A force varies with time and traces a sine curve. The peak value of force is `F_(0)`. The average force is given by

Options

A. `F_(0)`
B. `sqrt(F_(0))`
C. `sqrt(2).F_(0)`
D. zero

Explanation

To find the average force when the force varies with time and traces a sine curve, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Sine Curve**: The force \( F(t) \) varies with time and traces a sine curve. The sine function oscillates between a maximum value (peak) of \( F_0 \) and a minimum value of \( -F_0 \).

2. **Identify the Average Force Formula**: The average force over a time period \( T \) can be calculated using the formula:
\[ \text{Average Force} = \frac{1}{T} \int_0^T F(t) \, dt \]

3. **Set Up the Integral**: Since \( F(t) \) is a sine function, we can express it as:
\[ F(t) = F_0 \sin(\omega t) \]
where \( \omega \) is the angular frequency of the sine wave. The average force over one complete cycle (period \( T = \frac{2\pi}{\omega} \)) is:
\[ \text{Average Force} = \frac{1}{T} \int_0^T F_0 \sin(\omega t) \, dt \]

4. **Evaluate the Integral**: The integral of \( \sin(\omega t) \) over one complete cycle is zero because the positive area under the curve from \( 0 \) to \( \pi \) cancels out with the negative area from \( \pi \) to \( 2\pi \):
\[ \int_0^T \sin(\omega t) \, dt = 0 \]

5. **Calculate the Average Force**: Substituting the integral result back into the average force formula gives:
\[ \text{Average Force} = \frac{1}{T} \cdot 0 = 0 \]

6. **Conclusion**: Therefore, the average force over a complete cycle of the sine wave is:
\[ \text{Average Force} = 0 \]

### Final Answer:
The average force is \( 0 \).

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