If the work required to stretch a spring 2 ft beyond its natural length is 6 ft-lb, how much work is needed to stretch it 18 in. beyond its natural length?
To find the work required to stretch the spring 18 inches beyond its natural length, we can use the concept of Hooke's law. Hooke's law states that the force required to stretch or compress a spring is directly proportional to the displacement from its natural length. We can use the formula for work, which is given by:
Work = (1/2) * k * x^2
where k is the spring constant and x is the displacement.
From the given information, we know that the work required to stretch the spring 2 ft beyond its natural length is 6 ft-lb. Let's first convert the displacement from 2 ft to 24 inches.
Now, we can use the converted displacement value and the known work to find the spring constant, k.
6 ft-lb = (1/2) * k * (24 in)^2
Simplifying the equation further,
6 ft-lb = 12k * 576 in^2
Dividing both sides by 12 * 576,
6 ft-lb / (12 * 576 in^2) = k
k ≈ 0.008 lb/in^2
Now, we can calculate the work required to stretch the spring 18 inches beyond its natural length using the same formula.
Work = (1/2) * k * x^2
Where x is the displacement of 18 inches.
Work = (1/2) * 0.008 lb/in^2 * (18 in)^2
Simplifying the equation further,
Work = (1/2) * 0.008 lb/in^2 * 324 in^2
Work ≈ 1.296 lb-in
Therefore, the work needed to stretch the spring 18 inches beyond its natural length is approximately 1.296 lb-in.
To find the work required to stretch the spring 18 inches beyond its natural length, we can use the concept of proportionality in springs. The work done to stretch a spring is directly proportional to the displacement from its natural length.
We are given that the work required to stretch the spring 2 feet beyond its natural length is 6 ft-lb. Let's denote this as W_1 = 6 ft-lb and the corresponding displacement as x_1 = 2 ft.
To find the work required to stretch the spring 18 inches beyond its natural length, we need to find the displacement x_2 that corresponds to this work. Let's denote the work as W_2 and the displacement as x_2.
Now, we can set up a proportion using the given information:
W_1 / x_1 = W_2 / x_2
Substituting the known values:
6 ft-lb / 2 ft = W_2 / 18 in
Now, let's convert the units to have them consistent:
6 ft-lb / 2 ft = W_2 / (18 in * (1 ft / 12 in))
Simplifying:
3 ft-lb = W_2 / 1.5 ft
Multiply both sides by 1.5 ft:
W_2 = 3 ft-lb * 1.5 ft
W_2 = 4.5 ft-lb
Therefore, the work needed to stretch the spring 18 inches beyond its natural length is 4.5 ft-lb.