Two springs P and Q both obey Hooke’s law. They have spring constants 2k and k respectively.The springs are stretched, separately, by a force that is gradually increased from zero up to a certain maximum value, the same for each spring. The work done in stretching spring P is WP,and the work done in stretching spring Q is WQ.How is WP related to WQ?

Question

Two springs P and Q both obey Hooke’s law. They have spring constants 2k and k respectively.The springs are stretched, separately, by a force that is gradually increased from zero up to a certain maximum value, the same for each spring. The work done in stretching spring P is WP,and the work done in stretching spring Q is WQ.How is WP related to WQ?

A: Wp=0.25Wq
B: Wp=0.5Wq
C: Wp=2Wq
D: Wp=4Wq

I'm not sure of the answer and I need help.
Couldn't find the marking scheme :(
Its from nov 2002 paper.

Thanks!

Answer 1

The work done in stretching a spring is given by the formula W = (1/2)k*x^2, where k is the spring constant and x is the displacement of the spring.

Since the force applied to the springs is the same, we know that F = k * x. We can rewrite the applied force F as k * x = kQ * xQ = 2k * xP, where xP and xQ are the displacements of springs P and Q, respectively.

Now, we can find the ratio between the displacements of the springs by dividing the force equations:

xQ / xP = (2k * xP) / (kQ * xQ) = 2 / 1, so xQ = 2 * xP.

Now we can find the ratio between the work done on the springs:

Wp / Wq = ((1/2) * 2k * xP^2) / ((1/2) * k * (2 * xP)^2)
= (2k * xP^2) / (k * 4 * xP^2) = 2/4 = 1/2

So the answer is that Wp is half of Wq, i.e., Wp = 0.5 * Wq. The correct choice is option B: Wp = 0.5Wq.

Answer 2

To determine the relationship between WP and WQ, we can start by considering the work done in stretching each spring.

The work done in stretching a spring is given by the formula:
Work = (1/2)kx^2

Where:
- Work is the amount of energy transferred to the spring (measured in joules)
- k is the spring constant (measured in newtons per meter)
- x is the displacement of the spring from its equilibrium position (measured in meters)

In this case, we have two springs, P and Q, with spring constants 2k and k, respectively.

Let's assume that the same maximum force is applied to both springs, causing both of them to be stretched by the same amount, x.

For spring P, the work done (WP) can be calculated as:
WP = (1/2)(2k)(x^2) = k(x^2)

For spring Q, the work done (WQ) can be calculated as:
WQ = (1/2)(k)(x^2) = (1/2)k(x^2)

Now, we can compare WP and WQ:

WP = k(x^2)
WQ = (1/2)k(x^2)

By dividing WP by WQ, we get:
WP/WQ = (k(x^2))/(1/2)k(x^2)

The x^2 terms cancel out, leaving us with:
WP/WQ = 2/1

Simplifying the right-hand side, we have:
WP/WQ = 2

Therefore, the correct relationship between WP and WQ is:
WP = 2WQ

Hence, the answer is C: WP = 2WQ

Answer 3

To answer this question, we can use the formula for the work done in stretching a spring:

Work = (1/2)kx^2

where k is the spring constant and x is the displacement.

For spring P:
Work done = (1/2)(2k)x^2 = kx^2

For spring Q:
Work done = (1/2)(k)x^2 = (1/2)kx^2

Now, we can compare the work done for each spring.

WP = kx^2
WQ = (1/2)kx^2

We can see that WP is twice as large as WQ. So, the correct answer is C: WP = 2WQ.