Physics

You stand on a stool that is free to rotate about an axis perpendicular to itself and through its center. The stool’s moment of inertia around its central axis is 1.50 kg m² . Suppose you can model your body as a vertical solid cylinder (height = 1.80 m, radius = 20 cm, mass = 80 kg) with two horizontal thin rods as your arms (each:length = 80 cm, mass = 3 kg) that rotate at their ends, about the same axis, as shown. Suppose that your arms’ contribution to the total moment of inertia is negligible if you have them pressed against your body, but significant if you have them wide open. If you initially spin at 5 rad/s with your arms against your body, how fast will you spin once you stretch them wide open? (Note:The system has 4 objects (stool + body + 2 arms), but initially only stool + body contribute to its moment of inertia)

Suppose a diver spins at 8 rad/s while falling with a moment of inertia about an axis through himself of 3 kg•m². What moment of inertia would the diver need to have to spin at 4 rad/s? 

BONUS:How could he accomplish this?

Two astronauts, both 80 kg, are connected in space by a light cable. When they are 10 m apart, they spin about their center of mass with 6 rad/s. Calculate the new angular speed they’ll have if they pull on the rope to reduce their distance to 5 m. You may treat them as point masses, and assume they continue to spin around their center of mass.

A solid disc of mass M = 40 kg and radius R = 2 m is free to rotate about a fixed, frictionless, perpendicular axis through its center. You apply a constant, tangential force on the disc’s surface (as shown), to get it to spin. Calculate the magnitude of the force needed to get the disc to 100 rad/s in just one minute.

A 200 kg disc 2 m in radius spins around a perpendicular axis through its center, with a person on it, at 3 rad/s counter-clockwise. The person has mass 70 kg, is at rest (relative to the disc, that is, spins with it) at the disc’s edge, and can be treated as a point mass. If the person jumps tangentially out of the disc with 10 m/s (relative to the floor), as shown by the red arrow, what new angular speed will the disc have as a result? If the person steps out into ice with negligible speed of his/her own, what speed would it have upon exiting?

A small object (red, m) is on a smooth table top and attached to a light string that runs through a hole in the table. The other end of the spring attaches to a hanging weight (green, M). When the small object is given some speed, it spins in a circular path around the hole, with the tension from the hanging weight providing the centripetal force that keeps it spinning. If the object spins with angular speed ω when it is a distance R from the central role, what new angular speed (in terms of ω) does it have when this distance is halved? What new mass does the hanging weight need, in terms of M, to support a circular path at the new speed?

Two rotating doors, each 4 kg in mass and 6 m long, are fixed to the same central axis of rotation, as shown above (top view). Suppose a 4 kg bird flying with 30 m/s horizontal collides against the door and stays stuck to it, at a point 50 cm from one end. Calculate the angular speed with which the system (doors + bird) spin together.

When solid sphere 4 m in diameter spins around its central axis at 120 RPM, it has 1,000 kg m² / s in angular momentum. Calculate the sphere’s mass.

A composite disc is built from a solid disc and a concentric, thick-walled hoop, as shown below. The inner disc has mass 4 kg and radius 2 m. The outer disc (thick-walled) has mass 5 kg, inner radius 2 m, and outer radius 3 m. The two discs spin together and complete one revolution every 3 s. Calculate the system’s angular momentum about its central axis.

The Earth has mass 5.97 × 10²⁴ kg, radius 6.37 × 10⁶ m. The Earth-Sun distance is 1.5 × 10¹¹ m. Calculate its angular momentum as it spins around itself. Treat the Earth as a solid sphere of uniform mass distribution. 

BONUS 1:Treating the Earth as a point mass, calculate its angular momentum as it spins around the Sun. 

BONUS 2:Does the Earth have linear momentum as it spins around (i) itself; (ii) the Sun?

A system is made of two small, 3 kg masses attached to the ends of a 5 kg, 4 m long, thin rod. The system rotates with 180 RPM about an axis perpendicular to the rod and through one of its ends, as shown. Calculate the system’s angular momentum about its axis.