Question 1198759
**a) Linear Velocity of the Pulley**

* **Acceleration Phase:**
    * v = u + at 
    * v = 0 + (0.16 m/s²) * (32 s) 
    * v = 5.12 m/s 

* **Constant Velocity Phase:**
    * The linear velocity remains constant at 5.12 m/s during this phase.

* **Deceleration Phase:**
    * v = u + at 
    * 0 = 5.12 m/s + a * (15 s) 
    * a = -5.12 m/s / 15 s 
    * a = -0.3413 m/s²

* **Linear Velocity of the Pulley:** 5.12 m/s

**b) Angular Velocity of the Cable**

* **Linear Velocity (v):** 5.12 m/s
* **Radius of the Pulley (r):** 0.30 m / 2 = 0.15 m

* **Angular Velocity (ω):** ω = v / r 
    * ω = 5.12 m/s / 0.15 m 
    * ω = 34.13 rad/s

**c) Angular Acceleration of the Pulley**

* **Linear Acceleration (a):** 0.16 m/s² 
* **Radius of the Pulley (r):** 0.15 m

* **Angular Acceleration (α):** α = a / r 
    * α = 0.16 m/s² / 0.15 m 
    * α = 1.067 rad/s²

**d) Linear Acceleration of the Cable**

* **During Acceleration:** 0.16 m/s²
* **During Constant Velocity:** 0 m/s²
* **During Deceleration:** -0.3413 m/s²

**e) Torque Applied to the Shaft**

* **Torque (τ):** τ = I * α 
    * where I is the moment of inertia of the pulley and α is the angular acceleration.

* **Assuming the pulley is a solid disk:**
    * I = (1/2) * M * r² 
        * where M is the mass of the pulley and r is the radius.

* **To find the mass of the pulley, we need the density of the material.** 

**Dimensional Analysis**

* **v = u + at** 
    * [L/T] = [L/T] + [L/T²] * [T] 
    * [L/T] = [L/T] 

* **ω = v / r** 
    * [rad/s] = [L/T] / [L] 
    * [1/s] = [1/s] 

* **α = a / r** 
    * [rad/s²] = [L/T²] / [L] 
    * [1/s²] = [1/s²]

* **τ = I * α** 
    * [N.m] = [kg.m²] * [rad/s²] 
    * [kg.m/s²] * [m] = [kg.m²/s²] * [m] 
    * [kg.m²/s²] = [kg.m²/s²]

**Power Equations**

* **Power (P) = Force (F) * Velocity (v)** 

    * P = m * a * v 
    * [W] = [kg] * [L/T²] * [L/T] 
    * [kg.m²/s³] = [kg.m²/s³]

* **Power (P) = Torque (τ) * Angular Velocity (ω)** 

    * P = I * α * ω 
    * [W] = [kg.m²] * [rad/s²] * [rad/s] 
    * [kg.m²/s³] = [kg.m²/s³]

**Note:**

* This analysis assumes that the pulley is a solid disk.
* The mass of the pulley and the load it carries will affect the torque required and the overall power consumption.
* This analysis does not consider factors such as friction and efficiency losses.

**I hope this comprehensive analysis helps!**