# Worksheet: Rotational Variables

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

**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

**Q7: **

Starting from rest, a wheel spins with a constant angular
acceleration of 5.0 rad/s^{2}.
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?

**Q9: **

A wheel 1.0 m in diameter rotates with an
angular acceleration of 4.0 rad/s^{2},
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.

**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 5.0-m-diameter wheel 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 counter clockwise with a constant
angular acceleration of 2.0 rad/s^{2} 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?

**Q25: **

A particle is executing clockwise circular motion with a constant angular frequency rad/s. 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 ?