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 35 inches beyond its natural length? Your answer must include the correct units. (You may enter lbf or lb*ft for ft-lb.)
To solve this problem, we can use the concept of Hooke's Law, which states that the force required to stretch or compress a spring is directly proportional to the displacement from its equilibrium position. Mathematically, it can be expressed as:
F = k * x
Where:
F is the force applied to the spring (in pounds),
k is the spring constant (in pound-feet per inch),
x is the displacement from the natural length (in inches).
The work done in stretching a spring can be calculated by multiplying the force applied by the displacement:
Work = Force * Displacement
In the given problem, to determine the constant of proportionality (k), we can use the given information:
The work required to stretch the spring 4 feet beyond its natural length is 17 foot-pounds.
Work = 17 ft-lb
Displacement (x) = 4 ft
Using the formula:
Work = Force * Displacement
17 ft-lb = F * 4 ft
Now, let's calculate the force (F):
F = 17 ft-lb / 4 ft
F ≈ 4.25 lb
We now have the force (F) and displacement (x) required to stretch the spring 4 feet beyond its natural length.
Next, we can calculate the work needed to stretch the spring 35 inches beyond its natural length. The displacement (x) is given, so we just need to substitute the values into the work formula:
Work = Force * Displacement
Work = 4.25 lb * 35 inches
To get the answer in the correct units, we need to convert pounds and inches to foot-pounds:
1 lb*ft = 1 foot-pound
1 inch = 1/12 foot
Converting units:
Work = 4.25 lb * 35 inches * (1 lb*ft / 1 foot-pound) * (1/12 foot / 1 inch)
Work ≈ 4.25 * 35 / 12 foot-pounds
Therefore, the work needed to stretch the spring 35 inches beyond its natural length is approximately 12.4 foot-pounds.
To find the work required to stretch the spring 35 inches beyond its natural length, we need to use a proportion.
Let's set up the proportion using the given information:
(work for 4 feet) / (4 feet) = (work for 35 inches) / (35 inches)
Using this proportion, we can solve for the work needed to stretch the spring 35 inches beyond its natural length.
(work for 4 feet) = 17 foot-pounds
Let's substitute this value and solve for the unknown:
17 foot-pounds / 4 feet = (work for 35 inches) / 35 inches
To find the work for 35 inches, we can cross multiply:
17 foot-pounds * 35 inches = 4 feet * (work for 35 inches)
Now, let's solve for the work:
595 foot-inch-pounds = 4 feet * (work for 35 inches)
Divide both sides by 4 feet:
595 foot-inch-pounds / 4 feet = work for 35 inches
To convert foot-inch-pounds to foot-pounds, we divide by 12 inches:
(595 foot-inch-pounds / 4 feet) / 12 inches = work for 35 inches
Simplifying:
49.583 foot-pounds = work for 35 inches
Therefore, the work required to stretch the spring 35 inches beyond its natural length is 49.583 foot-pounds.