The problem is asking to calculate both the angular speed and angular displacement of an object after a hanging mass has dropped a given distance of $0.4$ meters.

Solution Steps:

1. Calculating Angular Displacement

To find angular displacement ($\theta$), use the relationship between linear displacement and angular displacement: $s = r \theta$ where:

• $s = 0.4 \, \text{m}$ is the linear displacement,
• $r$ is the radius of the rotational motion.

Thus: $\theta = \frac{s}{r}$

If the radius $r$ is known, plug it into this formula to find $\theta$.

2. Calculating Angular Speed

Angular speed ($\omega$) can be calculated if we know the angular acceleration ($\alpha$) and the time or displacement over which this acceleration applies.

If we assume uniform angular acceleration, we can use: $\omega^2 = \omega_0^2 + 2 \alpha \theta$ where:

• $\omega_0$ is the initial angular speed (which is 0 if the object started from rest),
• $\alpha$ is the angular acceleration.

From the image, it seems there’s an indication of $\alpha = 2.52 \, \text{rad/s}^2$ written.

So, if we have $\theta$ from Step 1, we can substitute it into this equation to solve for $\omega$: $\omega = \sqrt{2 \alpha \theta}$

Let me know if you'd like a detailed solution with specific values.

Further Questions

1. What is the radius $r$ in this scenario?
2. Was the initial angular velocity zero (starting from rest)?
3. Is there more information on how $\alpha = 2.52$ was determined?
4. Should the solution include time-based calculations if available?
5. Is there data on friction or external forces acting on the system?

Tip

Always check if units are consistent across calculations (meters, seconds, radians) to avoid conversion errors.