The hill is covered in gravel so that the truck's 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 mt, moving at an initial speed of v0, will be stopped. For this problem, it is easiest to define the system such that it contains everything: Earth, hill, truck, gravel, spring, etc. In all of the following questions, the initial configuration is the truck moving with a speed of v0 on the level ground, and the final configuration is the truck stopped on the hill with the spring compressed by an amount x. mt = 20000 kg v0 = 70 m/s x = 3.5 meters h = 55 meters μk = 0.78 θ = 32 degrees L = 11 meters I have calculated the total work done on the system to be 0 J. 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. Find the change in thermal energy of the system. NOTE: THE REGION UNDER THE SPRING ALSO HAS GRAVEL UNDER IT. What is the change in kinetic energy of the system?

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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The hill is covered in gravel so that the truck's 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 mt, moving at an initial speed of v0, will be stopped. For this problem, it is easiest to define the system such that it contains everything: Earth, hill, truck, gravel, spring, etc. In all of the following questions, the initial configuration is the truck moving with a speed of v0 on the level ground, and the final configuration is the truck stopped on the hill with the spring compressed by an amount x.

mt = 20000 kg
v₀ = 70 m/s
x = 3.5 meters
h = 55 meters
μ_k = 0.78
θ = 32 degrees
L = 11 meters

I have calculated the total work done on the system to be 0 J.

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.

Find the change in thermal energy of the system. NOTE: THE REGION UNDER THE SPRING ALSO HAS GRAVEL UNDER IT.

What is the change in kinetic energy of the system?

The diagram shows a small cart on a flat surface with an initial velocity \(v_0\), moving towards an inclined plane. Here’s a detailed description and explanation of the components:

1. **Cart**: The cart is depicted on the left, moving towards the right with an initial velocity denoted as \(v_0\).

2. **Inclined Plane**: The cart is headed towards an upward slope. This plane is marked by a rough surface labeled as "gravel," indicating increased friction or resistance as the cart ascends.

3. **Incline Angle (\(\theta\))**: The angle of the inclined plane with respect to the horizontal ground is marked as \(\theta\). This angle is crucial in determining the components of the gravitational force acting on the cart.

4. **Height (\(h\))**: The vertical height from the base to the top of the incline where the spring is located is labeled as \(h\).

5. **Length (\(L\))**: The horizontal distance covered on the incline before reaching the spring is denoted by \(L\).

6. **Spring**: At the top of the incline, there is a spring. The cart is expected to compress this spring once it reaches the top, provided it has sufficient velocity to overcome gravitational and frictional forces.

This setup can be used to study concepts such as energy conservation, dynamics on an inclined plane, and the effects of friction. It provides a practical frame for calculating the energy transformation from kinetic to potential, and then to elastic potential energy, as well as analyzing forces at play.

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