Questions: Suppose that 4 J of work is needed to stretch a spring from its natural length of 32 cm to a length of 49 cm. (a) How much work is needed to stretch the spring from 36 cm to 44 cm? (Round your answer to two decimal places.) J (b) How far beyond its natural length will a force of 35 N keep the spring stretched? (Round your answer one decimal place.) cm

Solution Steps

• Use Hooke's Law for work done on a spring: \[ W = \frac{1}{2} k x^2 \] where \( W \) is the work done, \( k \) is the spring constant, and \( x \) is the displacement from the natural length.

• Given: \( W = 4 \, \text{J} \), \( x = 49 \, \text{cm} - 32 \, \text{cm} = 17 \, \text{cm} = 0.17 \, \text{m} \).

• Solve for \( k \): \[ 4 = \frac{1}{2} k (0.17)^2 \] \[ k = \frac{4 \times 2}{0.17^2} \]

• Find the work needed to stretch the spring from 36 cm to 44 cm.

• Displacement from natural length: \[ x_1 = 36 \, \text{cm} - 32 \, \text{cm} = 4 \, \text{cm} = 0.04 \, \text{m} \] \[ x_2 = 44 \, \text{cm} - 32 \, \text{cm} = 12 \, \text{cm} = 0.12 \, \text{m} \]

• Work done from 36 cm to 44 cm: \[ W = \frac{1}{2} k (x_2^2 - x_1^2) \]

• Use Hooke's Law for force: \[ F = kx \]

• Given: \( F = 35 \, \text{N} \).

• Solve for \( x \): \[ x = \frac{F}{k} \]

• Convert \( x \) to cm for the final answer.

Final Answer