6.2 Torque and Work

Energy Transfer by Torque

Torque, the rotational equivalent of force, can transfer energy into or out of a rigid system when applied over an angular displacement. This energy transfer is fundamental to understanding how rotating objects gain or lose energy.

• When torque acts in the same direction as rotation, positive work is done (energy is added to the system)
• When torque opposes the direction of rotation, negative work is done (energy is removed from the system)
• The magnitude of energy transfer depends on both the torque applied and how far the object rotates

For example, when you apply torque to a bicycle pedal, you transfer energy into the bicycle's drivetrain, causing it to accelerate. Conversely, when brakes apply a torque that opposes wheel rotation, they remove energy from the system, causing deceleration.

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Work-Torque Relationship

The mathematical relationship between work and torque allows us to quantify the energy transfer in rotating systems. This relationship is analogous to the work-force relationship in linear systems.

To calculate the work done by a torque, we integrate the torque with respect to angular displacement:

$W=\int_{\theta_{1}}^{\theta_{2}} \tau d \theta$

Where:

• $W$ is the work done by the torque (measured in joules)
• $\tau$ is the torque as a function of angular position (measured in N·m)
• $\theta_{1}$ and $\theta_{2}$ are the initial and final angular positions (measured in radians)

For the special case of constant torque, the calculation simplifies to:

$W = \tau \Delta \theta$

This equation tells us that the work equals the product of the torque and the angular displacement through which it acts.

Graphical Work Analysis

Visualizing the relationship between torque and angular position provides valuable insights into the work done in rotational systems.

When we plot torque versus angular position:

• The area under the curve represents the work done by the torque
• Areas above the x-axis indicate positive work (energy added to the system)
• Areas below the x-axis indicate negative work (energy removed from the system)

This graphical approach is particularly useful for analyzing situations with variable torque, where the torque changes as the object rotates. By finding the area under the curve, we can determine the total work without needing to solve complex integrals analytically.

For example, in a car engine, the torque varies throughout the rotation cycle. By analyzing the torque-angle graph, engineers can calculate the total work output per cycle and optimize engine performance.

Practice Problem 1: Work Done by Constant Torque

Solution

Since the torque is constant, we can use the simplified equation for work:

$W = \tau \Delta \theta$

Substituting the given values: $W = 15 \text{ N·m} \times 2.5 \text{ rad}$ $W = 37.5 \text{ J}$

Therefore, the student does 37.5 joules of work on the wheel.

Practice Problem 2: Work from Torque-Angle Graph

Solution

To find the work, we need to integrate the torque function with respect to angular displacement:

$W = \int_{0}^{3} (20 - 5\theta) d\theta$

$W = [20\theta - \frac{5\theta^2}{2}]_{0}^{3}$

$W = (20 \times 3 - \frac{5 \times 3^2}{2}) - (20 \times 0 - \frac{5 \times 0^2}{2})$

$W = 60 - 22.5 - 0$

$W = 37.5 \text{ J}$

Therefore, the work done by the variable torque is 37.5 joules.