The small particle of mass `m` is given an initial high velocity in the horizontal plane and winds its cord around the fixed vertical shaft of radius `1m`. All motion occurs essentially in the horizontal plane. If the angular velocity of the cord is `0.8` rad/s when the distance from the particle to the tangency point is `5m`, determine the angular velocity `omega` (in rad//s) of the cord after it has turned through an angle 1 rad.

Explanation

To solve the problem step by step, we will analyze the motion of the particle as it winds the cord around the vertical shaft.

### Step 1: Understand the Initial Conditions
- The initial angular velocity of the cord, \( \omega_0 = 0.8 \, \text{rad/s} \).
- The initial distance from the particle to the tangency point (radius of the circular path), \( R = 5 \, \text{m} \).
- The radius of the shaft, \( a = 1 \, \text{m} \).
- The angle through which the cord has turned, \( \theta = 1 \, \text{rad} \).

### Step 2: Relate Linear and Angular Velocity
The linear velocity \( V \) of the particle can be expressed in terms of the angular velocity and radius:
\[ V = \omega_0 \cdot R \]
Substituting the known values:
\[ V = 0.8 \, \text{rad/s} \cdot 5 \, \text{m} = 4 \, \text{m/s} \]

### Step 3: Determine the New Radius After the Cord Turns
After the cord has turned through an angle \( \theta = 1 \, \text{rad} \), the new radius \( R' \) can be calculated as:
\[ R' = R - a \cdot \theta \]
Substituting the known values:
\[ R' = 5 \, \text{m} - 1 \, \text{m} \cdot 1 \, \text{rad} = 5 \, \text{m} - 1 \, \text{m} = 4 \, \text{m} \]

### Step 4: Calculate the New Angular Velocity
The new angular velocity \( \omega \) can be found using the relationship between linear velocity and the new radius:
\[ \omega = \frac{V}{R'} \]
Substituting the values we have:
\[ \omega = \frac{4 \, \text{m/s}}{4 \, \text{m}} = 1 \, \text{rad/s} \]

### Final Answer
The angular velocity \( \omega \) of the cord after it has turned through an angle of 1 radian is:
\[ \omega = 1 \, \text{rad/s} \]

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