The apparatus that was used in this laboratory is a mass spring system consisting of a rod clamped to the table with another rod clamped horizontally at the top of the vertical rod. Attached to the horizontal rod is a force sensor. Hanging from the force sensor we have a spring and a 50g hanging mass. Under the hanging mass we have our motion sensor. The purpose of the apparatus is to help us measure the force and position of the hanging mass and spring.
The first thing we did was find the spring constant "K" for the spring. To do this we pulled the hanging mass and let the spring oscillate and we recorded data for force and position of the spring.
We plotted the data in a Force vs. Position graph and added a linear fit. The slope of the linear fit gave us the spring constant "K" which turned out to be 9.1N/m.
Figuring out the value for our spring constant "K" we were able to find and equation for the elastic potential energy of our spring which is :
EPE = 1/2K(stretch)^2
Where the "stretch" is the distance of the unstretched position minus the distance of the position of the spring at any given time.
Next we had to derive an equation for the gravitational potential energy of the spring itself. To do this we:
1. Choose a representative piece dm of the spring and wrote an expression for it in terms of dy. the expression is dm = M/(H-yend)dy where M is the mass of the whole spring and H is the distance from the ground to the top of the spring. and yend is the distance from the bottom of the spring to the ground.
2. We wrote and expression for the GPE of the little piece of mass "dm" which was GPE = Mgy/(H-yend)dy where M is the mass of the whole spring g is acceleration of gravity y is the distance of dm to the floor, H is the distance of the top of the spring to the floor and yend is the distance from the bottom of the spring to the floor.
3. lastly we summed the gravitational potential energies of all the dm pieces in the spring and our expression for the gravitational energy of the spring turned out to be M/2gH + M/2gyend. which can also be written as Mg(H+yend)/2.
Next we had to derive an expression for the kinetic energy of the spring itself. To do so we followed the same three steps above only we changed our origin to the top of the spring and our positive direction was down.
1. Our expression for the small piece of the spring dm is dm = M/Ldy where L is the Length of the spring and M is the mass of the whole spring.
2.Our expression for the kinetic energy of the small piece of the spring is 1/2(M/L)dy(y/L)^2(vbottom)^2. where M is the mass of the whole spring, L is the length of the spring, y is the y is the distance of dm from the origin and v bottom is the velocity of the bottom of the spring.
3. Our expression for the kinetic energy of the spring is KE = 1/6M(vbottom)^2. where M is the mass of the whole spring and vbottom is the velocity of the bottom of the spring.
Other energy equations that we need and were given to us in this lab are:
Kinetic energy mass = 1/2(Mhanging + 1/3Mspring)v^2 where v is the velocity of the hanging mass.
Gravitational potential energy in mass = (Mhanging + 1/2Mspring)gy
What we did next was make calculated columns for the KEmass, EPEspring, GPEmass, KEspring, and GPEspring and the energy sum of all the energies, and we plotted the graphs in logger pro.
The black line on the graph represents the sum off all the energies while all the other graphs represent the individual energy by themselves. The fact that the black line is close to being a straight horizontal line shows that energy is conserved. Although the ideal result is to have a perfectly horizontal straight line for the sum of the energies there are error factors that affected the results.
for example I believe we were having trouble with the distance that the motion sensor was calculating, also when we forgot to write down the mass of the spring so we had to guess which spring we had the following day and we measured the mass of that spring. It could have been incorrect which could have led to error as well.
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