Apr-18-2015. Lab 11 : Conservation of energy----Mass-spring System

For this lab, We will be looking at the energy in a vertically-oscillating mass-spring system, where the spring has a non-negligible mass.

Before we can actually to do the lab there is some preliminary stuff to work out.

assume that the gravitational PE = 0 at the ground, and that you have a spring whose top is held fixed at a height H above the ground, and the bottom of the spring is at a position y above the ground.











there is the picture show that the GPE of the spring is mg(H+y)/2 be written as m(spring)/2 *g*H + m(spring)/2 *g*y:

Now put the origin at the top of the spring and call downward the positive direction, assume that the spring has a length L, the top of the spring is held at rest but that the bottom end of the spring is moving at a speed v downward.
there is the picture show that the KE of the moving spring is 1/2 *(1/3 *m(spring))*v^2.

then set up the spring, a 50-gram mass hanger, with the motion detector on the floor. we measured the length of the spring L = 0.485 m, the mass of the spring m(spring) = 0.24 kg.

First, we need to Determining the Spring constant of the spring.

mount a table clamp with a vertical rod to the table. mount a horizontal rod to the vertical rod. Put the Force sensor on the horizontal rod with the loop of the sensor pointing downward.


place a 50-gram the mass hanger so that it is vertical and the spring is just unstretched. zero the sensor with the mass hanger in this position.






Open the program, start collecting data and slowly pull down on the 50-gram mass . then we got the graph of Force vs. position.

after we liner fit the graph, we got the slope which is the constant of spring K = -8.03 N/m .


Right now, we need To do:
      Hang 250-grams on the mass hanger. After the spring is not stretch any more, we measured the y(0) = 0.73 m.

we have expressions of KE, GPE, PE that we worked out before:

we have the mass of the spring m(spring) = 0.24 kg  ,
the constant of the spring K = -8.03 N/m,    mass of hanging M = 0.25 kg.    y(0) = 0.73 m,
△y = 0.843 - "position"(we could measure it from the sensor)
the velocity v (we could measure it from sensor).

Under the Data menu in loggerPro, we created:
the New calculated column of KE = 1/2 *(M + 1/3 *m(spring))*v^2.
the New calculated column of GPE = (M + m(spring)/2) * g * "position" .
the New calculated column of EPE = 1/2 *K *△y^2 

Then, Pull the spring down about 10 cm and let go. we got the graphs:

this blue graph is the graph of KE vs. time.



this purple graph is the graph of GPE vs. time.



this green graph is the graph of EPE vs. time.

 this blue graph is the graph of KE vs. position.



this purple graph is the graph of GPE vs. position.



this green graph is the graph of EPE vs. position.

this blue graph is the graph of KE vs. velocity.



this purple graph is the graph of GPE vs. velocity.



this green graph is the graph of EPE vs. velocity.



Finally: create a new column called TE(Total Energy), which is the sum of the KE, GPE, EPE.

this is graph of TE vs. time.

the min of TE = 2.242 J
the mean of TE = 2.309 J.
the max of TE = 2.377 J


this is graph of TE vs. position.



this is graph of TE vs. velocity.





Conclusions:

From those graphs, we could know about the energy in a vertically-oscillating mass-spring system, where the spring has non-negligible mass.  The potential energy will be greatest when the spring is stretched or compacted. However, KE + GPE + EPE should be always same so the total energy is conserved. the graph of the total energy should be a straight line. Although our result of the graph of the total energy is a line which has small wave, the min of TE is very close to the max of TE, which means our measurements are not bad.