Question

Using Hooke’s Law, we can show that the work done in compressing a spring a distance of x feet from its at-rest position is W = 1/2 kx², where k is a stiffness constant depending on the spring. It can also be shown that the work done by a body in motion before it comes to rest is W' = w/2g * v², where w = weight of the object (lb),g = acceleration due to gravity (32.2 ft/sec²), and v = object’s velocity (in ft/sec). A parking garage has a spring shock absorber at the end of a ramp to stop run way cars. The spring has a stiffness constant k = 9450 lb/ft and must be able to stop a 4000-lb car traveling at 25 mph. What is the least compression required of the spring? Express your answer using feet to the nearest tenth.

Answer 1

To determine the least compression required of the spring, we need to calculate the work done by the body in motion using the given formula and equate it to the potential energy of the compressed spring. By rearranging the equation and solving for x, we can find the least compression required.

Step-by-step explanation:

According to Hooke's Law, the work done in compressing a spring is given by the formula W = 1/2 kx², where k is the stiffness constant of the spring and x is the distance of compression. On the other hand, the work done by a body in motion before coming to rest can be calculated using the formula W' = w/2g * v², where w is the weight of the object, g is the acceleration due to gravity, and v is the velocity of the object.

In this case, the car has a weight of 4000 lb and is traveling at 25 mph. We need to find the least compression required of the spring to stop the car. To do this, we first need to convert the weight of the car from lb to N by multiplying it by the acceleration due to gravity: w = 4000 lb * 9.8 m/s² = 39200 N. Next, we convert the velocity from mph to ft/s by multiplying it by 5280 ft/mi and dividing it by 3600 s/h: v = 25 mph * 5280 ft/mi / 3600 s/h = 36.67 ft/s.

Substituting the values into the formula for the work done by the body in motion, we have W' = 39200 N / (2 * 32.2 ft/s²) * (36.67 ft/s)². Calculating this expression gives us the work done by the body in motion before it comes to rest. To find the least compression required of the spring, we equate this work to the potential energy of the compressed spring: W' = 1/2 kx². Rearranging the equation and solving for x, we can determine the least compression required.