11.2 Angular Momentum

Which has greater angular momentum: a solid sphere of mass m rotating at a constant angular frequency $\omega_{0}$ about the z-axis, or a solid cylinder of same mass and rotation rate about the z-axis?

For a particle traveling in a straight line, are there any points about which the angular momentum is zero? Assume the line intersects the origin.

If a particle is moving with respect to a chosen origin it has linear momentum. What conditions must exist for this particle’s angular momentum to be zero about the chosen origin?

A bird flies overhead from where you stand at an altitude of 300.0 m and at a speed horizontal to the ground of 20.0 m/s. The bird has a mass of 2.0 kg. The radius vector to the bird makes an angle $\theta$ with respect to the ground. The radius vector to the bird and its momentum vector lie in the xy-plane. What is the bird’s angular momentum about the point where you are standing?

A particle of mass 5.0 kg has position vector $\overset{\rightarrow}{r} = \left(2.0 \hat{i} - 3.0 \hat{j}\right) \text{m}$ at a particular instant of time when its velocity is $\overset{\rightarrow}{v} = \left(3.0 \hat{i}\right) \text{m} / \text{s}$ with respect to the origin. (a) What is the angular momentum of the particle? (b) If a force $\overset{\rightarrow}{F} = 5.0 \hat{j} \text{N}$ acts on the particle at this instant, what is the torque about the origin?

Suppose the particles in the preceding problem have masses $m_{1} = 0.10 \text{kg}, m_{2} = 0.20 \text{kg}, m_{3} = 0.30 \text{kg},$ $m_{4} = 0.40 \text{kg}$. The velocities of the particles are $v_{1} = 2.0 \hat{i} \text{m} / \text{s}$, $v_{2} = \left(3.0 \hat{i} - 3.0 \hat{j}\right) \text{m} / \text{s}$, $v_{3} = −1.5 \hat{j} \text{m} / \text{s}$, $v_{4} = −4.0 \hat{i} \text{m} / \text{s}$. (a) Calculate the angular momentum of each particle about the origin. (b) What is the total angular momentum of the four-particle system about the origin?

An airplane of mass $4.0 \times 10^{4} \text{kg}$ flies horizontally at an altitude of 10 km with a constant speed of 250 m/s relative to Earth. (a) What is the magnitude of the airplane’s angular momentum relative to a ground observer directly below the plane? (b) Does the angular momentum change as the airplane flies along a constant altitude?

A boulder of mass 20 kg and radius 20 cm rolls down a hill 15 m high from rest. What is its angular momentum when it is half way down the hill? (b) At the bottom?

A propeller consists of two blades each 3.0 m in length and mass 120 kg each. The propeller can be approximated by a single rod rotating about its center of mass. The propeller starts from rest and rotates up to 1200 rpm in 30 seconds at a constant rate. (a) What is the angular momentum of the propeller at $t = 10 \text{s}; t = 20 \text{s}?$ (b) What is the torque on the propeller?

The blades of a wind turbine are 30 m in length and rotate at a maximum rotation rate of 20 rev/min. (a) If the blades are 6000 kg each and the rotor assembly has three blades, calculate the angular momentum of the turbine at this rotation rate. (b) What is the torque require to rotate the blades up to the maximum rotation rate in 5 minutes?

A mountain biker takes a jump in a race and goes airborne. The mountain bike is travelling at 10.0 m/s before it goes airborne. If the mass of the front wheel on the bike is 750 g and has radius 35 cm, what is the angular momentum of the spinning wheel in the air the moment the bike leaves the ground?