LAB 12 - Conservation of Energy: Mass-Spring System
Tony Wu
October 5, 2016
Lab Goal: To examine the energy in a vertically-oscillating mass-spring system, where the spring has a non-negligible mass.
Theory/Introduction: Prior to this lab, we have done spring problems with the assumption that the spring is ideal, meaning it has a negligible mass. However, in real life this is not the case. In this lab we will examine the total energy of the system. Because the spring's mass must be considered, we have to consider the gravitational potential energy due to the mass of the spring. The spring is also a length L. It is important to note that the bottom of the spring travels a much larger distance than the top of the spring as the spring oscillates. This means that each tiny piece of the spring has a different kinetic energy. We will have to use calculus in order to calculate potential energy and the kinetic energy.
We assumed that the gravitational potential energy is 0 at the ground. The top of the spring is held fixed at a height H above the ground and the bottom of the spring is at a position y above the ground.
Apparatus:
Theory/Introduction: Prior to this lab, we have done spring problems with the assumption that the spring is ideal, meaning it has a negligible mass. However, in real life this is not the case. In this lab we will examine the total energy of the system. Because the spring's mass must be considered, we have to consider the gravitational potential energy due to the mass of the spring. The spring is also a length L. It is important to note that the bottom of the spring travels a much larger distance than the top of the spring as the spring oscillates. This means that each tiny piece of the spring has a different kinetic energy. We will have to use calculus in order to calculate potential energy and the kinetic energy.
We assumed that the gravitational potential energy is 0 at the ground. The top of the spring is held fixed at a height H above the ground and the bottom of the spring is at a position y above the ground.
Forgot to take a photo of the actual apparatus. Here is a diagram of the apparatus.
There is a motion sensor placed directly below the spring in order to track the position and velocity of the end of the oscillating spring.
Experimental Procedure: We measure the mass of the spring, 0.087 kg. Using a force sensor, we determined the spring constant of the spring. We calibrated the force sensor using zero mass. The spring constant K was 8.417 N/m. The mass of the weight hanging at the bottom of the spring was 0.250 kg. We had to measure the length of the spring hanging on its own. The unstretched length was 1.05 m. The distance from the floor to the hanging mass on the spring was 0.719 m.
Data and Calculations:
The following calculation is using calculus to show that the GPE of the spring is mg(H+y)/2.
The following calculation is using calculus to show that the Kinetic Energy of the moving spring is 1/2(1/3 * M_spring) * v^2
Graphs of Position, Velocity, KE, GPE, and EPE.
Each of these quantities are related.
Conclusions:
We were able to find equations for the kinetic energy and gravitational potential energy of a spring with non-negligible mass. It is important to note the relationships between Position, Velocity, Kinetic Energy, Gravitational Potential Energy, and Elastic Potential Energy. The energy in the system is conserved and basically transferred as the spring oscillates. When the spring is fully stretched, the velocity is momentarily zero, thus no kinetic energy. The spring has the highest amount of EPE at this point, with the lowest GPE because the spring is closest to the ground. When the spring is unstretched at the top of the oscillation, the velocity is once again momentarily zero, thus no kinetic energy. At this point, the spring also has no EPE because the spring is not stretched at all. The spring does have the highest value of GPE when it is unstretched at the top of the oscillation because it is the furthest from the ground at this point.
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