KAHIBARO
Table of Contents
Rotational Work
In rotational motion, work is done when a torque causes an object to turn through an angle. This is the rotational version of linear work, where a force causes an object to move through a distance.
For straight line motion, work is
$$
W = Fd
$$
when the force is constant and acts along the direction of motion. In rotational motion, the matching idea is
$$
W = \tau \theta
$$
for a constant torque $\tau$ acting through an angular displacement $\theta$, measured in radians.
This formula shows a very important parallel between linear and rotational motion.
| Linear motion | Rotational motion |
|---|---|
| Force $F$ | Torque $\tau$ |
| Distance $d$ | Angular displacement $\theta$ |
| Work $W = Fd$ | Work $W = \tau \theta$ |
The angle must be in radians because radians connect naturally to arc length and rotational geometry.
For a constant torque, the rotational work is
$$
W = \tau \theta
$$
where $\tau$ is in newton meters, $\theta$ is in radians, and $W$ is in joules.
Why Rotation Can Transfer Energy
When a torque acts on a rotating object, it can change the object's rotational motion. If the object spins faster, its rotational kinetic energy increases. This means that work done by torque transfers energy into rotational kinetic energy.
This is closely related to the translational idea that work changes kinetic energy. Here the energy is associated with spinning.
If a wheel, disk, or pulley is pushed so that it rotates, the applied torque does work on it. If friction slows a rotating object, friction does negative work and removes rotational energy.
Deriving Rotational Work from Force
It is useful to see where the formula comes from. Suppose a force $F$ acts tangentially at a distance $r$ from the axis. Then the torque magnitude is
$$
\tau = rF
$$
If the object turns through an angle $\theta$, the point where the force acts moves along an arc of length
$$
s = r\theta
$$
The work done by the tangential force is
$$
W = Fs
$$
Substituting $s = r\theta$ gives
$$
W = F(r\theta) = (rF)\theta = \tau \theta
$$
So the rotational work formula is consistent with the ordinary definition of work.
Work Done by a Variable Torque
Sometimes torque is not constant. It may change with angle, just as force may change with position in linear motion. In that case, total work is found by adding small contributions of work over small angular displacements.
The small amount of work is
$$
dW = \tau \, d\theta
$$
So the total work is
$$
W = \int \tau \, d\theta
$$
This is the rotational analogue of work done by a variable force.
If torque changes with angular position, use
$$
W = \int_{\theta_1}^{\theta_2} \tau(\theta)\, d\theta
$$
This is the general formula for rotational work.
Power in Rotational Motion
Power tells us how fast work is done. In rotation, power depends on torque and angular velocity.
Starting from
$$
P = \frac{dW}{dt}
$$
and using
$$
dW = \tau \, d\theta
$$
we get
$$
P = \tau \frac{d\theta}{dt}
$$
Since angular velocity is
$$
\omega = \frac{d\theta}{dt}
$$
the power in rotational motion is
$$
P = \tau \omega
$$
This formula is one of the most useful results in rotational dynamics.
| Linear motion | Rotational motion |
|---|---|
| $P = Fv$ | $P = \tau \omega$ |
A motor that produces a large torque at a high angular speed delivers large power.
Rotational power is
$$
P = \tau \omega
$$
where $\tau$ is torque and $\omega$ is angular velocity in radians per second.
Meaning of the Power Formula
The formula
$$
P = \tau \omega
$$
shows that power increases when either torque or angular speed increases.
A machine can have high torque but low speed, or high speed but low torque. In both cases, the power may be moderate. Very powerful rotating machines combine both substantial torque and substantial angular speed.
This is why engine performance is often discussed in terms of both torque and rotational speed.
Units of Rotational Work and Power
Although torque has units of newton meters, and work also has units of newton meters, they do not mean the same thing physically.
Work is energy, measured in joules:
$$
1 \text{ J} = 1 \text{ N m}
$$
Torque is not energy. It is a turning effect, also written in newton meters, but it should not be confused with joules.
Power is measured in watts:
$$
1 \text{ W} = 1 \text{ J/s}
$$
Using $P = \tau \omega$, we can check units:
$$
(\text{N m})(\text{rad/s}) = \text{J/s} = \text{W}
$$
The radian is dimensionless, so the units are consistent.
Do not confuse torque and work.
Both may involve $\text{N m}$, but torque is a turning effect, while work is energy transferred.
Connection to Rotational Kinetic Energy
When net torque does work on a rigid body, the rotational kinetic energy changes. The rotational kinetic energy is
$$
K_{\text{rot}} = \frac{1}{2}I\omega^2
$$
So if a torque speeds up a rotating object, the work done appears as an increase in rotational kinetic energy. If the torque slows it down, the work is negative and the rotational kinetic energy decreases.
This leads to the rotational form of the work energy idea:
$$
W_{\text{net}} = \Delta K_{\text{rot}}
$$
This chapter focuses on work and power, so the broader energy interpretation is left at that key connection.
Example with Constant Torque
Suppose a wheel experiences a constant torque of
$$
\tau = 4.0 \, \text{N m}
$$
and turns through
$$
\theta = 3.0 \, \text{rad}
$$
Then the work done is
$$
W = \tau \theta = (4.0)(3.0) = 12.0 \, \text{J}
$$
So 12 joules of energy are transferred to the wheel.
Example of Rotational Power
Suppose a motor applies a torque of
$$
\tau = 6.0 \, \text{N m}
$$
to a rotating shaft whose angular velocity is
$$
\omega = 20 \, \text{rad/s}
$$
Then the power delivered is
$$
P = \tau \omega = (6.0)(20) = 120 \, \text{W}
$$
So the motor delivers 120 watts of power.
Sign of Work and Power
The sign matters.
If the torque and angular displacement are in the same rotational direction, the work is positive. Energy is added to the rotating object.
If the torque opposes the angular displacement, the work is negative. Energy is removed from the rotating object.
Similarly for power, if torque and angular velocity act in the same sense, power is positive. If torque opposes the rotation, power is negative.
A brake on a wheel usually exerts a torque opposite to the motion, so it does negative work and negative power on the wheel.
Practical Physical Situations
Many everyday systems involve rotational work and power. An electric fan motor does work on the blades and provides rotational power. A car engine turns the drive shaft with torque, and the rate of energy delivery depends on both torque and rotational speed. A person pushing a merry go round applies torque through an angle, doing rotational work.
In all these cases, the same core formulas apply:
$$
W = \tau \theta
$$
for constant torque, and
$$
P = \tau \omega
$$
for instantaneous rotational power.
Key Relations
The most important results of this chapter are the rotational analogues of linear work and power.
For rotational motion:
$$
W = \tau \theta \quad \text{(constant torque)}
$$
$$
W = \int \tau \, d\theta \quad \text{(variable torque)}
$$
$$
P = \tau \omega
$$
and the net rotational work changes rotational kinetic energy:
$$
W_{\text{net}} = \Delta K_{\text{rot}}
$$
These equations form the basic language for describing how energy is transferred in rotating systems.