If the work required to stretch a spring 4 feet beyond its natural length is 17 foot-pounds, how much work is needed to stretch it 14 inches beyond its natural length? Your answer must include the correct units. (You may enter lbf or lb*ft for ft-lb.)
To find the work needed to stretch the spring 14 inches beyond its natural length, we can use the principle of proportionality.
We can set up a proportion using the given information:
(Work1) / (Extension1) = (Work2) / (Extension2)
Where:
Work1 = 17 ft-lb (work required to stretch the spring 4 feet beyond its natural length)
Extension1 = 4 feet (extension of the spring in the first case)
Extension2 = 14 inches (extension of the spring in the second case; convert inches to feet by dividing by 12)
Work2 = unknown (work required to stretch the spring 14 inches beyond its natural length)
Plugging in the values, we have:
17 ft-lb / 4 ft = Work2 / (14 inches / 12)
Simplifying:
17/4 = Work2 / (14/12)
4.25 = Work2 / (1.17)
Now, let's solve for Work2:
Work2 = 4.25 * 1.17
Work2 ≈ 4.9725 ft-lb
Therefore, the work needed to stretch the spring 14 inches beyond its natural length is approximately 4.9725 ft-lb.
To solve this problem, we need to use Hooke's Law, which states that the force required to stretch or compress a spring is directly proportional to the displacement of the spring from its equilibrium position.
The formula for the work done on a spring is given by:
Work = (1/2) * k * (x^2)
Where:
- Work is the amount of work done on the spring (in foot-pounds)
- k is the spring constant (in lbf/foot)
- x is the displacement of the spring from its natural length (in feet)
Given that the work required to stretch the spring 4 feet is 17 foot-pounds, we can use this information to find the spring constant, k.
17 = (1/2) * k * (4^2)
17 = 8k
k = 17/8
k = 2.125 lbf/foot
Now, we can calculate the work required to stretch the spring 14 inches, which is equivalent to 14/12 = 1.17 feet.
Work = (1/2) * (2.125) * (1.17^2)
Work ≈ 1.15 foot-pounds
Therefore, the work needed to stretch the spring 14 inches beyond its natural length is approximately 1.15 foot-pounds.