Lab 9: Centripetal force with a motor

Title: Centripetal force with a motor

Purpose:
This lab aims to produce a relationship between theta and omega in a rotating system where the rotating object is affected only by gravity and centripetal force.

Theory:
As an object rotates in a circle, it will be affected by a centripetal force directed towards the center of its path. If this object is attached to a freely moving string, then the centripetal force can be related to the angle theta of the string from vertical as gravity, a component of the tension in the string, is constant and can thus be used with trigonometric identities to create a model describing angular velocity as it relates to theta. By finding the height to which the string rises above its base length while stationary, it is possible to find the angle at which the attached object rotates, calculate a value for centripetal force, and then calculate angular velocity of the object.

Apparatus:

• Powered rotating stand
• Adjustable power supply
• 2 meter stick (1)
• Stopwatch (1)
• String
• Small object (1)
• Stand with clamp and paper

Procedure:

Measure the overall height H of the rotating stand. Measure out a length l of string that nearly matches the height of the stand but does not touch the ground when tied to the top of the stand. Measure the length R of the horizontal rod on top of the powered stand from its axis of rotation to where the string is attached. Set the power supply to some output and measure the time taken for 10 rotations. Move the stand with paper clamped to it near the path of the rotating object, slowly moving the paper up until the object collides. Measure the height from the ground to the spot of collision on the paper.

Data and Graphs:

Table of collected data and calculations

H	1,79	m	rev	10
R	0,749	m	g	9,81
l	1,585	m
t (s)	ω (rad/s)	y (m)	ly (m)	θ (deg)	ω (calc)
32,92	1,908623	0,385	1,405	27,57133	1,858734
28,21	2,22729	0,623	1,167	42,58472	2,224776
22,93	2,740159	0,935	0,855	57,35501	2,710945
19,46	3,228769	1,16	0,63	66,57949	3,205976
16,41	3,828876	1,334	0,456	73,27986	3,795436
13,8	4,553033	1,422	0,368	76,57475	4,235706

Photo of experimental setup

Photo of derivation of formula relating angular velocity to angle theta

Data Analysis:

Calculations used to reach a value of omega from theta are shown in photos above.

Conclusion:

Our model produced values of omega that were near true values of omega, with only 2.06% error on average. These results suggest that this model is very accurate given the error prone experimental setup, particularly as the rotating stand was found to be unstable, tending to bend, tilt, and otherwise deviate from ideal conditions. The method for obtaining the height of the object as it rotated could also have been improved by using a more precise method of incrementing the height of the paper.