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In this lesson, we will learn how to calculate the variation of the moment of inertia of a rotating body with its angular velocity.

Q1:

Three children are riding on the edge of a merry-go-round that has a mass 100.0 kg, a 1.60-m radius, and is spinning at 20.0 rpm. The children have masses of 22.0, 28.0, and 33.0 kg. If the child who has a mass of 28.0 kg moves to the centre of the merry-go-round, what is the new angular velocity in rpm?

Q2:

Eight children, each of mass 40 kg, climb on a small merry-go-round. They position themselves evenly on the outer edge and join hands. The merry-go-round has a radius of 4.0 m and a moment of inertia 1 0 0 0 . 0 kg⋅m^{2}. After the merry-go-round is given an angular velocity of 6.0 rpm, the children walk inward and stop when they are 0.75 m from the axis of rotation. What is the new angular velocity of the merry-go-round? Assume there is negligible frictional torque on the structure.

Q3:

A merry-go-round has a radius of 2.0 m and a moment of inertia 300 kg⋅m^{2}. A boy of mass 50 kg runs tangent to the rim at a speed of 4.0 m/s and jumps on. If the merry-go-round is initially at rest, what is the angular velocity after the boy jumps on?

Q4:

A star collapses which greatly increases its density. The 2 . 0 × 1 0 3 0 -kg-mass of the star remains the same throughout the collapse, as does the star’s mass distribution, which is uniformly spherical. Before collapse, the star’s radius was 7 . 0 × 1 0 5 km and it had a rotational period of 28 days. The star’s radius is 3 . 5 × 1 0 3 km after the collapse, what is its rotational period? Consider a day to be equal to 8 . 6 4 0 0 × 1 0 4 seconds.

Q5:

A diver off the high board imparts an initial rotation to his fully extended body, goes into a tuck, executes three back somersaults, and then hits the water. His moment of inertia before the tuck is 16.9 kg⋅m^{2} and after the tuck during the somersaults is 4.2 kg⋅m^{2}. What initial rotation rate must he impart to his body directly off the board and before going into the tuck if he takes 1.4 s to execute the somersaults before hitting the water?

Q6:

A gymnast does cartwheels along the floor and then launches herself into the air, executing several flips in a tuck while she is airborne. Her moment of inertia when executing the cartwheels is 13.5 kg⋅m^{2} and her spin rate is 0.50 rev/s. If her moment of inertia in the airborne tuck is 3.4 kg⋅m^{2} and she completes the flips in 2.0 s, how many revolutions does she undergo in the air?

Q7:

A ride at a carnival has four 15-m-long, 200-kg-mass spokes attached to a central axis of rotation. At the end of each spoke is a pod of mass 100 kg that can hold two people. If the ride spins at 0.2 rev/s with each pod holding two 50-kg-mass children, what is the ride’s spin rate with empty pods?

Q8:

A satellite in the shape of a sphere of mass 2 0 0 0 0 kg and radius 5.0 m is spinning about an axis through its center of mass. It has a rotation rate of 8.0 rev/s. Two antennas deploy in the plane of rotation extending from the center of mass of the satellite. Each antenna can be approximated as a rod has mass 200.0 kg and length 7.0 m. What is the new rotation rate of the satellite?

Q9:

A space station consists of a giant, rotating, hollow cylinder of mass 1 . 0 0 × 1 0 6 kg, including the people on the station, and a radius of 100.00 m. It is rotating in space at 3.30 rpm in order to produce artificial gravity. If 100 people of an average mass of 65.00 kg spacewalk to an awaiting spaceship, what is the new rotation rate when all the people are off the station?

Q10:

A bug of mass 0.020 kg is at rest on the edge of a solid cylindrical disk ( 𝑀 = 0 . 1 0 , 𝑅 = 0 . 1 0 ) k g m rotating in a horizontal plane around the vertical axis through its center. The disk is rotating at 10 rad/s. The bug crawls to the center of the disk.

What is the new angular velocity of the disk?

What is the change in the kinetic energy of the system?

If the bug crawls back to the outer edge of the disk, what is the angular velocity of the disk then?

What is the new kinetic energy of the system?

Q11:

A particle of mass 4.0 kg moves in a circle of radius 2.0 m. The angular momentum of the particle varies in time according to 𝑙 = 5 . 0 𝑡 2 .

What is the magnitude of the torque on the particle about the centre of the circle at 𝑡 = 3 . 4 s ?

What is the angular speed of the particle at 𝑡 = 3 . 4 s ?

Q12:

The core of a star collapses during a supernova, forming a neutron star. Angular momentum of the core is conserved, so the neutron star spins rapidly. If the initial core radius is 5 . 0 0 × 1 0 5 km and it collapses to 10.0 km, find the neutron star’s angular velocity in revolutions per second, given the core’s angular velocity was originally 10 revolutions per 30.0 days.

Q13:

A 300 cm long uniform-density rod has a mass of 530 g. The rod rotates freely horizontally around a fixed vertical axis that passes through its center perpendicularly to its length. A groove of negligible thickness runs along the rod’s length and two small beads, each of mass 15 g, sit in the groove. Each bead is 12.0 cm from the axis of rotation and they are on opposite sides of the axis. Initially, the beads are held by catches. With the beads held in place, the rod rotates with an angular speed of . When the catches are released, the beads slide outward along the rod.

What is the rod’s angular speed when the beads reach the ends of the rod?

The beads fly from the rod’s ends when they reach them. What is the rod’s angular speed after the beads have lost contact with it?

Q14:

A merry-go-round has a mass of 110 kg and a radius of 1.30 m. The merry-go-round is rotating with an angular speed of 0.25 rev/s. A child of mass 25 kg, initially at rest, grabs the outer edge of the merry-go-round and begins to rotate with it. What is the angular speed of the merry-go-round after the child gets on?

Q15:

A disk has a mass of 3.6 kg and a radius of 65.0 cm. A small object of mass 0.42 kg is attached to the rim of the disk. The disk is rotating at 3.0 rev/s when the small object separates from the disk. What is the disk’s rotation rate after the small object separates?

Q16:

A cylinder with rotational inertia kg⋅m^{2} rotates clockwise about a vertical axis through its center with angular speed . A second cylinder with rotational inertia kg⋅m^{2} rotates counterclockwise about the same axis with angular speed . The cylinders are coupled, after which they have the same rotational axis.

What is the angular speed of the coupled cylinders?

What percentage of the cylinders’ original kinetic energy is lost to friction during coupling?

Q17:

A centrifuge has a radius of 7.400 m. At maximum rotation speed, the centrifuge produces forces on its payload of 15.0 Gs, meaning 15.0 times the force produced by gravity at Earth’s surface. A payload of mass 15.0 kg is rotated at the centrifuge’s maximum speed.

What is the angular momentum of the payload?

The centrifuge’s drive motor is turned off, and 7.5 kg of the payload is lost from the centrifuge. What rate does the centrifuge now spin at?

Q18:

A bug flying horizontally at 2.5 m/s collides and sticks to the end of a uniform stick hanging vertically. After the impact, the stick swings out to a maximum angle of 1 2 ∘ from the vertical before rotating back. If the mass of the stick is 15 times that of the bug, calculate the length of the stick.

Q19:

An ice-skater is spinning at 8.00 rev/s and his moment of inertia is 0.650 kg⋅m^{2}.

Calculate the angular momentum of the ice-skater.

The skater reduces his angular speed to 1.25 rev/s by extending his arms and increasing his moment of inertia. Find the value of the skater’s moment of inertia with outstretched arms.

If the skater did not stretch out his arms but rather allowed friction from contact with the ice to slow him to 2.80 rev/s in an 18.2 s interval, what average torque was exerted by friction?

Q20:

In 2015, in Warsaw, Poland, Olivia Oliver of Nova Scotia broke the world record for being the fastest spinner on ice skates. She achieved a record 342 rpm, beating the existing Guinness World Record by 34 rotations. If an ice skater extends her arms at that rotation rate, what would be her new rotation rate? Assume she can be approximated by a 45-kg rod that is 1.7 m tall with a radius of 15 cm in the record spin. With her arms stretched take the approximation of a rod of length 130 cm with 1 0 % of her body mass aligned perpendicular to the spin axis. Neglect frictional forces.

Q21:

A particle of mass 0.80 kg is traveling at a speed of 3.7 m/s along a circular trajectory for which the radius is 2.50 m. What is the angular momentum of the particle about the center of rotation?

Q22:

A gymnast with a mass of 80.0 kg and a height of 1.8 m swings on a 3.0 m tall high bar. The gymnast is initially at rest, is horizontally aligned and extended to his full body length, and has his hands on the high bar. In this position, the gymnast’s body can be approximated as a 1.8 m long thin rod. From this position, the gymnast swings about the high bar at a rotation rate of 0.865 rev/s. The gymnast releases his grip on the high bar at the instant that he is swinging vertically upward, his center of mass at that instant being 3.0 m displaced vertically above the ground. At the instant that the gymnast releases his grip, he tucks in his legs, taking a negligible time to do so. With his legs tucked in, the gymnast’s body can be approximated as a 0.90 m long thin rod. The gymnast rotates while in the air and continues to rotate until he is at a height above the ground where his extended legs would just reach the ground. At that moment, he extends his legs, taking a negligible amount of time to do so, and lands. Through how many revolutions does the gymnast turn before landing?

Q23:

A merry-go-round is rotating at 4.0 rpm. Some children jump onto the merry-go-round, increasing its moment of inertia by 2 5 % . What is the merry-go-round’s rotation rate after the children jump onto it?

Q24:

A satellite orbiting Earth has its apogee at km above the surface of Earth and perigee at 460 km above the surface of Earth. At apogee, its speed is 630 m/s. What is its speed at perigee? Use a value of km for the Earth’s radius.

Q25:

An ice-skater is preparing for a jump in which he will rotate while in the air. When he is on the ground, with his arms extended, his moment of inertia is 2.2 kg⋅m^{2}, and he is spinning at 0.31 rev/s. He launches himself into the air at a speed of 12.6 m/s at an angle of 4 5 ∘ above the horizontal. At the moment that he leaves the ground, he contracts his arms, taking negligible time, and changes his moment of inertia to 0.62 kg⋅m^{2}. How many revolutions can he complete while airborne?

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