In this worksheet, we will practice calculating the angular momentum of an object as the product of the object's moment of inertia and angular velocity.

**Q2: **

A particle of mass 5.0 kg has position vector at a particular instant of time, . At the instant , the velocity of the particle with respect to the origin is .

What is the angular momentum of the particle at the instant ?

- A
- B
- C
- D
- E

If a force acts on the particle at the instant , what is the torque about the origin at that instant?

- A
- B
- C
- D
- E

**Q3: **

An aeroplane of mass kg flies horizontally at an altitude of 10 km with a constant speed of 250 m/s relative to Earth. What is the magnitude of the aeroplane’s angular momentum relative to a ground observer directly below the plane?

- A
kg⋅m
^{2}/s - B
kg⋅m
^{2}/s - C
kg⋅m
^{2}/s - D
kg⋅m
^{2}/s - E
kg⋅m
^{2}/s

**Q4: **

A bird of mass 2.0 kg is in flight parallel to the ground. The bird is 300.0 m vertically above an observer and moving at 20.0 m/s. What is the bird’s angular momentum relative to the observer?

- A
kg⋅m
^{2}/s - B
kg⋅m
^{2}/s - C
kg⋅m
^{2}/s - D
kg⋅m
^{2}/s - E
kg⋅m
^{2}/s

**Q5: **

The blades of a wind turbine are cm in length and rotate at a maximum rotation rate of 0.333 rev/s. Each blade has a mass of kg, and a rotor assembly consists of three blades.

What is the magnitude of the angular momentum of the rotor assembly at its maximum rotation rate?

- A
kg⋅m
^{2}/s - B
kg⋅m
^{2}/s - C
kg⋅m
^{2}/s - D
kg⋅m
^{2}/s - E
kg⋅m
^{2}/s

What magnitude of torque is required to accelerate the rotor assembly from rest to its maximum rotation rate in a time of s?

- A
kg⋅m
^{2}/s - B
kg⋅m
^{2}/s - C
kg⋅m
^{2}/s - D
kg⋅m
^{2}/s - E
kg⋅m
^{2}/s

**Q6: **

A pulsar is a rapidly rotating neutron star. A nebula pulsar has a period of s, a radius of 9.250 km, and a mass of kg. The pulsar’s rotational period will increase over time due to the release of electromagnetic radiation, which reduces the pulsar’s rotational energy but does not change its radius.

What is the magnitude of the angular momentum of the pulsar?

- A
kg⋅m/s
^{2} - B
kg⋅m/s
^{2} - C
kg⋅m/s
^{2} - D
kg⋅m/s
^{2} - E
kg⋅m/s
^{2}

Suppose the pulsar’s angular velocity decreases at a rate of . What is the magnitude of the torque on the pulsar?

- A N⋅m
- B N⋅m
- C N⋅m
- D N⋅m
- E N⋅m

**Q7: **

A DVD that has a radius of 7.500 cm and a mass of 25.0 g is rotating at rpm. What is the magnitude of the angular momentum of the DVD?

- A
kg⋅m
^{2}/s - B
kg⋅m
^{2}/s - C
kg⋅m
^{2}/s - D
kg⋅m
^{2}/s - E
kg⋅m
^{2}/s

**Q8: **

A car in a roller coaster has mass kg. The car needs to safely travel along the track of a vertical circular loop of radius 60.00 m. What is the minimum angular momentum of the car at the bottom of the loop for it to safely travel along the track of the loop? Neglect friction on the track. Model the roller coaster as a point particle.

- A
kg⋅m
^{2}/s - B
kg⋅m
^{2}/s - C
kg⋅m
^{2}/s - D
kg⋅m
^{2}/s - E
kg⋅m
^{2}/s

**Q9: **

A small ball of mass 0.50 kg is attached by a string of negligible mass to a vertical rod that is spinning as shown. When the rod has an angular velocity of 6.0 rad/s, the angle between the string and the axis of the rod’s rotation is .

If the angular velocity of the rod is increased to 10.0 rad/s, what is the angle between the string and the rod’s axis of rotation.

What is the angular momentum of the ball when the angular velocity of the rod is 6.0 rad/s?

What is the angular momentum of the ball when the angular velocity of the rod is 10.0 rad/s?

**Q10: **

A particle has mass 0.82 kg and is traveling along the line at a speed of 3.5 m/s in the positive -direction. What is the particle’s angular momentum about the origin?

**Q11: **

A thin meter stick has a mass of 135 g. The stick rotates around an axis that is perpendicular to its length at an angular speed of 182 rpm.

What is the angular momentum of the stick if the rotation axis passes through the center of the stick?

What is the angular momentum of the stick if the rotation axis passes through one end of the stick?

**Q12: **

A satellite is spinning at 9.0 rev/s. The satellite consists of a main body in the shape of a sphere and two antennas projecting out from the center of mass of the main body in the plane of rotation. The sphere has a radius of 2.50 m and a mass of kg. Each antenna can be modeled as a 3.30 m long rod with a mass of 12 kg. What is the angular momentum of the satellite?

- A
kg⋅m
^{2}/s - B
kg⋅m
^{2}/s - C
kg⋅m
^{2}/s - D
kg⋅m
^{2}/s - E
kg⋅m
^{2}/s

**Q13: **

A proton is in a cyclotron that has a radius of 0.200 km. The proton is accelerated along a circular path to a speed of m/s in a time interval of 0.0300 s.

What is the magnitude of the angular momentum of the proton about the center at its maximum speed?

- A
kg⋅m
^{2}/s - B
kg⋅m
^{2}/s - C
kg⋅m
^{2}/s - D
kg⋅m
^{2}/s - E
kg⋅m
^{2}/s

What is the magnitude of the instantaneous torque on the proton about the center as it reaches its maximum speed?

- A N⋅m
- B N⋅m
- C N⋅m
- D N⋅m
- E N⋅m

**Q14: **

Due to a design flaw, a wheeled robot probe designed for operation on Mars loses
one of its wheels as it begins to move from rest. The wheel comes loose from the probe and
rolls without slipping down an incline that levels out to horizontal ground 25 m vertically below
where the wheel came loose. The wheel’s mass is 5.0 kg and its radius is
25 cm. Find the wheel’s speed
at the bottom of the incline, using a value of
3.71 m/s^{2} for the acceleration due to gravity on Mars’s surface.

**Q15: **

At an instant , a meteor in Earth’s atmosphere is seen by a ground-based observer, shortly before atmospheric friction causes the meteor to evaporate. Relative to the observer, the position vector for the asteroid . At the instant , the meteor is modeled as having a linear momentum and a linear acceleration .

What is the magnitude of the angular momentum of the meteor about the observer?

- A
kg⋅m
^{2}/s - B
kg⋅m
^{2}/s - C
kg⋅m
^{2}/s - D
kg⋅m
^{2}/s - E
kg⋅m
^{2}/s

What is the magnitude of the torque on the meteor about the observer?

- A N⋅m
- B N⋅m
- C N⋅m
- D N⋅m
- E N⋅m

**Q16: **

Three particles move independently of each other, and each particle is subject to a force perpendicular to the direction of its instantaneous velocity.

What is the total angular momentum about the origin of the particles?

- A
kg⋅m
^{2}/s - B
kg⋅m
^{2}/s - C
kg⋅m
^{2}/s - D
kg⋅m
^{2}/s - E
kg⋅m
^{2}/s

What is the rate of change of the total angular momentum about the origin of the particles?

- A N⋅m
- B N⋅m
- C N⋅m
- D N⋅m
- E N⋅m

**Q17: **

A robot probe used on Mars has a 1.0 m long rigid arm that ends in forceps designed for grasping rocks. The arm rotates about the opposite end to its end that has forceps attached. The arm has a mass of 2.0 kg and the forceps have a mass of 1.0 kg. The arm is initially raised to point upward. From the raised position, the arm accelerates toward the ground from rest to an angular speed of 0.10 rad/s in a time of 0.10 s. The arm then stops moving and the forceps pick up a rock of mass 1.5 kg. Finally, the arm is raised again, accelerating at the same rate as when lowering the arm. In modeling the motion of the arm, use a value of 3.71 m/s^{2} for the acceleration due to gravity at the Martian surface.

What is the magnitude of the angular momentum of the robot arm about its axis of rotation when it is at its maximum angular speed when the arm is lowering?

What is the magnitude of the angular momentum of the robot arm about its axis of rotation when it is at its maximum angular speed when the arm is rising?

What would be the magnitude of the torque applied to raise the arm if no rock was picked up by the forceps? Assume that the acceleration of the arm remains the same as if a rock had been picked up.

**Q18: **

A rotating solid cylinder and a rotating solid sphere have equal masses to each other and equal angular speeds to each other. What is the ratio of the angular momentum of the sphere to that of the cylinder? Use two significant figures in your answer.