To solve the problem, we need to analyze the forces and dynamics of the system. Here's a step-by-step solution:

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1. Kinematics to Determine Acceleration of the Block

The block falls with constant acceleration, so we can use the equation of motion: $s = ut + \frac{1}{2} a t^2$ where:

• $s = 3.00 \, \text{m}$ (distance fallen),
• $u = 0 \, \text{m/s}$ (initial velocity),
• $t = 2.00 \, \text{s}$ (time),
• $a$ is the acceleration.

Substitute values: $3.00 = 0 + \frac{1}{2} a (2.00)^2$ $3.00 = 2a$ $a = 1.50 \, \text{m/s}^2$

The acceleration of the block is $a = 1.50 \, \text{m/s}^2$.

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2. Forces Acting on the Block

The forces acting on the block are:

1. Gravitational force: $F_g = mg$,
2. Tension in the cable: $T$.

Using Newton's second law for the block: $mg - T = ma$ $T = mg - ma$ Substitute $m = 12.0 \, \text{kg}$, $g = 9.8 \, \text{m/s}^2$, and $a = 1.50 \, \text{m/s}^2$: $T = (12.0)(9.8) - (12.0)(1.50)$ $T = 117.6 - 18.0$ $T = 99.6 \, \text{N}$

The tension in the cable is $T = 99.6 \, \text{N}$.

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3. Rotational Dynamics of the Pulley

The pulley is a solid disk with radius $R = 0.60 \, \text{m}$. The torque ($\tau$) acting on the pulley is: $\tau = T R$ The rotational equation of motion for the pulley is: $\tau = I \alpha$ where:

• $I = \frac{1}{2} M R^2$ is the moment of inertia of the solid disk,
• $\alpha$ is the angular acceleration.

Relating angular acceleration to the block's linear acceleration: $\alpha = \frac{a}{R}$ Substitute $\alpha = \frac{1.50}{0.60} = 2.50 \, \text{rad/s}^2$.

Thus: $T R = \frac{1}{2} M R^2 \alpha$ Substitute $R = 0.60 \, \text{m}$, $T = 99.6 \, \text{N}$, and $\alpha = 2.50 \, \text{rad/s}^2$: $99.6 (0.60) = \frac{1}{2} M (0.60)^2 (2.50)$ $59.76 = \frac{1}{2} M (0.36) (2.50)$ $59.76 = 0.45 M$ $M = \frac{59.76}{0.45}$ $M = 132.8 \, \text{kg}$

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Final Answer:

The mass of the pulley is $M = 132.8 \, \text{kg}$.

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Do you want further clarification or detailed explanations for any step?

Here are some related questions for deeper understanding:

1. How would the solution change if the pulley was not a solid disk but a hoop?
2. What would happen to the acceleration of the block if the mass of the pulley were halved?
3. How is angular acceleration related to linear acceleration in systems like this?
4. What assumptions are made in this analysis, and how might they affect the real-world results?
5. Can you derive the tension formula directly from the system's constraints?

Tip: Always identify the type of pulley or rotational object to correctly calculate its moment of inertia!