1. **Problem Statement:** We have a spinning disk with angular acceleration given by $\alpha(t) = -ct$, where $c = 0.2$ rad/s$^3$. The initial angular velocity is $\omega_0 = 5$ rad/s. We need to find: A) Angular velocity $\omega(t)$ as a function of time. B) Angular displacement $\theta(t)$ as a function of time. C) The time when the disk stops rotating. 2. **Formulas and Rules:** - Angular acceleration is the derivative of angular velocity: $\alpha(t) = \frac{d\omega}{dt}$. - Angular velocity is the derivative of angular displacement: $\omega(t) = \frac{d\theta}{dt}$. - To find $\omega(t)$, integrate $\alpha(t)$ with respect to time. - To find $\theta(t)$, integrate $\omega(t)$ with respect to time. - Use initial conditions to find constants of integration. 3. **Find Angular Velocity $\omega(t)$:** Given $\alpha(t) = -ct = -0.2t$, $$\frac{d\omega}{dt} = -0.2t$$ Integrate both sides with respect to $t$: $$\omega(t) = \int -0.2t \, dt = -0.2 \frac{t^2}{2} + C_1 = -0.1 t^2 + C_1$$ Use initial condition $\omega(0) = 5$: $$5 = -0.1 \times 0^2 + C_1 \Rightarrow C_1 = 5$$ So, $$\boxed{\omega(t) = 5 - 0.1 t^2}$$ 4. **Find Angular Displacement $\theta(t)$:** Since $\omega(t) = \frac{d\theta}{dt} = 5 - 0.1 t^2$, Integrate with respect to $t$: $$\theta(t) = \int (5 - 0.1 t^2) dt = 5t - 0.1 \frac{t^3}{3} + C_2 = 5t - \frac{0.1}{3} t^3 + C_2$$ Assuming initial angular displacement $\theta(0) = 0$: $$0 = 5 \times 0 - \frac{0.1}{3} \times 0^3 + C_2 \Rightarrow C_2 = 0$$ So, $$\boxed{\theta(t) = 5t - \frac{0.1}{3} t^3 = 5t - \frac{t^3}{30}}$$ 5. **Find Time When Disk Stops Rotating:** Disk stops when angular velocity is zero: $$0 = 5 - 0.1 t^2$$ Solve for $t$: $$0.1 t^2 = 5 \Rightarrow t^2 = \frac{5}{0.1} = 50 \Rightarrow t = \sqrt{50} = 5 \sqrt{2} \approx 7.07 \text{ seconds}$$ **Final answers:** - Angular velocity: $\omega(t) = 5 - 0.1 t^2$ - Angular displacement: $\theta(t) = 5t - \frac{t^3}{30}$ - Time when disk stops: $t = 5 \sqrt{2} \approx 7.07$ seconds **Note:** A free body diagram is not applicable here because the problem involves rotational kinematics with given angular acceleration, not forces or torques acting on the disk.