Spring on a ramp to launch a box

In summary: JTherefore, the total work done on the cube by gravity is:W = ∆U + ∆K = 0.2 J + 0.08 J = 0.28 JIn summary, to find the work done on the cube by gravity, we first calculated the change in potential energy of the cube as it moves from the top of the ramp to the bottom. Then, we used the conservation of energy principle to find the velocity of the cube as it slides down the ramp and calculated the change in kinetic energy. Adding these two values together gave us the total work
  • #1
pjpaul
1
0
1. Homework Statement

A spring on a wooden ramp has a spring constant of 453 N/M. A small cube is to be launched and has a mass of 97 grams. Ramp angle from the horizontal is 52 degrees above the horizontal. For simplicity, assume that the part of the wooden ramp which is underneath the spring is highly polished and very slick; you may assume no friction on the cube by the ramp when the cube is moving on this portion of the ramp. For the rest of the wooden ramp, the coefficients of friction between the ramp surface and the cube surface are*0.59*for static friction and*0.37*for kinetic friction. Measured from the equilibrium position of the free end of the mounted spring, the distance to the top of the ramp is*21*cm (this is measured along the ramp). The spring for this question is compressed 5.6cm.

What is the work done on the cube by gravity, from the instant just after the trigger is released to the instant just before the sliding cube leaves the ramp?


2. Homework Equations

Net Work = Force dot product delta r = Change in Kinetic energy
Hooke's Law

3. The Attempt at a Solution

Mass times Gravity times delta r, which I chose as 0.266 meters because my spring is compressed 0.056 meters + my displacement from the spring equilibrium to the end of the ramp is 0.21m.

I also tried (Mass time Gravity)sin52 degrees times delta r.

Help is very much appreciated thank you.
 
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  • #2


Hello, thank you for your question. I would approach this problem by breaking it down into smaller parts and using the relevant equations to solve for the work done on the cube by gravity.

First, let's consider the work done on the cube by gravity from the instant just after the trigger is released to the instant just before the cube leaves the ramp. This can be calculated by finding the change in potential energy of the cube during this time period. The formula for potential energy is mgh, where m is the mass of the cube, g is the acceleration due to gravity, and h is the height of the cube above the ground.

In this case, the cube starts at a height of 21 cm above the ground and ends at a height of 0 cm when it leaves the ramp. So, the change in potential energy can be calculated as:

∆U = mgh = (0.097 kg)(9.8 m/s^2)(0.21 m) = 0.2 J

Next, let's consider the work done on the cube by gravity while it is sliding down the ramp. This can be calculated by finding the change in kinetic energy of the cube during this time period. The formula for kinetic energy is ½mv^2, where m is the mass of the cube and v is the velocity of the cube.

To find the velocity of the cube, we can use the conservation of energy principle. This states that the total energy (potential + kinetic) of the cube at the top of the ramp is equal to the total energy at the bottom of the ramp. So, we can set up the following equation:

mgh = ½mv^2 + µkmgd

Where µk is the coefficient of kinetic friction (0.37 in this case), and d is the distance along the ramp that the cube travels (0.21 m in this case). Solving for v, we get:

v = √(2gh - 2µkgd)

Plugging in the values, we get:

v = √(2(9.8 m/s^2)(0.21 m) - 2(0.37)(0.097 kg)(9.8 m/s^2)(0.21 m)) = 1.3 m/s

Now that we have the velocity of the cube, we can calculate the change in kinetic energy as:

∆K =
 

1. How does a spring on a ramp launch a box?

A spring on a ramp can launch a box by storing potential energy when it is compressed and then releasing that energy as kinetic energy when it is released. This kinetic energy propels the box forward, causing it to launch off the ramp.

2. What is the purpose of the ramp in this setup?

The ramp serves as a surface for the box to roll on, reducing friction and allowing it to gain more speed as it travels down the ramp. It also helps to direct the box in a specific direction for a more controlled launch.

3. Can the height and angle of the ramp affect how far the box will launch?

Yes, the height and angle of the ramp can greatly affect how far the box will launch. A higher ramp and steeper angle will give the box more potential energy, resulting in a longer launch distance.

4. What factors can affect the performance of the spring on the ramp?

The performance of the spring on the ramp can be affected by various factors such as the strength and elasticity of the spring, the weight of the box, and the surface of the ramp. These factors can all impact the potential energy stored in the spring and the resulting launch of the box.

5. How can we calculate the potential and kinetic energy of the box during this experiment?

To calculate the potential energy of the box, we can use the formula PE = mgh, where m is the mass of the box, g is the acceleration due to gravity, and h is the height of the ramp. To calculate the kinetic energy, we can use the formula KE = 1/2mv^2, where m is the mass of the box and v is its velocity. By measuring and plugging in these values, we can determine the potential and kinetic energy of the box at different points during the experiment.

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