1-April-2015 Centripetal force with a motor

Purpose: To show mathematically (through the slope of a chart) that as the apparatus spins, thus increasing its centripetal acceleration, the omega (rotations per second converted to radians per second) increases, and so to will the angle the tether makes with the vertical axis.

Theory: As the centripetal acceleration increases on a mass tethered to the apparatus, so to will the angle made between the tether and vertical axis. The omega (w), which will be determined separately, can be expressed using that angle and measurements of the apparatus and forces on the mass. The expression should equal the w that was determined separately, and so the chart of w vs. the expression should make a straight line with 1/1 slope.

Procedure: Measure the height (H) of the apparatus, the Length of the tether (L), the distance between the mass and center of the apparatus at rest (R). Have the apparatus spin at differing speeds six separate times, timing how long it takes to complete ten rotations for each spin speed, and measure the height the mass makes with the ground during each of the spin speeds (h). Calculate w, and then find an expression for w using the measurements taken, using known formulas.

*The Apparatus

*H is measured to be 2 meters +/- 0.0005 meters

*L is measured to be 1.66 meters +/- 0.001 meters

*R is measured to be  0.96 meters +/- 0.0005 meters

*w is calculated by taking the 10 revolutions dividing by the time and multiplied by 2Pi of each spin speed.

times	Rad/s
39.79	1.579087
33.8	1.858931
28.7	2.189263
25.34	2.479552
21.39	2.937441
17.71	3.547818

*h will vary at different spin speeds.

h values (m)

0.455

0.578

0.832

1.02

1.26

1.48

* The angle L makes with the vertical axis at any spin speed can be determined by realizing a 90 degree triangle is made during each spin speed.

* The angle at each spin speed is than (arc cos ((H-h)/L))

angle (rad)

0.3744118

0.5421012

0.7903249

0.9392897

1.1087474

1.2521798

*Additionally the distance of the mass from the center of the apparatus will be (R+L sin (angle))

radius (m)

1.567104

1.816455

2.139566

2.299851

2.445934

2.536452

* The forces on the mass.

*T can be found to be mg/cos (angle), and so a_c=g*Tan (angle) with the mass canceled out.

*Centripetal acceleration (a_c) is equal to radius*w^2

* The radius was determined to be the distance= (R+L sin angle)

*Plugging all three equations into one another, and solving for w, gives us the expression
(w=Sq_root of (g*Tan (angle)/(R+L sin (angle))

 

Calculated w

1.567586033

1.80261615

2.15074983

2.413672734

2.836446352

3.422460172

Graph of Calculated w (using angle and other measurements) vs actual w

*We get a line that can be expressed as y=0.9455x+0.0661, meaning the slope is 0.9455

Conclusion

* There was only a 5.45% difference between the "actual" value of the expected w.

*Problems with the experiment are, the "actual" values of w were determined using the human eye and a stop watch, error could have come from human error here.

*The measurements did have uncertainty, which could have been accounted for in the actual experiment if I was more certain on how to calculate error beyond using propagated error from partial derivatives.

*Overall I am satisfied with the results, and feel experiment showed through the equations a relation between the angle formed by the centripetal acceleration and omega.