13-April-2015 Magnetic Potential Energy

Purpose: To create an expression for potential energy from a two same charged magnets (U_mag) using what we know of kinetic energy (KE) and the forces involved on an air track.

Theory: Because energy is conserved within a system, the kinetic energy of a moving cart on an air track should transfer to the potential energy created by the magnets (assuming no other energy is added or removed from the system), the relationship graphed should look something like this over time.

Procedure: Set up an air track with a magnet on one end, with a cart with another magnet on it. Weigh the cart for its mass, once its on the track incline the track at various steeper angles and measure the distance between the magnets, make a free body diagram of the forces on the cart, graph these points and find a line that expresses this Force vs distance between magnets. The expression for the line once integrated with regards to distance between the magnets and multiplied by negative one is our model for U_mag, we test the accuracy of this model by moving the cart along the track (creating KE with out too much friction), and seeing if the energy is conserved or not (since it must for our model to be accurate).

Apparatus

Forces on the Cart

*So the force of the magnets is Mass of cart * gravity * the sine of the angle the track makes with the horizontal.

*We measured the distance between the magnets at varying angles.

*Mass of the cart was weighed to be 0.34 kg.

Results

Angle (Rad)	Force Magnets (N)	Distance (M)
0.06108472	0.203407742	0.023
0.09075444	0.301978875	0.02
0.11693361	0.388735483	0.017
0.14485806	0.480980775	0.015
0.1727825	0.572851035	0.014
0.26877278	0.884807519	0.011

Graph of Force vs Distance

*The best fit line is A*y^B.

*A is a constant and = 0.0001316

*B is the power value of the line and is = -1.956

*y is the distance between the magnets and is the value we integrate with respect to (dy).

* Once integrated our model for (limits of integration 0 to y) and multiplied by negative one is

                                             U_mag= -Int(0.0001316*y^-1.956 dy) from 0 to y

                                           U_mag= -0.0001316/(-1.956+1) (y^-1.956+1.0)

                                                 U_mag= (-0.0001316/-0.956)*y^-0.956

                                                     U_mag=0.000137657*y^--0.956

*We now test this model with the KE (0.5*Mass of cart*Velocity^2) of the cart as it approaches the magnet, the position sensor records the distance between the magnets, and we graph the results.

Graph KE and U_mag vs time

*Are graph isn't ideal; although there is an inverse relationship between KE and U_mag, they aren't equal, suggesting that something was contributing to the KE as it approached the magnets.

*The slant from the KE line suggests the track was at an angle, potential energy from gravity was effecting the system on its way towards and away from the magnets.

Conclusion

*Although I still believe the methods were valid, too much human error caused unsatisfactory results.

*Our final graph has an obvious slant, meaning the track wasn't level and energy due to gravity effected the results.

*The experiment ignores friction caused by the air against the cart, though because of the nature of an air track there was no friction between the cart and the track.

*We relied on the human eye and a ruler to measure the distance between the magnets in the first part of the lab, meaning our results could there could only be accurate up to +/- 0.0005 meters, though the U_mag that was derived from those results behaved as expected.