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.

June-04-2015 Lab 19: Conservation of Energy/Conservation of angular momentum

This experiment goes like:
We release a meter stick, pivoted at or near one end, from a horizontal position. Right when the meter stick reaches the bottom of its swing it collides inelastically with a blob of clay. The meter stick and clay continue to rotate to some final position. Like the picture:

Purpose: 
Measure the mass of meter stick and clay, and length of meter stick to work out how high the clay-stick combination should rise by using Laws of Conservation of Energy and Conservation of angular momentum. Then capture the the experiment on video and compare our actual results with our predictions.

The calculation of Predict: 
1. We measured the length of meter stick L = 1 m, the mass of meter stick M = 77 g. the mass of clay m = 30 g.

Because we did not pivot the meter stick actually at one end (There is 2 cm away from one end). We had to work out the new stick of moment of inertia first. After calculated, we got the new stick of moment of inertia I(stick) = 0.02416 kg*m^2.


2. Before and after collision :
Before collision: We release the meter stick from a horizontal position. At the moment that before meter stick reaches the bottom of its swing to hit the clay, the meter stick will have a angular speed w(0). We could calculate w(0) by using laws of conservation of energy.
At first, we set the position of zero Gravity Potential Energy at middle of pivot point between bottom which is at 48cm on meter stick. So meter stick have M*g*0.48 GPE before we released it. Because rotating objects have rotational Kinetic Energy, we know GPE all transfer to rotational Kinetic Energy which is GPE = KE from laws of conservation of energy.
we got
           M*g*0.48 = 1/2 * I(stick)*w(0)^2.
We know I(stick) from step 1.  We calculated w(0) = 5.81 rad/s.

After collision: After meter stick hit the clay, clay and meter stick will stick together. The mass of system will be different (mass of system = m(mass of clay) + M(mass of meter stick)), the system of moment of inertia will be different (I(system) = I(stick) + m*(L-0.02)). So the angular speed is not w(0) any more. There is a angular speed of system w(f) that we could calculate it by using laws of conservation of angular momentum. Here is calculation:

We got w(f) = 2.6499 rad/s.

3. Then the clay-stick combination continue to rotate together to a final position. There is a angle θ between the position of combination at bottom and the final position of combination. We set the position of zero GPE at bottom of combination.

From Law of conservation of energy, we got
       1/2 * I(system) * w(f)^2 = M*g*(0.48 - 0.48cosθ) + m*g*(0.98 - 0.98cosθ)

We could calculate cosθ. Once we know cosθ, we can work out the h' which is the highest position of clay-stick combination. Here is calculation:

We got h'= 0.28024 m.

4. We already have our result with prediction. Now we need to capture the experiment on video.
Here is the Set up:

One meter stick, pivoted at near one end.

One clay which could stick with things if that thing hit it.

A laptop which connect with a video to capture the whole experiment.


We measured the length of meter stick L = 1 m, the mass of meter stick M = 77 g. the mass of clay m = 30 g.






We released that meter stick from horizontal position, then we captured the whole process of experiment by video. After processed the video by using the advanced program in the laptop, We easily got the height of the clay-stick combination rose:

We got h' = 0.279 m from this experiment.

There is uncertainty by Comparing our actual result with our prediction:

Conclusion:

We came up with a prediction for how high the clay-stick combination should rise by using laws of Conservation of Energy or angular momentum which states total energy or angular momentum of a system remains constant.    Then we obtained the actual result from the experiment.    After compared our actual result with our prediction, there is 0.4% uncertainty.   For this experiment, I think there are some energy lost when meter stick collided with clay. That means the total energy of that system are not perfectly constant. Air resistance will also cause energy lose. In addition, we may not release the meter stick from a perfectly horizontal position. All of things could effect the actual result.