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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.
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
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?
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
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.
A merry-go-round has a radius of 2.0 m and a moment
of inertia 300 kg⋅m2. 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?
A star collapses which greatly increases its density.
-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
it had a rotational period of 28 days.
The star’s radius is
after the collapse, what is its rotational period?
Consider a day to be equal to
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⋅m2 and after the tuck during the somersaults
is 4.2 kg⋅m2.
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?
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⋅m2
and her spin rate is 0.50 rev/s.
If her moment of inertia in the airborne tuck is 3.4 kg⋅m2
and she completes the flips in 2.0 s, how many revolutions does she undergo in the air?
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?
A satellite in the shape of a sphere of mass
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?
A space station consists of a giant, rotating, hollow cylinder of mass
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?
A bug of mass 0.020 kg is at rest on the edge of a solid
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
What is the change in the kinetic energy of
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?
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
What is the magnitude of the torque on the particle about the centre of the circle at
What is the angular speed of the particle at
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
km and it collapses to
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.
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?
A merry-go-round has a mass of 110 kg and a radius of
The merry-go-round is rotating with an angular speed of
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?
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?
A cylinder with rotational inertia
clockwise about a vertical axis through its center with angular speed
A second cylinder with rotational inertia
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?
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?
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?
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
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.
An ice-skater is spinning at 8.00 rev/s and his moment
of inertia is 0.650 kg⋅m2.
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?
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
of her body mass
aligned perpendicular to the spin axis. Neglect frictional
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?
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?
A merry-go-round is rotating at 4.0 rpm. Some children jump onto the
merry-go-round, increasing its moment of inertia by
. What is the merry-go-round’s
rotation rate after the children jump onto it?
A satellite orbiting Earth has its apogee at
km above the surface of
Earth and perigee at 460 km above the surface of Earth.
its speed is 630 m/s. What is its speed at perigee?
Use a value of
km for the Earth’s radius.
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
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
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
How many revolutions can he complete while airborne?
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