11.1.6 Work and Power

Work Done

Work Done in the context of rotational motion is defined as the force that causes rotation multiplied by the angular displacement of the object. To calculate the work done on a rotating object, you can find the product of the torque $(T)$ and angular displacement $(\theta)$:

$$W = T\theta$$

This is similar to linear motion, where work is force multiplied by distance. In rotational motion, applying torque increases the rotational kinetic energy of an object, but first, the torque must overcome any frictional torque that resists the movement.

Calculating Frictional Torque Experimentally

The frictional torque on a rotating wheel can be determined through a series of steps:

1. Apply an accelerating torque to bring the wheel up to a specific angular velocity.
2. Remove the accelerating torque and measure the time it takes for the wheel to stop.
3. Calculate the average deceleration (assuming constant frictional torque) with the equation:

$$\alpha = \frac{\omega_{\text{initial}} - \omega_{\text{final}}}{t}$$

where $\alpha$ is the angular deceleration, $\omega_{\text{initial}}$ is the initial angular velocity, and $t$ is the time taken to stop.

4. Use the moment of inertia $(I)$ of the wheel and deceleration $(\alpha)$ to calculate the frictional torque with:

$$T = I\alpha$$

Frictional Torque in Practical Applications

In many practical applications, minimising frictional torque is important to reduce energy losses as heat and sound. However, in some tools, frictional torque can be advantageous. For instance, using a screwdriver involves applying frictional torque to prevent slipping and increase rotational energy.

Work done can also be found by calculating the area under a torque-angular displacement graph.

Power in Rotational Systems

Power (P) represents the rate of energy transfer. Since work involves energy transfer, power can be considered the rate of doing work. For rotational motion, you can calculate power as:

$$P = \frac{W}{t} = \frac{T \theta}{t}$$

Since $\frac{\theta}{t}$ equals angular velocity $(\omega)$, we can simplify this to:

$$P = T \omega$$

Where $T$ is torque and $\omega$ is angular velocity. This equation allows you to determine the power output in rotating systems, such as engines and turbines.