PHYS4A/ Fall 2014; Conservation of Energy

Purpose: To experimentally show conservation of energy by attaching a mass to a metal spring and force sensor and allowing it to oscillate over a motion sensor. The motion sensor then measures the changes in distance to calculate kinetic, potential, and total energy of the system.

Experiment: To start the experiment, we attached a metal spring to a force sensor that was pointing down and zeroed. We calculated the difference between the spring length stretched and at rest. By knowing the total change in distance and the force exerted by the spring, the spring was calculated to have a spring constant of 20.86.

Experimental set-up of spring on force sensor as the mass is oscillating over the motion sensor.

Graph of force vs position where the slope of the graph is the equivalent to the spring constant. Due to the spring constant being equal to Force/ Displacement. 

Once the spring constant was calculated, the mass was set to oscillate over the motion sensor in a vertical direction. By recording the change in distance, energy for the spring and mass could be calculated using various equations. 

Kinetic Energy (Mass) = (1/2)mv^2

Potential Energy (Mass) = mgh

Elastic Potential Energy (Spring) = (1/2)kx^2

Elastic Kinetic Energy (Spring) = (1/2)mv^2

Gravitational Potential Energy (Spring) = mgh

Total Energy = Sum of Energies calculated above

By inputting these formulas into loggerpro and making newly calculated columns, we were able to get several energy curves for analysis.

Graphs of the different types of energy for the calculated columns.

As seen above, the graphs oscillate according to they type of energy it corresponds with. Based on conservation of energy and the potential energy of the mass and the elastic potential energy of the spring should oscillate with relation to each other. By knowing that the total energy should be constant throughout the system, if potential energy of the mass goes up, elastic potential should go down. This is apparent in the graphs above as you see the graphs oscillating and canceling out any changes that are done to total energy. Overall, total energy is constant as seen in the first graph with very little oscillation change thus supporting the theory of conservation of energy. 

PHYS4A/ Fall 2014; Work-Energy Theorem

Purpose: We looked at the work-energy theorem which equates the area under a kinetic energy graph to total work being done.

To set up the experiment, a cart was attached to a metal spring and placed on a metal track. The cart was positioned to be at rest with the spring at a neutral length, where it is at rest and neither stretched or unstretched. This was the zero position that was used to calibrate a motion sensor attached to the end of the track. Once calibrated and setting a positive axis on loggerpro, the cart was pulled back a small length to measure a positive distance. The cart was then released and loggerpro was used to measure the force vs time and plot it in a graph.

Cart and mass at rest while spring is unstretched

Cart and mass after moved a distance x so that spring is stretched and containing potential energy.

Once the data was collected, a new calculated column was created for kinetic energy using the formula K=(1/2)mv^2. Kinetic force vs time and kinetic energy vs time were both graphed. The integral of kinetic force vs time was equivalent to the total kinetic energy over the same interval of time.

As seen in the figure above, the total area under the graph is the integral of kinetic force. The integral is equal to the calculated kinetic energy for the region.

The three points are three different intervals that correlate with kinetic energy intervals with a % error of less than 1%.