A body rotating at 20 rad/s is acted upon by a constant torque providing it a deceleration of `2(rad)/s^2`. At what time will the body have kinetic energy same as the initial value if the torque continues to act?

Explanation

To solve the problem step by step, we will analyze the motion of the body under the influence of torque and calculate the time at which the kinetic energy of the body becomes equal to its initial value.

### Step 1: Identify the initial conditions
The initial angular velocity (ω₀) of the body is given as:
\[ \omega_0 = 20 \, \text{rad/s} \]

The angular deceleration (α) due to the torque is:
\[ \alpha = -2 \, \text{rad/s}^2 \]

### Step 2: Calculate the initial kinetic energy
The initial kinetic energy (K) of the rotating body can be calculated using the formula:
\[ K = \frac{1}{2} I \omega_0^2 \]
Since we are interested in the ratio of kinetic energies, we can represent the initial kinetic energy as:
\[ K = \frac{1}{2} I (20)^2 = 200 I \]

### Step 3: Determine the time taken to come to rest
To find the time taken for the body to come to rest, we can use the equation of motion:
\[ \omega = \omega_0 + \alpha t \]
Setting the final angular velocity (ω) to 0 (when the body comes to rest):
\[ 0 = 20 + (-2)t \]
Solving for t:
\[ 2t = 20 \]
\[ t = 10 \, \text{seconds} \]

### Step 4: Analyze motion after coming to rest
After 10 seconds, the body comes to rest (ω = 0). The torque continues to act, causing the body to rotate in the opposite direction.

### Step 5: Determine the condition for kinetic energy to be equal to the initial value
The kinetic energy will be the same as the initial value when the angular velocity becomes:
\[ \omega = -\omega_0 = -20 \, \text{rad/s} \]

### Step 6: Calculate the time taken to reach -20 rad/s
Using the equation of motion again:
\[ \omega = \omega_0 + \alpha t' \]
Here, the initial angular velocity (ω₀) is now 0 (since it just came to rest), and we want to find the time (t') to reach -20 rad/s:
\[ -20 = 0 + (-2)t' \]
Solving for t':
\[ -20 = -2t' \]
\[ t' = \frac{20}{2} = 10 \, \text{seconds} \]

### Step 7: Calculate the total time
The total time (T) taken for the body to have the same kinetic energy as the initial value is the sum of the time taken to come to rest and the time taken to reach -20 rad/s:
\[ T = t + t' = 10 + 10 = 20 \, \text{seconds} \]

### Final Answer
The total time at which the body will have kinetic energy equal to its initial value is:
\[ T = 20 \, \text{seconds} \]

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