A circular disk of radius R meter and mass M kg is rotating around the axis perpendicular to the disk. An external torque is applied to the disk such that $$\theta (t)=5t^{2}-8t$$, where $$\theta (t)$$ is the angular position of the rotating disc as a function of time t. How much power is delivered by the applied torque, when t = 2 s ?

Given the angular position:

$$\theta(t) = 5t^2 - 8t$$

Differentiating with respect to time to find the angular velocity:

$$\omega = \frac{d\theta}{dt} = 10t - 8$$

$$At\ t=2\text{ s},\ \omega=10(2)-8=12\text{ rad/s}$$

$$\alpha = \frac{d\omega}{dt} = 10 \text{ rad/s}^2$$

$$\tau = I\alpha$$

$$I=\frac{1}{2}MR^2$$ for a circular disk rotating about an axis perpendicular to its plane through the center

$$\tau = \left( \frac{1}{2}MR^2 \right) \times 10 = 5MR^2$$ 

Power Delivered ($$P$$):

$$P = \tau \cdot \omega$$

$$P = (5MR^2) \times 12$$
$$P = 60MR^2$$