The centripetal force acting on a body of mass 4 kg moving in a circle of radius 0.3 m with a unifrom speed 5 m/s is calculated using the equation `F=(mv^(2))/r`. If m,v and r are measured to accuracies of 0.04 kg, 0.01 m/s and 0.009 m respectively, find the percentage error in the force?

Explanation

To find the percentage error in the centripetal force acting on a body, we can follow these steps:

### Step 1: Write down the formula for centripetal force
The centripetal force \( F \) is given by the formula:
\[ F = \frac{mv^2}{r} \]
where:
- \( m \) = mass of the body
- \( v \) = speed of the body
- \( r \) = radius of the circular path

### Step 2: Identify the given values
From the problem, we have:
- Mass \( m = 4 \, \text{kg} \)
- Speed \( v = 5 \, \text{m/s} \)
- Radius \( r = 0.3 \, \text{m} \)

The accuracies (uncertainties) in the measurements are:
- \( \Delta m = 0.04 \, \text{kg} \)
- \( \Delta v = 0.01 \, \text{m/s} \)
- \( \Delta r = 0.009 \, \text{m} \)

### Step 3: Calculate the percentage errors in the measurements
The percentage error in a quantity can be calculated using the formula:
\[ \text{Percentage Error} = \left( \frac{\Delta x}{x} \right) \times 100 \]

1. **Percentage error in mass \( m \)**:
\[ \text{Percentage error in } m = \left( \frac{\Delta m}{m} \right) \times 100 = \left( \frac{0.04}{4} \right) \times 100 = 1\% \]

2. **Percentage error in speed \( v \)**:
\[ \text{Percentage error in } v = \left( \frac{\Delta v}{v} \right) \times 100 = \left( \frac{0.01}{5} \right) \times 100 = 0.2\% \]

3. **Percentage error in radius \( r \)**:
\[ \text{Percentage error in } r = \left( \frac{\Delta r}{r} \right) \times 100 = \left( \frac{0.009}{0.3} \right) \times 100 = 3\% \]

### Step 4: Calculate the total percentage error in force \( F \)
The formula for the total percentage error in \( F \) is given by:
\[ \text{Percentage error in } F = \text{Percentage error in } m + 2 \times \text{Percentage error in } v + \text{Percentage error in } r \]
Substituting the values we calculated:
\[ \text{Percentage error in } F = 1\% + 2 \times 0.2\% + 3\% \]
\[ = 1\% + 0.4\% + 3\% = 4.4\% \]

### Step 5: Conclusion
The percentage error in the centripetal force is:
\[ \text{Percentage error in } F = 4.4\% \]

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