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Work and Power in Rotation

Just as forces do work in linear motion, torques do work in rotational motion. When a torque acts through an angular displacement, it transfers energy to the rotating object. The rate at which this work is done is the rotational power.

Work Done by Torque

$$W = \tau \Delta \theta$$

$W$: Work done by torque (Joules, J)
$\tau$: Torque (Newton-meters, N·m)
$\Delta \theta$: Angular displacement (radians, rad)

Power from Torque

$$P = \tau \omega$$

$P$: Rotational power (Watts, W)
$\tau$: Torque (Newton-meters, N·m)
$\omega$: Angular velocity (radians per second, rad/s)

Calculus-Based Definitions

When torque is not constant, we use calculus to find the total work done and instantaneous power in rotational motion.

$$ W = \int_{\theta_i}^{\theta_f} \tau(\theta) \, d\theta $$

This sums up the small amounts of work done by torque over each tiny angular displacement.

$$ P(t) = \frac{dW}{dt} = \tau(t) \omega(t) $$

Instantaneous power is the product of torque and angular velocity at each moment.

For variable torque, work is the area under the torque vs. angle graph.
For variable torque or angular velocity, power can change with time.

Physical Meaning

Work: When you apply a torque (like turning a wrench), you do work equal to the torque times the angle through which you turn. This work can increase the rotational kinetic energy of the object.

Power: The rate at which work is done. A motor that applies a constant torque while rotating at a constant angular velocity delivers constant power.

Compare: Linear vs. Rotational Work and Power

Linear	Rotational
Work: $W = F \Delta x$	Work: $W = \tau \Delta \theta$
Power: $P = Fv$	Power: $P = \tau \omega$
Force ($F$)	Torque ($\tau$)
Displacement ($\Delta x$)	Angular displacement ($\Delta \theta$)
Velocity ($v$)	Angular velocity ($\omega$)

Interactive Simulation: Rotational Impulse and Angular Momentum

Explore how torque impulse affects angular momentum and angular velocity.

Moment of Inertia (kg·m²): 5.0

Initial Angular Velocity (rad/s): 2.0

Impulse Torque (N·m): 0

Impulse Duration (s): 1.0

Angular Impulse: 0 N·m·s

Angular Momentum: 10.0 kg·m²/s

Angular Impulse Bar:

Example 1: Work Done by a Wrench

A mechanic applies a torque of $25\,\text{N} \cdot \text{m}$ to tighten a bolt, rotating it through $0.5\,\text{rad}$. How much work is done?

Solution:

$W = \tau \Delta \theta = 25 \times 0.5 = 12.5\,\text{J}$

The mechanic does 12.5 Joules of work to tighten the bolt.

Example 2: Motor Power

An electric motor applies a constant torque of $15\,\text{N} \cdot \text{m}$ while rotating at $120\,\text{rad/s}$. What is the motor's power output?

Solution:

$P = \tau \omega = 15 \times 120 = 1800\,\text{W} = 1.8\,\text{kW}$

The motor delivers 1.8 kilowatts of power.

Example 3: Variable Torque

A wind turbine blade experiences a torque that varies with time: $\tau(t) = 50t\,\text{N} \cdot \text{m}$ for $0 \leq t \leq 10\,\text{s}$. If it rotates at constant $\omega = 2\,\text{rad/s}$, what is the average power over 10 seconds?

Solution:

Average torque: $\tau_{\text{avg}} = \frac{1}{10} \int_0^{10} 50t \, dt = \frac{1}{10} \times 2500 = 250\,\text{N} \cdot \text{m}$

Average power: $P_{\text{avg}} = \tau_{\text{avg}} \omega = 250 \times 2 = 500\,\text{W}$

Real-World Applications

Electric Motors: Convert electrical energy to rotational work and power.
Wind Turbines: Extract work from wind to generate electrical power.
Automotive Engines: Produce torque and power to drive vehicles.
Power Tools: Drills, saws, and grinders use rotational work and power.
Industrial Machinery: Conveyors, pumps, and compressors rely on rotational power.

Energy Conservation

The work-energy theorem applies to rotation: the net work done by torques equals the change in rotational kinetic energy.

$$W_{\text{net}} = \Delta K_{\text{rot}} = \frac{1}{2}I\omega_f^2 - \frac{1}{2}I\omega_i^2$$

We will go into more detail about energy conservation in the next concept.

Summary / Takeaways

Torque does work when it acts through an angular displacement: $W = \tau \Delta \theta$
Rotational power is the rate of doing work: $P = \tau \omega$
These formulas are the rotational analogs of linear work ($W = F \Delta x$) and power ($P = Fv$)
Work can change rotational kinetic energy according to the work-energy theorem
Understanding rotational work and power is crucial for engineering applications like motors, turbines, and machinery