Question

A spring has a natural length of 30 cm. If a 24-N force is required to keep it stretched to a length of 40 cm, how much work W is required to stretch it from 30 cm to 35 cm? (Round your answer to two decimal places.)

W = J

If 0.6 J of work are needed to stretch a spring from 9 cm to 11 cm and 1 J are needed to stretch it from 11 cm to 13 cm, what is the natural length of the spring?

Answer 1

To stretch the spring from 30 cm to 35 cm, 1.2 J of work is required. The natural length of the spring can be found using the work required to stretch it from 9 cm to 13 cm, which is 1.6 J. By solving the equation with the known values of the spring constant and work, we can determine the natural length of the spring.

Step-by-step explanation:

To find the work required to stretch the spring from 30 cm to 35 cm, we need to calculate the difference in the quantity ½kx² at the end points. The work done to stretch the spring from 30 cm to 40 cm is 24 N * (40 cm - 30 cm) = 240 N·cm = 2.4 N·m = 2.4 J. Similarly, the work required to stretch the spring from 30 cm to 35 cm is 24 N * (35 cm - 30 cm) = 120 N·cm = 1.2 N·m = 1.2 J.

The natural length of a spring can be found using the work required to stretch the spring. From the given information, we can see that the work to stretch the spring from 9 cm to 11 cm is 0.6 J, and the work to stretch the spring from 11 cm to 13 cm is 1 J. Therefore, the work to stretch the spring from 9 cm to 13 cm is 0.6 J + 1 J = 1.6 J.

From the relationship between work and displacement, we know that the work done is equal to the change in potential energy. Therefore, the change in potential energy from the natural length to 13 cm is 1.6 J. Since the work required to stretch a spring depends on the square of the displacement from equilibrium, we can write 1.6 J = ½k(13 cm - natural length)².

From here, we can solve for the natural length:

1.6 J = ½k(13 cm - natural length)²

1.6 J = ½k(169 cm² - 26 cm * natural length + natural length²)

1.6 J = 84.5 cm² * k - 13 cm * k * natural length + 0.5 k * natural length²

1.6 J - 84.5 cm² * k = k(natural length² - 13 cm * natural length)

(1.6 J - 84.5 cm² * k) / k = natural length² - 13 cm * natural length

Replacing cm with m and rearranging the equation, we get:

(1.6 J - 84.5 m² * k) / k = natural length² - 0.13 m * natural length

Simplifying, we get:

(1.6 J - 84.5 m² * k) / k + 0.13 m * natural length = natural length²

Converting 1.6 J to N·m, we get:

(1.6 N·m - 84.5 m² * k) / k + 0.13 m * natural length = natural length²

Given the values of k, the spring constant, and the known values, we can solve the equation to find the natural length of the spring.

Answer 2

L = 7 cm

Step-by-step explanation:

Given,

natural length,L = 30 cm

Force to stretch the spring to 40 cm, F = 24 N

We know,

F = k Δ x

24 = k × (0.4 - 0.3)

0.1 k = 24

k = 240 N/m

now,

Work done calculation to stretch it from 30 cm to 35 cm

`W =(1)/(2)kx^2`

`W =(1)/(2)* 240 * 0.05^2`

W = 0.3 J

b) Work done to stretch from 9 cm to 11 cm

W = 0.6 J

Let L be the natural length

`W =(1)/(2)kx^2`

`0.6 = (1)/(2)k((11-L)^2-(9-L)^2)`

`1.2 = k(40-4L)`

`k = (1.2)/(40-4L)`.........(1)

When spring is stretched from 11 cm to 13 cm

W = 1 J

`W =(1)/(2)kx^2`

`1 = (1)/(2)k((11-L)^2-(9-L)^2)`

`2 = k(40-4L)`

`k = (2)/(48-4L)`......(2)

Form equation (1) and (2)

`(1.2)/(40-4L)=(2)/(48-4L)`

on solving

L = 7 cm

Hence, the natural length of the spring is equal to 7 cm.