# Worksheet: Rotational Kinematics

In this worksheet, we will practice modeling the change of the position with time of objects that move along circular paths.

**Q1: **

A car drives repeatedly around a roundabout. The car has a constant angular velocity of 1.4 rad/s. What is the period of the car’s circular motion?

**Q2: **

A helicopter flies in a circle of radius 375 m, taking a time of 42 s to complete one revolution. What is the magnitude of the acceleration of the helicopter toward the center of its circular path?

**Q3: **

An angular displacement of is equal to an angular displacement of 7.25 rad.

**Q4: **

The wheels of a moving car rotate 13.5 times per second. What is the angular velocity of a point on the wheel that is not at the axis of the rotation of the wheel?

**Q5: **

A drill bit is initially at rest. When the drill is activated, the drill bit rotates 47.5 times per second. The drill bit reaches this speed in a time of 175 ms. What is the angular acceleration of the drill bit?

**Q6: **

An angular displacement of is equal to revolutions.

**Q7: **

The blades of a large wind turbine rotate fully in a time of
3.25 s when it is operating at its full speed. The angular acceleration of the
turbine while increasing to its full speed is 0.124 rad/s^{2}. How much
time is required to bring an initially inactive turbine to its full operating
speed?

**Q8: **

A felled tree trunk rolls down a slope in a time of 7.2 s. The trunk is initially at rest at the top of the slope and has an angular velocity of 12 rad/s at the base of the slope. How many times does the trunk completely rotate as it rolls down the slope?

**Q9: **

A wheel of a moving car comes to rest in a time of 8.4 s
when a brake pad is applied to it. The magnitude of the wheel’s initial angular velocity is 5.5 rad/s,
and its angular deceleration is 2.2 rad/s^{2}.
How many complete rotations did the wheel make before coming to rest?

**Q10: **

An ice-skater spins around her axis of rotation 2.5 times per second. Her arms are outstretched to different lengths while she spins. The skater’s fingertip on one arm is 1.3 m from the skater’s axis of rotation, and her fingertip on the other arm is 0.45 m from her axis of rotation. What is the difference between the angular velocity of the skater’s fingertips on one of her arms compared to her other arm?

**Q11: **

A satellite has a circular orbit around Earth with a period of 1.6 hours.

How many degrees is the angular displacement of the satellite during a single Earth day to the nearest degree?

- A
- B
- C
- D
- E

How many radians is the angular displacement of the satellite during a single Earth day to the nearest radian?

- A47 rad
- B9 rad
- C94 rad
- D2 rad
- E5 rad

**Q12: **

An angular displacement of radians is equal to an angular displacement of .

**Q13: **

An angular displacement of 0.45 rad is equal to revolutions.

**Q14: **

A ball rolls along a spiral track with deep grooves, as shown in an overhead view in the diagram. Normal reaction force on the ball from the grooves of the track causes the ball to accelerate uniformly perpendicularly to its horizontal motion. At point A, the horizontal distance to the center of the track is 18.2 m, and at point B, the horizontal distance is 5.5 m. The length of the track between the points is 350 m. The angular velocity of the ball at point A is 0.28 rad/s. How much does the angular velocity of the ball change as the distance it has rolled along the track changes?

- A rad/m⋅s
- B0.229 rad/m⋅s
- C rad/m⋅s
- D0.509 rad/m⋅s
- E1.43 rad/m⋅s

**Q15: **

A tornado has an outer radius of 1.20 km. The air at the tornado’s outer radius follows a circular path that returns an air particle to a point on the path every 525 seconds. The speed , in kilometers per hour, is the speed of an air particle in the tornado in a direction perpendicular to the direction of a line that intersects the particle and the center of the tornado.

Find at the outer radius of the cyclone.

At a distance of only 275 m from the center of the tornado, is 144 km/h. How much time is required for such an air particle to revolve around the center of the tornado?

**Q16: **

Two identical racing cars have centers of mass that intersect with a line that passes through the center of a 6.0 m wide circular race track that has an inner radius m, as shown in the diagram. Each car drives in a circular path around the track, always remaining 1.0 m from the edge of the track nearest to the car. Each car drives at a constant speed, but the two cars do not drive at the same speed as each other. After a time interval, the car nearer to the outer edge has driven a distance of 150 m and the car nearer to the inner edge has driven a distance of 130 m.

Find the difference between the angular displacements of the position of the center of mass of each car.

When the car nearer to the inner edge of the track has returned to its initial position, how many meters farther has the car nearer to the outer edge of the track driven?