# Conservation of Angular Momentum Practice Problems

A researcher studying rotational inertia and angular velocity constructs an object to simulate the human body and the effects of arms and hands. The object has a central trunk and rods representing arms/hands. When the rods are outstretched, they are treated like thin rods rotating about an axis centered on the trunk. When the rods are lowered, they are treated like hollow cylinders. The rods representing arms/hands have a total mass of 10.2 kg, each 0.750 m long. When lowered, they become a cylinder of radius 30.0 cm. The trunk has a constant moment of inertia equal to 8.10 kg•m^{2}. If the object has an initial speed of 1.20 rad/s, determine its speed when the rods are lowered.

A child's toy has a cube that rotates on a level board with negligible friction. The cube has a mass of 0.350 kg and is rotating on one end of a string of negligible mass passing through a hollow at the center of the board. The cube is rotating at 3.20 rad/s and is located at a distance of 0.450 m from the hollow when the child reduces the cube's radius of rotation to 0.220 m by pulling the string from beneath the board. Calculate the work done by the child in shortening the string. Treat the cube like a point mass.

A block of mass 0.0200 kg is rotating on a flat bench with negligible friction. It is held in the circle by a massless string running through a hollow in the bench. At first, the block is rotating at a radius of 0.350 m from the hollow at an angular speed of 1.95 rad/s. The string is shortened by pulling from below the bench, reducing the rotational radius to 0.180 m. Determine the change in kinetic energy of the block. Treat the block like a particle.

During an experiment, the wheel of a laboratory gyroscope precesses in a horizontal plane at 30 revolutions per minute. The wheel (m_{wheel} = 200 g) that is fixed to a frame is free to spin about the shaft (m_{shaft} = 25 g) of the frame. The moment of inertia of the wheel about the shaft is 10^{–}^{4} kg•m^{2} The gyroscope is supported by a pivot near the right-hand end of the shaft located 5 cm away from its center of gravity, as shown in the figure. Determine the force exerted by the pivot on the shaft.

A spacecraft traveling from Earth to Mars uses a gyroscope to maintain its orientation and angular velocity. What will be the precession rate of the gyroscope near Mars's surface If the precession rate near Earth's surface was 1 rad/s? The surface gravity on Mars is 38% of the surface gravity on Earth.

A uniform rod of length 0.80 m is fixed to a massless axle on one of its ends such that its rotational axis is perpendicular to its length. The rod's moment of inertia about the axle is 0.0410 kg•m^{2}. The rod is rotating at 1.40 rads/s when a remote-controlled movable mass slides from very close to the axle to the opposite end of the rod. The mass has a linear speed of 0.980 m/s when located at the opposite end. Determine the mass of the rod. You may treat the mass as a particle.

Neutron stars are believed to form when a massive star depletes its fuel and collapses. The collapse crushes together all protons and electrons into neutrons. The density of a neutron star is about 10^{14} times greater than the density of the sun. Imagine that the sun (with a radius of 6.96 *×* 10^{5} km) collapses into a neutron star of radius 15 km. Using the sun's average rotational speed of one rotation every 27 days, what would be the rotational speed of the neutron star formed? Model the sun and the neutron star formed as uniform solid spheres.

A family dining table has two circular wooden table tops. A small top spins about a vertical axis through its center on top of a large and stationary tabletop below it. Meals are taken from the larger top while Foods and drinks are circulated from the rotating smaller top. Suppose the smaller top has a mass of 11.4 kg and a radius of 2.00 m and is rotating at 2.30 rad/s when a pot of mass of 4.50 kg is placed near its outer edge. Determine the system's kinetic energy before and after the pot is placed down. Treat the small top like a disk and the pot as a point mass.

A wooden dining table has two concentric circular surfaces. People feed on the outer stationary surface. Foods and drinks are placed on the inner surface (radius 1.20 m and mass 6.75 kg) and then circulated to members on the dining table. The inner surface is rotating at 2.50 rad/s about its center on a vertical axis when a dish of mass 5.00 kg is placed on it gently at a point very close to its outer edge. Calculate the angular speed of the inner surface once the dish has been placed on it. You may treat the dish as a point mass.

A double-sided tape covers the circular surface of a plate rotating about its central axis. The plate has a mass of 750 g and a radius of 7.5 cm. An experimental setup is designed in such a way that two spheres of mass 125 g each are dropped simultaneously from a height of h and stuck to the plate. The location of impact is precisely controlled at the opposite ends of a diameter. The plate spins initially at 185 rpm. Find the plate's rotational speed after the impact.