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The image contains a physics problem involving a cylindrical reel, a cord, and a hanging bucket. Let's extract the key information and solve the questions step by step.

### Problem Details:
1. **Given:**
   - Radius of the reel, \( r = 0.45 \, \text{m} \)
   - Mass of the reel, \( M = 50 \, \text{kg} \)
   - Moment of Inertia of the reel, \( I = 5 \, \text{kg} \cdot \text{m}^2 \)
   - Mass of the bucket, \( m = 3 \, \text{kg} \)

2. **Questions:**
   - (a) Draw the free body diagram for the cylindrical reel and the bucket.
   - (b) Calculate the linear acceleration of the falling bucket.
   - (c) Calculate the angular acceleration of the reel.
   - (d) Calculate the tension in the cord.
   - (e) Calculate how far the bucket has moved.

### Solutions:

Let's start solving the problems step-by-step:

#### (a) Free Body Diagrams
1. **For the bucket:**
   - Downward force: Weight \( W = m \cdot g \)
   - Upward force: Tension \( T \) in the cord.

2. **For the reel:**
   - Torque due to tension in the cord: \( \tau = T \cdot r \)
   - Moment of inertia \( I \) of the reel.
   - Angular acceleration \( \alpha \) of the reel.

#### (b) Linear Acceleration of the Falling Bucket
The forces on the bucket can be analyzed using Newton's second law:

\[
m \cdot g - T = m \cdot a
\]

The torque on the reel can be expressed as:

\[
T \cdot r = I \cdot \alpha
\]

Since the linear acceleration \( a \) of the bucket and the angular acceleration \( \alpha \) of the reel are related by:

\[
a = r \cdot \alpha
\]

Substitute \( \alpha \) in the torque equation:

\[
T \cdot r = I \cdot \frac{a}{r} \implies T = \frac{I \cdot a}{r^2}
\]

Substitute \( T \) back into the equation for the bucket:

\[
m \cdot g - \frac{I \cdot a}{r^2} = m \cdot a
\]

Solve for \( a \):

\[
a = \frac{m \cdot g}{m + \frac{I}{r^2}}
\]

Plugging in the values:

\[
a = \frac{3 \cdot 9.81}{3 + \frac{5}{0.45^2}}
\]

#### (c) Angular Acceleration of the Reel
\[
\alpha = \frac{a}{r}
\]

#### (d) Tension in the Cord
\[
T = m \cdot (g - a)
\]

#### (e) Distance Moved by the Bucket
Using the kinematic equation:

\[
s = \frac{1}{2} a t^2
\]

where \( s \) is the distance moved, \( a \) is the linear acceleration, and \( t \) is the time taken.

Would you like me to calculate these values step-by-step? Or do you need help with a specific part of the problem?

A cylindrical reel with a hanging bucket is given, and several physical properties like radius, mass of the reel, and moment of inertia are provided. The questions ask for the free body diagram, linear and angular acceleration of the bucket and reel, tension in the cord, and distance the bucket has moved.

This problem involves calculating the dynamics of a cylindrical reel and a hanging bucket. Learn how to draw free body diagrams, calculate linear and angular accelerations, tension in the cord, and the distance the bucket has moved using key concepts from mechanics and rotational dynamics. Suitable for undergraduate students.

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