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Step-by-Step Solution
Step 1: Identify the Given Information
• The length of the string (which is the radius of the circular path) is 180 cm = 1.8 m.
• The stone makes 28 revolutions per minute.
• We need to find the value of x if the acceleration a = \frac{1936}{x}\,\text{m/s}^2 .
We are given \pi = \frac{22}{7} .
Step 2: Express Angular Velocity
The stone completes 28 revolutions in 60 seconds. Each revolution corresponds to 2\pi radians. Hence, the angular velocity \omega is
\omega = \frac{\text{(Number of revolutions)} \times 2\pi}{\text{(Time in seconds)}}
= \frac{28 \times 2 \times \frac{22}{7}}{60}.
Step 3: Write the Formula for Centripetal Acceleration
For uniform circular motion, centripetal acceleration a is given by
a = \omega^2 R,
where R is the radius (1.8 m) and \omega is the angular velocity.
Step 4: Substitute the Values
Substitute \omega and R = 1.8 \text{ m} into the formula:
\begin{aligned}
a &= \left(\frac{28 \times 2 \times \frac{22}{7}}{60}\right)^2 \times 1.8.
\end{aligned}
Step 5: Simplify the Expression
Carrying out the multiplication and simplification carefully, we obtain:
a = \frac{1936}{125} \,\text{m/s}^2.
Hence, comparing a = \frac{1936}{x} \,\text{m/s}^2 with \frac{1936}{125} \,\text{m/s}^2 , we see that
x = 125.
Final Answer
x = 125.
