6:

A phonograph turntable rotating at rpm slows down and stops in 1.00 min.

What is the turntable’s angular acceleration assuming it is constant?

How many complete revolutions does the turntable make while stopping?

Q7:

Starting from rest, a wheel spins with a constant angular acceleration of 5.0 rad/s2. The wheel reaches its final angular velocity at the instant that it has turned through a rad angular displacement.

What is the final angular velocity of the wheel?

How much time is taken for the wheel to reach its final angular velocity?

Q8:

A circular disk of radius 10 cm has a constant angular acceleration of 1.0 rad/s2. At its angular velocity is 2.0 rad/s.

Determine the disk’s angular velocity at .

What is the angle it has rotated through during this time?

What is the tangential acceleration of a point on the disk at ?

Q9:

A wheel 1.0 m in diameter rotates with an angular acceleration of 4.0 rad/s2, starting with an initial angular speed of 2.0 rad/s and accelerating for 10 seconds in the direction that it is was initially turning.

What is the wheel’s angular speed after accelerating?

Through what angle does the wheel rotate during accelerating?

What is the tangential speed of a point on the rim of the wheel after the wheel has accelerated?

What is the tangential acceleration of a point on the rim of the wheel after the wheel has accelerated?

Q10:

A piece of dust falls onto a spinning compact disk and sticks in place. The spin rate of the disk is rpm, and the piece of dust lands 4.30 cm from the disk’s center. What is the total distance traveled by the piece of dust in s of rotation? Assume that the disk is negligibly slowed by the dust falling on it and that the time taken for the dust to accelerate to the same speed as the part of the disk that it lands on is negligible.

Q11:

A track star runs a 400 m race on a circular track of length 400 m in 45 s. What is his angular velocity assuming a constant speed?

Q12:

What is the angular velocity of a 75.0 cm diameter tire on an automobile traveling at 90.0 km/h?

Q13:

The angular velocity of a rotating rigid body increases from 500 rpm to 1,500 rpm in 120 s.

What is the angular acceleration of the body?

Through what angle does it turn in this 120 s?

Q14:

On takeoff, the propellers on a UAV (unmanned aerial vehicle) increase their angular velocity from rest at a rate of for 3.0 s.

What is the instantaneous angular speed of the propellers at ?

What is the magnitude of the angular acceleration of the propellers?

Q15:

A rod 2.35 m long is rotating at 1.4 rev/s about an axis at one of its ends.

What is the centripetal acceleration of a point on the rod that is 0.25 m from the rod’s axis of rotation?

What is the centripetal acceleration of a point on the rod that is 1.2 m from the rod’s axis of rotation?

What is the centripetal acceleration of a point on the rod that is 2.0 m from the rod’s axis of rotation?

Q16:

A wheel has a diameter of 5.0 m and is suspended vertically above the ground. While the wheel is at rest, the point at the bottom of the wheel, nearest to the ground, is marked. The wheel then rotates counterclockwise with a constant angular acceleration of 2.0 rad/s2 around a fixed axis through its center.

After 3.0 s of rotation, what is the magnitude of the vertical displacement of the marked point from the base of the wheel?

After 3.0 s of rotation, what is the magnitude of the linear acceleration of the marked point?

Q17:

A compact disc rotates at 350 rpm. The radius of the disc is 150 mm.

What is the tangential speed of a point at the edge of the disc?

What is the tangential speed at a point halfway to the center of the disc?

Q18:

A flywheel slows from 500 rpm to 300 rpm while rotating through 30 revolutions.

What is the magnitude of the angular acceleration of the flywheel?

How much time elapses during the 30 revolutions?

Q19:

A bicycle wheel with radius 0.50 m rotates from rest to an angular speed of 11 rev/s in a time of 7.0 s. What is the magnitude of the total acceleration vector at the edge of the wheel 3.0 s after the wheel starts to accelerate?

Q20:

With the aid of a string, a gyroscope is accelerated uniformly from rest to an angular speed of 56 rad/s, taking 0.50 s.

What is the gyroscope’s angular acceleration?

How many revolutions does the gyroscope go through during its acceleration?

Q21:

A wheel rotates at a constant rate of rpm.

What is the wheel’s angular velocity?

Through what angle does the wheel turn in 5.0 s?

Q22:

A flywheel has a radius of 20.00 cm. What is the speed of a point on the edge of the flywheel if it experiences a centripetal acceleration of 9.00 m/s2?

Q23:

A proton in a synchrotron is moving in a circle of radius 825 m. The proton is increasing its speed according to , where m/s and m/s3. What is the magnitude of the proton’s total acceleration at s?

• A m/s2
• B m/s2
• C m/s2
• D m/s2
• E m/s2

Q24:

A flywheel is rotating at 18.7 rev/s. What is the total angle through which a point on the flywheel rotates in 106 s?

Q25:

A particle is executing clockwise circular motion with a constant angular frequency . The instant corresponds to the particle having an -direction displacement of 2.17 m and a -direction displacement of 0.00 m.

What is the particle’s -direction displacement at the instant ?

What is the particle’s -direction displacement at the instant ?

What is the magnitude of the particle’s centripetal acceleration at the instant ?