Question

1. a force of 10 n is required to stretch a spring a distance of 0.40 m from its rest position. what force (in newtons) is required to stretch the same spring the following distances?

a. twice the distance

b. three times the distance

c. one - half the distance

Explanation:

To solve this problem, we use Hooke's Law, which states that the force \( F \) required to stretch a spring is proportional to the displacement \( x \) from its rest position, i.e., \( F = kx \), where \( k \) is the spring constant. First, we find the spring constant \( k \) using the given information.

Step 1: Find the spring constant \( k \)

Given \( F = 10 \, \text{N} \) and \( x = 0.40 \, \text{m} \), from \( F = kx \), we can solve for \( k \):
\[
k = \frac{F}{x} = \frac{10 \, \text{N}}{0.40 \, \text{m}} = 25 \, \text{N/m}
\]

Part (a): Twice the distance

The new distance \( x_a = 2 \times 0.40 \, \text{m} = 0.80 \, \text{m} \). Using \( F = kx \):
\[
F_a = kx_a = 25 \, \text{N/m} \times 0.80 \, \text{m} = 20 \, \text{N}
\]

Part (b): Three times the distance

The new distance \( x_b = 3 \times 0.40 \, \text{m} = 1.20 \, \text{m} \). Using \( F = kx \):
\[
F_b = kx_b = 25 \, \text{N/m} \times 1.20 \, \text{m} = 30 \, \text{N}
\]

Part (c): One - half the distance

The new distance \( x_c=\frac{1}{2}\times0.40 \, \text{m} = 0.20 \, \text{m} \). Using \( F = kx \):
\[
F_c = kx_c = 25 \, \text{N/m} \times 0.20 \, \text{m} = 5 \, \text{N}
\]

Final Answers

a. The force required to stretch the spring twice the distance is \(\boldsymbol{20 \, \text{N}}\).
b. The force required to stretch the spring three times the distance is \(\boldsymbol{30 \, \text{N}}\).
c. The force required to stretch the spring one - half the distance is \(\boldsymbol{5 \, \text{N}}\).

Answer:

To solve this problem, we use Hooke's Law, which states that the force \( F \) required to stretch a spring is proportional to the displacement \( x \) from its rest position, i.e., \( F = kx \), where \( k \) is the spring constant. First, we find the spring constant \( k \) using the given information.

Step 1: Find the spring constant \( k \)

Given \( F = 10 \, \text{N} \) and \( x = 0.40 \, \text{m} \), from \( F = kx \), we can solve for \( k \):
\[
k = \frac{F}{x} = \frac{10 \, \text{N}}{0.40 \, \text{m}} = 25 \, \text{N/m}
\]

Part (a): Twice the distance

The new distance \( x_a = 2 \times 0.40 \, \text{m} = 0.80 \, \text{m} \). Using \( F = kx \):
\[
F_a = kx_a = 25 \, \text{N/m} \times 0.80 \, \text{m} = 20 \, \text{N}
\]

Part (b): Three times the distance

The new distance \( x_b = 3 \times 0.40 \, \text{m} = 1.20 \, \text{m} \). Using \( F = kx \):
\[
F_b = kx_b = 25 \, \text{N/m} \times 1.20 \, \text{m} = 30 \, \text{N}
\]

Part (c): One - half the distance

The new distance \( x_c=\frac{1}{2}\times0.40 \, \text{m} = 0.20 \, \text{m} \). Using \( F = kx \):
\[
F_c = kx_c = 25 \, \text{N/m} \times 0.20 \, \text{m} = 5 \, \text{N}
\]

Final Answers

a. The force required to stretch the spring twice the distance is \(\boldsymbol{20 \, \text{N}}\).
b. The force required to stretch the spring three times the distance is \(\boldsymbol{30 \, \text{N}}\).
c. The force required to stretch the spring one - half the distance is \(\boldsymbol{5 \, \text{N}}\).