Question

Two metallic wires \( P \) and \( Q \) have the same volume and are made of the same material. If their areas of cross-section are in the ratio \( 4 : 1 \) and a force \( F_1 \) is applied to \( P \), producing an extension of \( \Delta l \), the force required to produce the same extension in \( Q \) is \( F_2 \).The value of \( \frac{F_1}{F_2} \) is _______.

Answer 1

Correct Answer: 16

Given that wires \( P \) and \( Q \) have equal volumes and are composed of the same material, their volume \( V \) can be defined as \( V = A \times l \), where \( A \) denotes the cross-sectional area and \( l \) represents the length.

Let \( A_P \) and \( A_Q \) be the cross-sectional areas of \( P \) and \( Q \), respectively, and \( l_P \) and \( l_Q \) be their corresponding lengths. With the given ratio \( \frac{A_P}{A_Q} = 4 \), it follows that:

\( A_P = 4A_Q \)

Since the volumes are equal:

\( A_P \cdot l_P = A_Q \cdot l_Q \)

Upon substituting \( A_P = 4A_Q \):

\( 4A_Q \cdot l_P = A_Q \cdot l_Q \)

This leads to the relationship \( l_Q = 4l_P \).

Considering the formula for Young's modulus:

\( \text{Stress} = \frac{F}{A} \) and \( \text{Strain} = \frac{\Delta l}{l} \)

Young's modulus is \( Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/l}{A/\Delta l} \), which can be rearranged to \( F = \frac{YA\Delta l}{l} \).

For wire \( P \): \( F_1 = \frac{YA_P \Delta l}{l_P} \)

For wire \( Q \) to exhibit the same extension \( \Delta l \): \( F_2 = \frac{YA_Q \Delta l}{l_Q} \)

Substituting \( A_Q = \frac{A_P}{4} \) and \( l_Q = 4l_P \):

\( F_2 = \frac{Y(A_P/4)\Delta l}{4l_P} = \frac{YA_P \Delta l}{16l_P} \)

Therefore, the ratio \( \frac{F_1}{F_2} = \frac{\frac{YA_P \Delta l}{l_P}}{\frac{YA_P \Delta l}{16l_P}} = 16 \).

The calculated value is \( 16 \).