Theory: Since energy is conserved, the graphs can represent the relationship between the potential energy (elastic from the spring and gravitational on both the spring and hanging mass) and kinetic energy (of both the spring and the mass) in a manner that makes sense from what we know of the initial energy of a system must equaling the finial energy of a system.
Procedure: We determine the mass of our spring, we set up our apparatus to hang a spring off of (attached to a force gage) and a position sensor on the floor. Determine the K constant of our spring (we will need this for when we determine the elastic potential energy (EPE)) by attaching a mass to stretch the string and measure the resulting force and stretch. Derive an expression for the kinetic energy of the spring and potential energy due to gravity on the spring. Stretch the spring, with the mass attached, and allow it to oscillate at a nice steady rate. Graph results and compare.
Apparatus
*We weighed our spring to have a mass of 0.121 kg
*We attached a heavy enough mass to stretch the string, and graphed the resulting force vs. stretch
*The slope is our K constant for the spring 18.79 N/M.
*We know EPE is determined by 0.5*K*(stretch)^2, so we make a column that will determine this for us by imputing the stretch during the oscillation
*For the hanging mass we already have expressions to calculate energy (KE=0.5*Mass*Velocity^2, GPE=Mass*gravity*height).
*We calculate the KE and GPE for the spring by taking a representative piece and rewriting it in terms of dy (dm/dy = Mass/(Height- y end of spring), limits of integration y end to Height, GPE then should be = dm*g*y.
*Fully calculated GPE_spring= Mass_spring*gravity*(Height -y end)*0.5, and Height - yend is merely the position read by the position sensor.
*We calculate the KE of the spring by once again rewriting everything in terms of dy(dm/Mass = dy/Length of spring), limits of integration 0 to Length, KE then should be = 0.5*dm* y/L*velocity. (velocity of the piece is equal to y/Length * velocity of the mass)
*Fully calculated KE_spring = 1/6 * Mass_spring * Velocity^2
*We can now add a 0.2 kg mass to the spring (hanging mass), and begin oscillating
Graphs
GPE_spring, KE_mass, and KE_spring
*We see that when the GPE_spring is at its max, both the KE_spring and KE_mass are at zero
*This makes sense as GPE_spring's max energy is when the system is at its highest point and about to fall, meaning the velocity is zero as their is no motion yet. If the velocity is zero, both KE of the spring and mass must also be zero.
EPE and GPE_mass
*We again see graphs that are mirror opposites of each other.
*This again makes sense as the EPE is at its max when the spring is stretched to its max, this point's position however is considered the zero mark of our height, therefore the GPE_mass is zero as well.
GPE_mass, GPE_spring, KE_mass, KE_spring, EPE, Sum of all Energy vs time
*In general we do see the sum of energy remaining the same throughout the oscillations, returning to the previous peaks and valleys over time, as well as all the other energy graphs
*Although not very apparent, we do see smaller peaks and valleys of the graphs over time.
*This makes sense as energy is being lost during each oscillation due to friction from the air.
GPE_mass, GPE_spring, KE_mass, KE_spring, EPE, Sum of all Energy vs position
*We see some very straight(ish) lines from the sum of forces
*Straight lines would make sense as the energy is never really being lost throughout the process, meaning the energy is conserved.
*I believe the reasoning for any "slants" within the sum is due to a scaling the graphs.
Conclusion
*All around the graphs behaved consistently with what I would expect if the conservation of energy is true.
*Any "leveling off" of the graphs can be explained as due to air friction doing work against the system, stealing energy.
*There was some inconstancy with the Sum of energy vs position graph, as it seems to gain energy over time, but I believe that was due to me not properly scaling the graph
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