Lab 12: Conservation of Energy - Mass-Spring System

Lab 12:

April 14, 2015

Brandon Elder

Conservation of Energy - Mass-Spring System

Purpose: 

To observe the energy in a system, to show that the energy is conserved by determining all the different places that energy is used in the system and to sum those energies. The kinetic and potential energy of the mass, the kinetic, potential, and elastic energy of the spring, and the gravitational potential energy of the spring will be totaled.

Set-Up and Procedure:

Mount a table clamp with a vertical rod to the table, then mount a horizontal rod to the vertical rod. From that horizontal rod a spring will be attached with a mass hanging from the end of it. A force sensor will be required at first along with the motion detector, so Logger Pro will also be needed to be setup. Reference Fig. 1 for proper set up of the experiment. The spring's natural length will be measured and then the stretch position will be measured in order to determine the constant of the spring. Once the lab is set up, the actual experiment is very short. We will be measuring the distance that the spring oscillates back and forth and the velocity at which this takes place. This will help us determine the kinetic energy.

Fig. 1 The spring here is being stretched
(with mass) above the motion detector while 
connected to the force sensor.

Stretch: Record the position reading when the spring is un-stretched. This value will be used to calculate a column for the "stretch". The unstretched position minus the stretched position will determine the stretched length. Our experiment's data was: 0.618 m (unstretched).

Determine "k":  The slope of the Force vs postion graph will be the k value. The mass is pulled down slowly to stretch the spring. Logger Pro will record the force that is applied in stretching the spring. This data gives us the necessary info to determine the k value from the graph. This will be important later for when we set up the calculated columns. Our "k" value was: 29.32 (see Fig 1a).

Fig 1a Our chart of Force vs. Position. The slope of the best fit line gave us our K value for the spring.

Measure the mass of the spring and the mass of the hanging weight as well as the hanger that is used to hang the mass on. Record these values for insertion into Logger Pro later.

Determine calculated columns for Logger Pro:

Stretch: Use the information above to caluclate the column for stretch.

Elastic Potential Energy in the Spring: 1/2 * "k" * (stretch)^2. K is a value that will be inputted based on the calculations above. Stretch is a column that will be referenced from above.

Gravitational Potential Energy of the Mass: mass (hanging) * gravity * Position. Position will be the column measured in Logger Pro for position.

Kinetic Energy of the Mass: 1/2 * Mass (hanging) * velocity^2. The velocity will be a column that is automatically calculated by Logger Pro during the experiment.

Kinetic Energy of the Spring: In order to determine this it is a three step process. First, choose a representative piece (dm) of spring and write an expression in terms of dy. dm/M = dy/L. Therefore, dm = M/L * dy. Next, write an expression for the KE of that little piece. (1/2)(M/L *dy)(y/L*V(end))^2. This is equivalent to the equation 1/2*m*v^2. The third step requires you to sum the KE's of all the DM's in the spring. This formula will reduce to (1/2)(M(spring)/3)V(end)^2. This is the formula that will be input into Logger Pro.

Gravitational Potential Energy for the Spring: The same steps as above will be followed to derive the formula for GPE spring. The formula that will be input is: Mass (spring)/2 * gravity * y (end of spring). The y value will be the position column from the experiment recorded in Logger Pro. The formula is derived in Fig. 2. 

Sum of All Energy: This column will be a sum of all the columns created above. This will be the column that we use on the graph to show that the energy is conserved in this system.

Fig. 2 The formula derived for the Gravitational Potential Energy of the Spring.

Now that all columns are calculated and prepared for data collection run the experiment. Pull the spring down with the mass on the hanger about 10 cm and let go. Set up a graph to show all of the columns from above on it. They will all be a bit wavy (see Fig. 3).

Fig. 3 All the columns added are shown on the graph here. The top graph (purple) is the sum of all the energy. This graph should be a straight line as we were trying to show that energy is conserved and this is the result of all our energy columns being summed up.

Conclusion:

According to the graph above the GPE of the mass was at it's lowest when the Elastic PE of the spring was at it's highest. Also, when the GPE of the mass was at it's highest, the Elastic PE of the spring was at it's lowest. This makes sense because when the string is fully stretched the mass would have very little gravitational potential energy. As is evident by the above graph, the energy is conserved, even though the individual readings have giant peaks and valleys, the overall sum is relatively staying steady. No amount of energy left the system, and no amount of energy entered the system. All energy was accounted for. Some slight error could have been introduced through errors in the accuracy of the readings of the spring constant measurement. The height may not have been perfectly calibrated with the motion sensor and the spring itself isn't perfectly out of the box in brand new ideal conditions.