Problem 24 - Calculus Early Transcendentals

Question

Calculate the work required to stretch the following springs \(0.4 \mathrm{m}\) from their equilibrium positions. Assume Hooke's law is obeyed. a. A spring that requires a force of \(50 \mathrm{N}\) to be stretched $0.1 \mathrm{m}$ from its equilibrium position. b. A spring that requires 2 J of work to be stretched \(0.1 \mathrm{m}\) from its equilibrium position.

Step-by-Step Solution

Answer

Question: Calculate the work required to stretch two different springs (a and b) 0.4 meters each. For spring a, the force required to stretch it 0.1 meter is 50 N. For spring b, the work required to stretch it 0.1 meter is 2 J. Answer: The work required to stretch spring a 0.4 meters is 40 J, and the work required to stretch spring b 0.4 meters is 32 J.

Step 1: Part a: Calculate the spring constant k

Using Hooke's Law (F = k * x), we can calculate the spring constant k as k = F / x. We know the force F is 50 N and the distance x is 0.1 m, so the spring constant is: k = \frac{50 \mathrm{N}}{0.1 \mathrm{m}} = 500 \frac{\mathrm{N}}{\mathrm{m}}

Step 2: Part a: Calculate the work required to stretch the spring 0.4 m

Now we can use the work formula (W = (1/2) * k * x^2) to calculate the work required to stretch the spring 0.4 m. We know the spring constant k is 500 N/m and the distance x is 0.4 m, so the work is: W = \frac{1}{2} * 500 \frac{\mathrm{N}}{\mathrm{m}} * (0.4 \mathrm{m})^2 = 40 \mathrm{J}

Step 3: Part b: Calculate the spring constant k from the work information

We are given the work required to stretch the spring 0.1 m is 2 J. We can rearrange the work formula (W = (1/2) * k * x^2) to solve for the spring constant k: k = \frac{2 * W}{x^2} = \frac{2 * 2 \mathrm{J}}{(0.1 \mathrm{m})^2} = 400 \frac{\mathrm{N}}{\mathrm{m}}

Step 4: Part b: Calculate the work required to stretch the spring 0.4 m

Now we can use the work formula (W = (1/2) * k * x^2) to calculate the work required to stretch the spring 0.4 m. We know the spring constant k is 400 N/m and the distance x is 0.4 m, so the work is: W = \frac{1}{2} * 400 \frac{\mathrm{N}}{\mathrm{m}} * (0.4 \mathrm{m})^2 = 32 \mathrm{J}