June 3rd, 2015; Physical Pendulum Lab

Purpose:

Derive an expression for period of oscillation for 3 different geometric shapes (also finding moment of inertia) and compare the value to the experimental data.

Part 1: Finding metal ring's period of oscillation

Experiment:

Finding predicted value for "T":
- Using T= 2π / ω, we have to find ω to find T. In simple harmonic motion, ω^2 comes from 
α = - constant * θ, where constant = ω^2.

- Moment of inertia of this geometric shape around its center is (1/2)M (R^2 +  r^2)

- We use net torque equation as this experiment is related to rotational motion. The rigid body diagram (not shown here but similar to those of the triangle and semicircle) gives us mgsinθ, the force perpendicular to the distance from the pivot to center of mass. That distance, therefore is R-average = (outer radius + inner radius)/2. 

- Use Parallel Axis Theorem to shift axis from the center to where distance of shift = R-average.

The experiment
- Weigh the ring's mass and use caliper for diameter.
- The ring has a hole that could go in the pivot point.
- Open "Pendulum Timer.cmbl" to set up photogate to record the period of the metal ring. Attach a thin stick note to the bottom of the ring so that when oscillation takes place, the photogate could detect from the stick note.
- Gently tap the ring so the oscillation occurs at a small angle displacement; collect data.

Data and Analysis:

- The period data should be consistent of one another (represented by its straight line from the graph).

- Our metal ring's period is T= 0.7196 sec. Calculating percent error gives us ~ 0.3%.

Part 2: Triangle and semicircle period of oscillation:

Experiment:

- Cut out the shape of isosceles triangle and semi circle.

- Measure the dimensions of the shapes and their mass.

- Attach masking tape to electrical connectors, one on each side so the hole allows for a pivot point roughly close to the tip of the object.Use paper clip to pivot the objects.

- Set up photogate and have a sticker (attached from bottom of the object) go through the photogate.

- Measure the period of the pendulum at pivot:

a). isosceles triangle oscillating about its apex

b). isosceles triangle oscillating about the midpoint of its base

c). semicircular plate oscillating about the midpoint of its base

d). semicircular plate oscillating about a point of its edge, directly above the midpoint of its base.

Data and Analysis:

M-triangle = 0.0093 +/- 0.001 kg; base = 0.15 +/- 0.001 m; height = 0.175 +/- 0.001 m

M- semicircle = 0.0112 +/- 0.001 kg; radius = 0.108 +/- 0.001 m

- Below is one of the four screenshots taken after period data is collected.

T (semicircle, pivot at midpoint of base) = 0.695931 s

a). and b).  

- Since we're dealing with oscillation, we can write a net torque equation (and Newton's second law) to get the equation to be in a form of: acceleration = -constant * displacement. We also know that one side of torque equation has Iα in it. So moment of inertia is needed.

-  Finding center of mass (left-hand side), and moment of inertia of triangle (pivot at apex) respectively.

- For "I", use dI(rod) around its center and use Parallel Axis Theorem To shift every dm that is a distance "y" from the new pivot point to that pivot point.

- Deriving moment of inertia for triangle pivot at middle of its base shown below,

- Coming up with an expression for "T' for triangle pivot at midpoint of base and comparing this "T" to the one from LoggerPro.

- Similar process here for "T" of triangle pivot at apex. Error this time turns out to be ~ 2.6%

c). and d).
- On left-hand side, moment of inertia of semicircle at midpoint of its base, middle: finding center of mass with setting origin at midpoint of the base, right-hand side: expression for moment of inertia of semicircle pivot directly above midpoint of base.

- Calculating period and comparing it to experimental value for semicircle pivot at midpoint of base,

- Calculating period and comparing it to experimental value for semicircle pivot at point directly above midpoint of base,

- Despite the written order of this lab, deriving an expression for the wanted value first is always a good idea before doing the experiment.

Conclusion:

From knowing the radius of the semicircle and base and height of a triangle, we could derive an equation for period. From both parts of the lab, we see that period of physical pendulum is independent on mass of the oscillating object. Our range of error and uncertainty is between 0.6 to 3%. In most cases, experimental values should be slightly larger and that brings us to the discussion of errors and uncertainty. The larger values result from the neglected friction at pivot point and the fact that the pivot is not exactly where it's meant to be ideally but above it. This unaccountable shift of distance would alter everything including the period of oscillation. Other usual uncertainty is from mass and length measurement. Nonetheless, within our range of uncertainty, our derived expression agrees with the experimental values of "T".

May 20th, 2015: Lab 19: Conservation of Energy and Angular Momentum

Purpose:

Given a clay+meter stick system, by releasing the meter stick to collide inelastically with the clay, show that there's conservation of energy and conservation of momentum in the system.

Experiment:

Data: M(stick+ clay) = 0.113 +/- 0.001 kg

m-meter stick = 0.080 +/- 0.001 kg

Coming up with predicted "h"
- Noticing that there's 3 parts to this lab to find "h",

• conservation of energy of meter stick before released to meter stick right before collision
• conservation of angular momentum of meter stick right before collision and the system right after collision
• conservation of energy of the system from right after inelastic collision to the maximum height reached.

* Energy and momentum approaches are more doable and kinematics wouldn't be possible because angular acceleration is not constant at all time.

- Note that at pivot on meter stick cannot be exactly at the edge. Using Parallel Axis Theorem gets us the pivot that, in our case, is 0.026 m away from the edge. So our "d" of shifting is 0.5-0.026= 0.474

- the three parts shown in calculation below,

- Note that GPE = 0 is set at the pivot for part 1 and 3. After finding cosθ, we can now find "h":

The experiment
- Take mass measurements of meter stick and clay(also with the paper clips and tape around it).
- Set up the apparatus. Place one farthest end of meter stick to the pivot point that's attached a standing rod.

camera seen attached to the bottom of chair

- Connect camera to Loggerpro for data collection. Place the camera such that the field of view show the collision and the system raised to maximum height.
- Test out the setup by releasing the meter stick, held horizontally, and make sure that it stays with the clay (after collision) all the way till they reach maximum height.
- For video capture setup, set scale and axis. Set scale by measuring the width of the whiteboard behind the setup. The axis should preferably be centered at where the clay is, for easier video analysis.

- Since the only data we mainly want from the experiment is the maximum height reached, when adding point series, we can just put two points, blue dots from the screenshot above.
- The lower dot won't be exactly at zero so we have to subtract between the two dots to find experimental value for "h".

Data Analysis:

- From the y-axis of point series data, we get h= 0.2419 - 0.007486 = 0.234414 m.

 - Comparing to predicted value and calculation of error and uncertainty:

- Experimental "h" should be smaller then predicted value due to other unaccounted error and certainty during calculation.

Conclusion

Our error and uncertainty include friction at pivot (we test it out with only meter stick released from one end doesn't travel to the same height at the other end), kinetic friction from the floor during the collision (from scraping of clay with the floor tends to slow system down), imperfect stability at the pivot point. The usual systematic error result from measurement equipments and poor video quality. Nonetheless, our experimental "h" is smaller than the predicted due to the errors and uncertainty described, and so it's safe to conclude that there's conservation of energy and conservation of angular momentum in the meter stick+ clay system.