A wheel starting from rest is uniformly acceleration with angular acceleration of `4 "rad"//s^(2)` for 10 seconds . It is then allowed to rotate uniformly for next 10 seconds and finally brought to rest in next 10 seconds by uniform angular retardation. Total angle rotated is (100 n) radian. Find value of n.

Answer

To solve the problem step by step, we will analyze the motion of the wheel in three distinct phases: acceleration, uniform rotation, and deceleration.

### Step 1: Calculate the final angular velocity after the first 10 seconds of acceleration.

The wheel starts from rest, so the initial angular velocity (ω₀) is 0. The angular acceleration (α) is given as 4 rad/s², and the time (t) is 10 seconds.

Using the formula:
\[ \omega = \omega_0 + \alpha t \]
Substituting the values:
\[ \omega = 0 + (4 \, \text{rad/s}^2)(10 \, \text{s}) = 40 \, \text{rad/s} \]

### Step 2: Calculate the angular displacement during the first 10 seconds.

We can use the formula for angular displacement (θ) under constant acceleration:
\[ \theta_1 = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Substituting the values:
\[ \theta_1 = 0 \cdot 10 + \frac{1}{2} \cdot 4 \cdot (10)^2 = \frac{1}{2} \cdot 4 \cdot 100 = 200 \, \text{radians} \]

### Step 3: Calculate the angular displacement during the next 10 seconds of uniform rotation.

During uniform rotation, the angular acceleration is 0, and the initial angular velocity is the final angular velocity from the previous phase (40 rad/s).

Using the formula:
\[ \theta_2 = \omega_0 t \]
Substituting the values:
\[ \theta_2 = 40 \cdot 10 = 400 \, \text{radians} \]

### Step 4: Calculate the angular displacement during the last 10 seconds of deceleration.

In this phase, the wheel comes to rest, so the final angular velocity (ω_f) is 0. The initial angular velocity (ω₀) is still 40 rad/s, and we need to find the angular retardation (α).

Using the equation:
\[ \omega_f = \omega_0 + \alpha t \]
Substituting the values:
\[ 0 = 40 + \alpha (10) \implies \alpha = -4 \, \text{rad/s}^2 \]

Now, we can calculate the angular displacement during this phase:
\[ \theta_3 = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Substituting the values:
\[ \theta_3 = 40 \cdot 10 + \frac{1}{2} \cdot (-4) \cdot (10)^2 \]
Calculating:
\[ \theta_3 = 400 - 200 = 200 \, \text{radians} \]

### Step 5: Calculate the total angular displacement.

Now, we can find the total angular displacement (θ_total):
\[ \theta_{\text{total}} = \theta_1 + \theta_2 + \theta_3 = 200 + 400 + 200 = 800 \, \text{radians} \]

### Step 6: Relate the total angle to the given expression.

According to the problem, the total angle rotated is given by:
\[ 100n = 800 \]
To find n, we divide both sides by 100:
\[ n = \frac{800}{100} = 8 \]

### Final Answer:
The value of n is **8**.

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