3-June 2015 Physical Pendulum Lab

Purpose: Derive an expression for the period of a half circle acting as a pendulum, and verify my predicted period by experiment.

Theory: Using the formula of Torque= Inertia*Angular acceleration, I can "rework" the expression into Angular acceleration= - angular frequency^2*Theta, and assuming theta is very small, I can find the period (T) as the period is just 2*pi/angular frequency.

Procedure: Cut out a semicircle "disk," measure its mass and radius, set it up to act as a pendulum once as swinging from the "flat bottom" and once from the curve. Give it a slight nudge to begin oscillating, and have the computer measure the period for each oscillation (add thin stripe of tape to the semi circle disk so it quickly breaks the "beam counter" of the computer for each oscillation of the pendulum, record actual result and compare with calculated result.

Apparatus

Period Results

*From the Curve

*Period  0.600427 seconds

*From the "flat bottom"

*Period 0.604880 seconds

Dimensions of Semi circle

Calculation to determine Center of mass in the vertical position

*This will be needed to determine the distance from the pivot as the semi circle acts as a pendulum

*Plugging in the Radius we get 0.003289202 meters from the bottom.

*So the distance between the pivot, when its at the flat bottom, is the same as the center of mass (d_1 = 0.003289202)

*And the distance between the pivot and the center of mass when the semi circle is pivoted at the curve is the Radius minus the distance of the center of mass (d_2=0.0044607980

Calculations of moment of Inertia of Semi-Circle

Calculation to determine period (From the flat bottom)

*Period from flat bottom: 0.14571 s, difference from actual 75.9%

*Period from the curve: 0.161102 s, difference from actual 73.2%

Conclusion

*Because of the amount of error I received, I do not consider this a very good experiment for confirming the period from an unusual pendulum.

Error Analysis

*Because we had to create our own semi-circle pendulums, I do not know if my semi-circle pendulum was ideal. It was traced from a protractor and cut into shape, so "grooves" and dimensions of the semi-circle may have invalidated my assumption from my calculations that I was working with an actual semi-circle.

*Theta needs to be extremely small for my calculations to reflect reality, since the actual periods were so much larger than my calculated amount, I believe theta was far too large during the experiment. Perhaps I added to much force to begin the oscillations.

*Measurements were all done with a ruler for the Semi-circle, causing uncertainty in measurements.

1-June 12015 Conservation of Energy/Conservation of Angular Momentum

Purpose: To show that energy and angular momentum are conserved, just like regular momentum.

Theory: We can determine the angular velocity (W_1) of a mass/ruler (M_r) just before it strikes another mass/clay (M_c), with the conservation of energy formula (energy initial = energy final), then find another angular velocity at the moment of collision, with the conservation of angular momentum formula (Inertia*W_1= Inertia* W_2), and finally use another conservation of energy set up to determine the max height (h) the clay reaches, when attached to the ruler, after the collision.

Procedure: Get a meter stick and pivot it in such away that it will swing like a pendulum, weigh for mass, and determine where the pivot will be in relation to the center of mass (parallel axis theorem). Have some clay that will stick to the meter stick after collision, weigh it for mass, and set it up so the "pendulum stick" will hit the approximate center of mass of the clay when released. Use logger pro and video caption, so the max height the clay/system reaches can be accurately measured.

Apparatus

System in Motion

Measurements

*Mass of Ruler/Meter Stick (M_r): 0.077 kg

*Length of Ruler: 1.0 m

*Distance of pivot in relation to the entire Ruler: 0.01 m

*Mass of Clay (M_c): 0.0373 kg

Data collected

*Max height (h) of clay was 0.1294 m

Calculations using conservation of energy

*We will treat the "at rest" point of the ruler's swing as the zero point for gravitational potential energy.

*Because the center of mass of the meter stick is 0.5 meters, and the pivot is 0.01 meters in relation to the ruler, the center of mass will be at -0.49 m from the zero point for gravitational potential energy

*We will use the parallel axis theorem when calculating the moment of inertia for the kinetic energy portion of the problem, since the ruler isn't rotating about its center of mass and the distance from the pivot is 0.49 meters.

*The angular velocity before collision (W_1) was determined to be 5.45 rad/s

Calculations using conservation of angular momentum

*This set up works so long as the clay is struck at the center of mass, so its distance from the pivot will be 0.99 meters.

*Angular velocity after collision (W_2) was determined to be 2.208 rad/s

Calculations using conservation of energy

*We will place the zero point of potential gravitational energy at the lowest point of the swing for easier calculations.

*Max height was determined to be 0.1337 meters

*Difference percentage from actual result: 3.32%

Conclusion

*Angular momentum is conserved, just like "regular" momentum, and can be proven with the conservation of angular momentum equation (Inertia*W_1= Inertia* W_2).

*This was a very good experiment for confirming this principle

Error Analysis

*Although minor there was some error/difference from the real world result and calculated result.

*The experiment didn't account for friction due to air resistance during the motion of the system, which would influence the conservation of energy equations.