Rotational Kinematics and Energy
Answers to Even-numbered Conceptual Questions
The angular speed of an object with constant angular acceleration changes linearly with
Therefore, at the time
/2 the angular speed of this object is
Jason and Betsy both complete one revolution in the same amount of time. Therefore,
their periods of rotation are the same.
Yes. In fact, this is the situation whenever you drive in a circular path with constant
Every point on Earth has the same angular speed.
Therefore, the smallest linear speed
occurs where the distance from the axis of rotation is smallest; namely, at the poles.
At the top of these building, your distance from the axis of rotation of the Earth is greater
than it was on the ground floor.
Therefore, your linear speed at the top of the building is
greater than it is on the ground floor.
The moment of inertia is greater when the axis of rotation is on the rim of the wheel.
reason is that much of the wheel’s mass is now at a significantly greater distance from the
axis of rotation, compared with the case where the axis is at the center of the wheel.
The long, thin minute hand – with mass far from the axis of rotation – has the greater
moment of inertia.
The moment of inertia of an object changes with the position of the axis of rotation
because the distance from the axis to all the elements of mass have been changed.
It is not
just the shape of an object that matters, but the distribution of mass with respect to the axis
Spin the two spheres with equal angular speeds.
The one with the larger moment of
inertia – the hollow sphere – has the greater kinetic energy, and hence will spin for a
longer time before stopping.
The moment of inertia of the Earth was increased slightly, because mass (water in the
dam) was moved from a lower elevation to a higher elevation as the dam was filled.
Raising this mass to a higher elevation moves it farther from the axis of rotation, which
increases the moment of inertia.
When the chunky stew is rolled down the aisle, all of the contents of the can roll together
with approximately the same angular speed.
This is because the chunky stew is thick and
The beef broth, however, is little more than water.
Therefore, when the
broth is rolled down the aisle, almost all that is actually rolling is the metal can itself.
follows that the stew has the greater initial kinetic energy, and hence it rolls a greater
Assuming the ball starts spinning immediately on encountering the no-slip surface, with
no loss of energy, it will rise to the same height from which it was released. However,
some energy will be lost in a real system as the ball begins to spin; therefore, the ball
should reach a height slightly less than its release height.