June-05-2015 Lab 20: Physical Pendulum

Purpose:  Derive expressions for the period of various physical pendulums and verify our predicted periods by experiment.

Part 1: predict the oscillation period of steel ring, and compare the result with the actual oscillation period by experiment.

We use the setup with a photogate (like the left picture) to determine the actual oscillation period of the steel ring.

At first, We measured the radius of outside ring is 0.06955 m.

the radius of inner ring is 0.06355 m.


the mass of steel ring is 0.405 kg.




we calculated the system of moment of inertia is 0.0035911 kg * m^2





We know net torque = moment of inertia times angular acceleration which from Newton's 2nd law.  Here is calculation:

we came up a equation. When the angle is very small, sinθ = θ.
then we got a equation of simple harmonic motion.

we got
angular speed = sqrt(mg(R+r)/2I)

and we know angular speed = 2pi/period

We predicted a period is 0.7235 s.

Then we use the set up like the left picture to measure the period.

We got the actual oscillation period 0.7218 s.















Here is calculation for uncertainty:

It's pretty nice to get only 0.2% uncertainty off.


Part 2:  Derive expressions for the period of the various physical pendulums.

Procedure: 
1. At first, we cut out or find objects of the appropriate shapes. Measure the appropriate dimensions of each one.   We measured base B = 10 cm, height H= 15.6 cm, mass of triangle m = 4 g, radius of semicircular plate R = 11.75 cm, mass of semicircular plate M = 11 g.

2. Attach a thin piece of masking tape to each object. Attach a pivot to the appropriate location in each object.
3. Use the same setup of part 1 with a photogate to determine the actual oscillation period of each object.
4. Compare the actual and theoretical values for the periods using the actual measured dimensions for the various objects.
  
Derive expressions for the period of the various physical pendulums:
a) Isosceles triangle, base B, height H, oscillating about its apex. 

We got the actual period T' = 0.7001 s from the program by experiment .

To determine the theoretical values for the periods, we need to find center of mass of the isosceles triangle :

After we got the position of center of mass, we could work out the moment of inertia of triangle oscillating about its apex I(top) = 0.000050338667 kg*m^2:

Then, it's easy to calculate the period T by using our measured dimensions from Newton's 2nd law:

Uncertainty of period is only 0.3% and less than 1%.

b) Isosceles triangle, base B, height H, oscillating about the midpoint of its base:

We got the actual period T' = 0.6039 s from the program by experiment.

We already know the position of center of mass and the moment of inertia of triangle oscillating about its apex I(top). We could calculate the I(cm) = 0.000007074667 kg*m^2 from the parallel axis theorem:

In addition, we could know the moment of inertia of triangle oscillating about the midpoint of its base I(bot) = 0.000017890667 kg*m^2 by using the parallel axis theorem: 

Here is result for the periods using the actual measured dimensions:

Uncertainty of period is only 0.3% and less than 1%.

c) a semicircular plate of radius R, oscillating about the midpoint of its base:

We got the actual period T' = 0.7433 s from the program by experiment.

To determine the theoretical values for the periods, we need to find center of mass of the semicircular plate :

After we got the position of center of mass, we could work out the moment of inertia of semicircular plate oscillating about the midpoint of its base I(top) = 0.000075934375 kg*m^2:

Then, we could predict for the period T using measured dimensions from Newton's 2nd law:

Uncertainty of period is only 0.4% and less than 1%.

d) a semicircular plate of radius R, oscillating about a point on its edge, directly above the midpoint of its base:

We got the actual period T' = 0.7433 s from the program by experiment:

We already know the position of center of mass and the moment of inertia of semicircular plate oscillating about the midpoint of its base I(top). We could calculate the moment of inertia of center of mass I(cm) = 0.0000048578781 kg*m^2 from the parallel axis theorem:

In addition, we could know the moment of inertia of semicircular plate oscillating about a point on its edge, directly above the midpoint of its base I(bot) = 0.000098892926 kg*m^2 by using the parallel axis theorem:

Then, we could predict for the period T using measured dimensions from Newton's 2nd law:

Uncertainty of period is only 0.1% and less than 1%.

Conclusion: 
This lab is to determine the actual oscillation periods of those five objects first, then using the knowledge we learned at class to calculate the periods with measured dimensions. At last, compare the actual period with calculated period for each object.  For those 5 objects, all of uncertainty is pretty low, they are less than 1%. We could say we did a successful experiment.