1201

Experiment:  Chain between 5 and 10 rubber bands together.

Observe the maximum unstretched length of the chain, and its length when it supports your assigned mass.

Observe the natural frequency of your assigned mass when the oscillation has amplitude about 1 cm.

Repeat for amplitude of oscillation approximately .1 times the unstretched length of the chain.

Repeat for amplitude of oscillation approximately .2 times the unstretched length of the chain.

Repeat for amplitude of oscillation approximately .3 times the unstretched length of the chain.

Determine the restoring force constant k for the chain, from the mass and oscillation frequency for each trial.

Determine the restoring force constant k for the chain, from the amount the chain stretched when the weight was added.

Analysis:

omega = sqrt(k / m) so k = omega^2 * m.  If m is in kg and omega in rad/s, we get k in N / m.

If you counted cycles then each cycle corresponds to 2 pi radians, and you can easily get omega in rad / sec.

From the extra length when the weight was suspended, and from the mass of the weight, you can easily calculate k. 

• The force stretching the rubber band chain is F_grav = m g, or just the weight of your mass.
• You measured the change in the length.
• So just divide the weight in Newtons by the change in length in meters and you get the force constant.

e.g., if a weight of .95 Newtons results in a stretch of .04 meters, we have k = .95 N / (.04 m) = 24 N / m, approx..

Where does omega = sqrt(k / m) come from?

If Fnet = - k x, then since Fnet = m a we have

m a = - k x so that

a = - k / m * x.

a and x are the acceleration and position functions. 

• If we had a graph of x vs. t we could find the slopes of this graph and construct a graph of v vs. t.
• If we had a graph of v vs. t we could find the slopes of this graph and construct a graph of a vs. t.

So

• The v function is the derivative with respect to clock time of the x function and
• The a function is the derivative with respect to clock time of the v function so
• The a function is the second derivative with respect to clock time of the x function.

So

• a = x ''

and our equation

• a = - k / m * x

becomes

• x '' = -k / m * x.

Everyone should understand this development, whether you've had calculus or not.  You know that slope-taking is referred to a taking the derivative.  a = x '' just means you do this process twice to the x vs. t graph and end up with the a vs. t graph.

University Physics question:

What sorta functions have second derivatives that are multiples of the functions themselves?  Research this question, be sure you understand all possible answers, be sure you understand the chain rule, and be prepared next time.

What was the velocity with which the pearl pendulum struck the meter stick?

What was the angular momentum of the pearl before collision, relative to the axis of rotation?

What was the angular momentum of the meter stick immediately after collision?

What was the angular momentum of the pearl immediately after collision?

Was angular momentum conserved?