As you read, pay attention to the diagram of a spinning disk in Figure 10.15. For the disk to spin, work must be done on the disk. The force acting upon the disk must be perpendicular to the radius of the disk, which we know is torque. We also know torque
is related to moment of inertia. We can relate the work done on the disk to moment of inertia using the equation .
Example 10.8 shows how to calculate the net work for a rotating disk using this work equation. In the second part of the example, the rotational velocity is determined using the equation for rotational acceleration and moment of inertia. Lastly, it uses this equation to calculate the rotational kinetic energy.
Rotational Kinetic Energy: Work and Energy Revisited
In this module, we will learn about work and energy associated with rotational motion. Figure 10.14 shows a worker using an electric grindstone propelled by a motor. Sparks are flying, and noise and vibration are created as layers of steel are pared from
the pole. The stone continues to turn even after the motor is turned off, but it is eventually brought to a stop by friction. Clearly, the motor had to work to get the stone spinning. This work went into heat, light, sound, vibration, and considerable
rotational kinetic energy.
Figure 10.14 The motor works in spinning the grindstone, giving it rotational kinetic energy. That energy is then converted to heat, light, sound, and vibration.
Work must be done to rotate objects such as grindstones or merry-go-rounds. Work was defined in Uniform Circular Motion and Gravitation for translational motion, and we can build on that knowledge when considering work done in rotational motion. The simplest
rotational situation is one in which the net force is exerted perpendicular to the radius of a disk (as shown in Figure 10.15) and remains perpendicular as the disk starts to rotate. The force is parallel to the displacement, and so the net work done
is the product of the force times the arc length traveled:
To get torque and other rotational quantities into the equation, we multiply and divide the right-hand side of the equation by , and gather terms:
We recognize that and
, so that
This equation is the expression for rotational work. It is very similar to the familiar definition of translational work as force multiplied by distance. Here, torque is analogous to force, and angle is analogous to distance. The equation n is valid in general, even though it was derived for a special case.
To get an expression for rotational kinetic energy, we must again perform some algebraic manipulations. The first step is to note that net so that
Figure 10.15 The net force on this disk is kept perpendicular to its radius as the force causes the disk to rotate. The net work done is thus . The net work goes into rotational kinetic energy.
Making Connections
Work and energy in rotational motion are completely analogous to work and energy in translational motion, first presented in Uniform Circular Motion and Gravitation.
Now, we solve one of the rotational kinematics equations for We start with the equation
Substituting this into the equation for net and gathering terms yields
This equation is the work-energy theorem for rotational motion only. As you may recall, net work changes the kinetic energy of a system. Through an analogy with translational motion, we define the term to be rotational kinetic energy
for an object with a moment of inertia
and an angular velocity
The expression for rotational kinetic energy is exactly analogous to translational kinetic energy, with being analogous to
and
to
. Rotational kinetic energy has important effects. Flywheels, for example, can be used to store
large amounts of rotational kinetic energy in a vehicle, as seen in Figure 10.16.
Figure 10.16 Experimental vehicles, such as this bus, have been constructed in which rotational kinetic energy is stored in a large flywheel. When the bus goes down a hill, its transmission converts its gravitational potential energy
into. It can also convert translational kinetic energy, when the bus stops, into
. The flywheel's energy can then be used to accelerate, to go up another hill, or to keep the bus from slowing
down due to friction.
Source: Rice University, https://openstax.org/books/college-physics/pages/10-4-rotational-kinetic-energy-work-and-energy-revisited
This work is licensed under a Creative Commons Attribution 4.0 License.