Mr. Burtness Name______
Lab
Simple Harmonic Motion - Bouncing mass on a Spring
The velocity of an object in Simple Harmonic Motion as a function of position is:
v=+/-vmaxÖ(1-x2/A2), where x is the displacement from the equilibrium position, and A is the amplitude of oscillation, xmax ( ½ peak to peak distance). This comes from conservation of energy principles:
Uspring = ½ kx2 (k = spring constant from Hooke’s Law: F=-kx). K.E. = ½ mv2
Since mechanical energy is conserved: ½ mv2 + ½ kx2 = ½ kA2 + ½ m[v(=0)]2 = ½ mvmax2
- Solve this underlined equation for: v= ___ P.E.max = K.E.max, so ½ kA2 = ½ mvmax2
- Solve this for k/m=, and substitute into equation #1.
Laboratory Experiment:
· Hang a mass from a spring at least 70 cm above a rangefinder. Note the stretch a 500g mass gives the spring to determine the spring constant, k.
· Use the range finder to collect data (ball-bounce setting on the CBR) on the bouncing of the mass. The data should look very smooth.
- Record A, vmax , and T, the period of oscillation.
- Calculate vmax =2pA/T and compare to your experimental value.
- Substitute vmax /A=Ö(k/m) into the above equation and solve for T. This is the equation for the period of motion.
- To see if your data agrees with the equation (#2) above, you can calculate v from the position graph as follows. First, shift the entire position graph down so it is symmetric with the x-axis. Do this by noting a time where velocity has reached a maximum. Next read your position graph at this time, xmax . QUIT the RANGER program, and “ENTER” L2-xmax STOà L2. This will re-center your position graph.
On your TI calculator: L2 = position, L3 = velocity, L2=x (displacement from equil.)
v=vmax{abs[1-(L2)2/A2].5} àL5.
Graph L1 vs. L3, and L1 vs. L5. This will show you how well they agree. (Note: absolute value is required in the calculation to prevent negative square roots that would result from experimental data.) An actual comparison of these velocities is shown below.- To see that the motion is sinusoidal, set Y1=vmaxsin{(k/m).5x} You may have to shift the curve left or right to get complete agreement, but the shape will be the same as the above picture.
SHMCBR.doc