The hill is covered in gravel so that the trucks wheels will slide up the hill instead of rolling up the hill. The coefficient of kinetic friction between the tires and the gravel is μk . This design has a spring at the top of the ramp that will help to stop the trucks. This spring is located at height h. The spring will compress until the truck stops, and then a latch will keep the spring from decompressing (stretching back out). The spring can compress a maximum distance x because of the latching mechanism. Your job is to determine how strong the spring must be. In other words, you need to find the spring constant so that a truck of mass m, moving at an initial speed of v0, will be stopped.
For this problem, it is easiest to define the system such that the system contains everything: the earth, hill, truck, gravel, spring, etc. In all of the following questions, the initial configuration is the truck moving at an initial speed of v0, and the final configuration is the truck stopped on the hill with the spring compressed by an amount x. The truck is still in contact with the spring. Solve all of the questions algebraically first. Then use the following values to get a number for the desired answer.
m = 14000.0 kg
v0 = 67.0 m/s
x = 3.1 meters
h = 59.0 meters
μk = 0.69
θ = 30.4 degrees
L = 11.3 meters
If the total work is equal to 0 and the change in kinetic energy is -31420000 J, then find the following:
1. Find the change in gravitational potential energy. Since the final goal of this problem is to find the minimum spring constant, assume that the spring will compress to its maximum value.
2. Find the change in thermal energy of the system. Note: The region under the spring also has gravel under it.
3. Using all of the information from above, determine the minimum spring constant necessary to stop the truck.
4. In this configuration, what force will the latch have to withstand to keep the spring compressed?
