A stone tied to the end of string 1m long is whirled in a horizontal circle with a constant speed. If the stone makes 22 revolution in 44s, What is the magnitude and direction of acceleration of the ston is ?

Options

A. `(pi^(2))/4 ms^(-2)` and direction along the radius towards the centre
B. `pi^(2)ms^(-2)` and direction along the radius away from centre
C. `pi^(2)ms^(-2)` and direction along the radius towards the centre
D. `pi^(2)ms^(-2)` and direction along the tangent to the circle.

Explanation

To solve the problem of finding the magnitude and direction of the acceleration of a stone tied to a string and whirled in a horizontal circle, we can follow these steps:

### Step 1: Calculate the Frequency of Rotation
The stone makes 22 revolutions in 44 seconds. The frequency \( f \) (in hertz) can be calculated using the formula:
\[ f = \frac{\text{Number of revolutions}}{\text{Time in seconds}} = \frac{22}{44} = 0.5 \, \text{Hz} \]

### Step 2: Calculate the Angular Velocity
Angular velocity \( \omega \) (in radians per second) is related to frequency by the formula:
\[ \omega = 2\pi f \]
Substituting the value of \( f \):
\[ \omega = 2\pi \times 0.5 = \pi \, \text{rad/s} \]

### Step 3: Calculate the Centripetal Acceleration
Centripetal acceleration \( a_c \) is given by the formula:
\[ a_c = \frac{v^2}{r} \]
where \( v \) is the linear velocity and \( r \) is the radius of the circle. The linear velocity \( v \) can be calculated using:
\[ v = r \cdot \omega \]
Given that the radius \( r \) is 1 meter:
\[ v = 1 \cdot \pi = \pi \, \text{m/s} \]
Now substituting \( v \) into the centripetal acceleration formula:
\[ a_c = \frac{(\pi)^2}{1} = \pi^2 \, \text{m/s}^2 \]

### Step 4: Determine the Direction of Acceleration
The direction of the centripetal acceleration is always directed towards the center of the circular path. Therefore, the direction of the acceleration of the stone is inward, towards the center of the circle.

### Final Answer
The magnitude of the acceleration of the stone is \( \pi^2 \, \text{m/s}^2 \) and the direction is towards the center of the circle.

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