If and object completes 49 revolutions in a minute around a circular path with a speed ` 7 ms ^(-1)` find the radius of the path.

Answer

To find the radius of the circular path for an object completing 49 revolutions in one minute with a speed of 7 m/s, we can follow these steps:

### Step 1: Determine the time period of one revolution
The object completes 49 revolutions in 1 minute (60 seconds). Therefore, the time period \( T \) for one revolution can be calculated as:
\[ T = \frac{60 \text{ seconds}}{49 \text{ revolutions}} = \frac{60}{49} \text{ seconds} \]

### Step 2: Calculate the angular velocity \( \omega \)
The angular velocity \( \omega \) can be calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the value of \( T \):
\[ \omega = \frac{2\pi}{\frac{60}{49}} = 2\pi \cdot \frac{49}{60} \]
Using \( \pi \approx \frac{22}{7} \):
\[ \omega = 2 \cdot \frac{22}{7} \cdot \frac{49}{60} \]
Calculating this gives:
\[ \omega = \frac{44 \cdot 49}{7 \cdot 60} = \frac{2156}{420} = \frac{1078}{210} \text{ rad/s} \]

### Step 3: Relate linear speed \( v \) to angular velocity \( \omega \) and radius \( r \)
The relationship between linear speed \( v \), angular velocity \( \omega \), and radius \( r \) is given by:
\[ v = \omega r \]
We know \( v = 7 \text{ m/s} \), so we can rearrange the formula to find \( r \):
\[ r = \frac{v}{\omega} \]

### Step 4: Substitute the values to find \( r \)
Substituting the known values:
\[ r = \frac{7}{\frac{1078}{210}} = 7 \cdot \frac{210}{1078} \]
Calculating this gives:
\[ r = \frac{1470}{1078} \approx 1.36 \text{ meters} \]

### Final Answer
Thus, the radius of the path is approximately:
\[ r \approx 1.36 \text{ meters} \]

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