Amplitude of vibration is `A = (F_(0))/(p-q+r)` . Resonance will occur when

Options

A. `p=0,q=r`
B. `p=q=r`
C. `p= -r, q=0`
D. Both (1) & (3)

Answer

To determine when resonance occurs in the given equation for amplitude \( A = \frac{F_0}{p - q + r} \), we need to analyze the conditions under which the amplitude becomes maximum, or theoretically infinite.

### Step-by-Step Solution:

1. **Understanding Resonance**: Resonance occurs when the amplitude of the oscillation reaches its maximum value. Mathematically, this can be expressed as \( A_{max} = \infty \).

2. **Setting Up the Equation**: From the given equation, we have:
\[ A = \frac{F_0}{p - q + r} \]
For resonance (maximum amplitude), we set:
\[ \frac{F_0}{p - q + r} = \infty \]

3. **Analyzing the Condition**: The above equation implies that the denominator must approach zero:
\[ p - q + r = 0 \]

4. **Rearranging the Equation**: We can rearrange the equation to find the condition for resonance:
\[ p + r = q \]

5. **Testing Options**: We will now test the provided options to see which satisfy the condition \( p - q + r = 0 \).

- **Option 1**: \( p = 0, q = r \)
\[ 0 - q + q = 0 \quad \text{(Valid)} \]

- **Option 2**: \( p = q = r \)
\[ p - p + p = p \quad \text{(Not valid, as it does not equal 0)} \]

- **Option 3**: \( p = -r, q = 0 \)
\[ -r - 0 + r = 0 \quad \text{(Valid)} \]

- **Option 4**: Both options 1 and 3 are correct.

6. **Conclusion**: Therefore, resonance will occur under the conditions specified in options 1 and 3.

### Final Answer:
Resonance will occur when \( p - q + r = 0 \). The valid options are 1 and 3.

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