A shaft initially rotating at 1725 rpm is brought to rest uniformly in 20s. The number of revolutions that the shaft will make during this time is

Options

A. 1680
B. 575
C. 287
D. 627

Explanation

To solve the problem step by step, we will follow the outlined approach in the video transcript.

### Step 1: Convert RPM to Radians per Second
The initial angular velocity (ω₀) is given as 1725 rpm. We need to convert this to radians per second using the conversion factor \( \frac{2\pi \text{ radians}}{1 \text{ revolution}} \) and \( \frac{1 \text{ minute}}{60 \text{ seconds}} \).

\[ \omega_0 = 1725 \, \text{rpm} = 1725 \times \frac{2\pi}{60} \, \text{rad/s} \]

### Step 2: Calculate Angular Deceleration (α)
Since the shaft is brought to rest uniformly in 20 seconds, the final angular velocity (ω) is 0. We can use the first equation of motion for rotation:

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

Substituting the known values:

\[ 0 = \omega_0 + \alpha \cdot 20 \]

From this, we can solve for α:

\[ \alpha = -\frac{\omega_0}{20} \]

### Step 3: Calculate the Angular Displacement (θ)
We can use the second equation of motion for rotation to find the angular displacement:

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

Substituting for α from Step 2:

\[ \theta = \omega_0 \cdot 20 + \frac{1}{2} \left(-\frac{\omega_0}{20}\right) \cdot (20^2) \]

Simplifying this expression:

\[ \theta = 20\omega_0 - \frac{1}{2} \cdot \frac{\omega_0}{20} \cdot 400 \]
\[ \theta = 20\omega_0 - 10\omega_0 \]
\[ \theta = 10\omega_0 \]

### Step 4: Substitute ω₀ and Calculate θ
Now substitute the value of ω₀ from Step 1 into the equation for θ:

\[ \theta = 10 \left(1725 \times \frac{2\pi}{60}\right) \]

### Step 5: Calculate the Total Number of Revolutions
To find the total number of revolutions (N), we divide the angular displacement θ by \( 2\pi \):

\[ N = \frac{\theta}{2\pi} = \frac{10 \left(1725 \times \frac{2\pi}{60}\right)}{2\pi} \]

The \( 2\pi \) cancels out:

\[ N = \frac{10 \times 1725}{60} \]
\[ N = \frac{17250}{60} = 287.5 \]

### Final Answer
The total number of revolutions made by the shaft is **287.5 revolutions**.

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