A flywheel is rotating at the rate of 100 rpm and slows down at a constant rate of `1 " rad"//"s"^2`. Calculate the time required to stop the flywheel and the number of rotations made by the flywheel before coming to rest?

Explanation

To solve the problem step by step, we will first convert the given values and then use the appropriate equations of motion for rotational motion.

### Step 1: Convert the initial angular velocity from rpm to rad/s
The initial angular velocity (ω₀) is given as 100 revolutions per minute (rpm). We need to convert this to radians per second (rad/s).

\[ \text{Angular velocity in rad/s} = \text{rpm} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} \]

\[ \omega_0 = 100 \, \text{rpm} \times \frac{2\pi}{60} = \frac{100 \times 2\pi}{60} = \frac{100\pi}{30} = \frac{10\pi}{3} \, \text{rad/s} \]

### Step 2: Identify the angular deceleration
The angular deceleration (α) is given as -1 rad/s² (negative because it is slowing down).

### Step 3: Use the equation of motion to find the time to stop
We know that the final angular velocity (ω) when the flywheel stops is 0 rad/s. We can use the equation:

\[ \omega = \omega_0 + \alpha t \]

Substituting the known values:

\[ 0 = \frac{10\pi}{3} + (-1) t \]

Rearranging gives:

\[ t = \frac{10\pi}{3} \]

Calculating the value:

\[ t \approx \frac{10 \times 3.14}{3} \approx 10.47 \, \text{s} \]

### Step 4: Calculate the angle covered before coming to rest
We can use the equation for angular displacement (θ):

\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]

Substituting the known values:

\[ \theta = \left(\frac{10\pi}{3}\right) \left(10.47\right) + \frac{1}{2} \left(-1\right) (10.47)^2 \]

Calculating each term:

1. First term:
\[ \frac{10\pi}{3} \times 10.47 \approx \frac{10 \times 3.14}{3} \times 10.47 \approx 109.1 \, \text{rad} \]

2. Second term:
\[ \frac{1}{2} \times (-1) \times (10.47)^2 \approx -\frac{1}{2} \times 109.6 \approx -54.8 \, \text{rad} \]

Now, combining both terms:

\[ \theta \approx 109.1 - 54.8 \approx 54.3 \, \text{rad} \]

### Step 5: Calculate the number of revolutions
To find the number of revolutions (n), we use:

\[ n = \frac{\theta}{2\pi} \]

Substituting the value of θ:

\[ n = \frac{54.3}{2\pi} \approx \frac{54.3}{6.28} \approx 8.65 \, \text{revolutions} \]

### Final Answers
- Time required to stop the flywheel: **10.47 seconds**
- Number of rotations made by the flywheel before coming to rest: **8.65 revolutions**

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