8 April 2015. Conservation of Energy. Mass-Spring System.

Purpose

to prove the theory of conservation of energy: that the initial energy is always going to be in the same amount of the final energy by calculating the energy of a moving "mass" spring (in addition to some masses attached to it).

Apparatus:

1. Spring and some masses. In this experiment, we basically use these aparratus as an object to determine its potential and kinetic energy
2. Motion detector, logger pro, laptop. For recording and analyzing data.

Procedure:

1. Set up the clamp,  the spring and the mass as shown in the picture. (pic)

2. Measure the spring constant by hanging some masses and record the distance stretched. Then you will be able to plot the distance vs mass, and the slope is your spring constant.
3. Then, attached the mass hanger so that it is vertical and the spring is just unstretched. Zero the motion sensor with the mass hanger in this position. Don't forget to reverse the direction, so that towards the motion sensor is positive. Also measure the this distance from the floor.
4. To verify that the motion detector is working, pull the spring down slowly, and check if the graph make sense.
5. Now, record the spring movement. Collect the data for at least 5  or 10 seconds. Make sure that the motion detector could read the movement perfectly by attaching a post-notes below the mass and carefully pull the spring in a perfectly vertical position. Otherwise, the spring will sway away.

Base Theory

• From the procedure, we do realize that we are going to plot every energy there are in the moving spring, and calculate the TOTAL ENERGY of all - expecting a constant value (based on the theory of conservation of energy).
• Let's us evaluate the energy in this moving "mass" spring.
• The mass: Gravitational Potential Energy and Kinetic Energy.
• The spring: Elastic Potential Energy, Gravitational Potential Energy, and Kinetic Energy.
• How do we calculate the Gravitational Potential Energy and Kinetic Energy of moving spring? We usually neglect that the spring actually has mass, but now we are not going to ignore it!
• This is how to calculate Gravitational Potential Energy of the moving spring.

• This is how to calculate Kinetic Energy of the moving spring.

Data Analysis:

• We measure the spring constant by hanging some masses and record the distance stretched (look at the pictures below). First we measure the length of the spring at equilibrium. Then add some masses and measure the additional distance created.

x= 37.1 cm

• From this data, we plot our graph of force vs distance; and the slope is our spring constant!

k = 11.01 N/m

• Then, we measure the mass of the spring itself. And we also measure the distance from the floor when the spring is unstretched.

• Realize that now, we have these data:
• spring constant = 11.01 N/m
• m spring= 0.064 kg
• m mass= 0.2 kg
• distance when the spring is equilbrium from the floor= 0.887 m
• Next we record the spring movement. The motion detector will give us data, including position and velocity over time (these are the rest data that we need).
• Now, we create several new calculated column, which are all the energy that varies with the movement of the spring (picture).

• This is how we input it into logger pro.

• Before we actually graph the real condition into logger pro. Let us predict several graph first.

• Now let's plot the energy playing in this system and the TOTAL ENERGY for real.

• Yes! We can see that the total energy is constant! (with some errors, calculation below)

EXTRA:

• Let us predict these graphs (Explanation in the pic).

• Turns out, the prediction matches the real graph.
• Now, we plot all the energy vs position and time.

• Form these graphs, we can see that the energy always go through a cycle, an in every point of each cycle energy is conserved! Look at the "almost" constant value of total energy playing in this system.

Conclusions:

• In this experiment, we manage to proof that the theory of conservation of energy is true and amazingly applicable in every second!
• There are indeed some errors in this experiment. The possible sources are:
• When we acquire the spring constant, we notice that the correlation of the line is not one. Thus its already producing some errors.
• When we zero the motion sensor we didn't actually get the value zero precisely but near it. 
• There are some air resistance that we do not calculate.
• There are some human errors in measuring the distance etc.
• Mass of the spring can affect the calculation quite massively. The fact that we always omit the mass of the spring as negligible when we calculate is quite disturbing.