Lab 16: Angular Acceleration

Title: Angular Acceleration

Purpose: Find the relationship between torque and angular acceleration and determine the moment of inertia of a disc rotated by a constant force by measuring the magnitude of force used to accelerate the disc, the radius and mass of the disc, and the angular acceleration of the disc.

Theory: Newton's second law F=ma describes the motion as an object undergoes translational motion. This equation can also be related to rotational motion where torque t equals intertia I multiplied by angular acceleration a for t=Ia. By combining the use of these two equations where F=ma describes a falling mass and t=Ia describes a rotating object accelerated by the falling mass, an equation for inertia of the rotating object can be derived. This can then be compared to the mathematical solution to inertia where I=(1/2)mr^2 for a rotating disc.

Apparatus:

• Steel disc (2)
• Aluminum disc (1)
• 50g masses (2)
• 50g hanging mass (1)
• Digital scale
• Vernier caliper
• String
• Small radius pulley
• Large radius pulley
• Supplied experimental apparatus (see picture)
• LoggerPro

Procedure:
Using the caliper and digital scale, measure the diameter and masses of the 2 steel and 1 aluminum discs. Using the small radius pulley, record the angular acceleration of a single rotating steel disc as it is pulled by masses of 50, 100g, and 150g (To rotate a single disc, insert a solid pin into the center of the disc). Then, using the large radius pulley and a 50g mass, measure and record the angular acceleration of a steel disc, an aluminum disc, and two steel discs. Using measurements of angular acceleration, calculate a value of inertia for each disc and compare to the true value calculated using mathematical methods.

Data and Graphs:

Photo of experimental setup and apparatus:

Table of collected and calculated data:

Msteel top	1,362	kg		rsteel top	0,0632	m
Msteel bot	1,348	kg		rsteel bot	0,0632	m
Maluminum	0,466	kg		raluminum	0,0632	m
Mpulley-s	0,01003	kg		rpulley-s	0,0124	m
Mpulley-l	0,03625	kg		rpulley-l	0,02495	m
Mass (kg)	Pulley	Disc(s)	α up (rad/s)	α down	α avg	Inertia (kgm^2)	True Inertia	%Error
0,05	Small	Steel	1,165	-1,315	1,240	0,002721	0,00272	0,0239
0,1	Small	Steel	2,417	-2,614	2,516	0,002678	0,00272	-1,5469
0,15	Small	Steel	3,556	-3,955	3,756	0,002686	0,00272	-1,2371
0,05	Large	Steel	2,323	-2,671	2,497	0,002706	0,00272	-0,5350
0,05	Large	Aluminum	6,591	-7,197	6,894	0,000969	0,00093	4,1102
0,05	Large	2 Steel	1,179	-1,297	1,238	0,005475	0,00541	1,1517

Graph of angular acceleration vs hanging mass (torque)

Graph of angular acceleration vs disc inertia

Data Analysis:
The equation used to calculate inertia was derived in class, the derivation is also included in the lab manual. Graphs were made based on the variable being changed (mass and inertia). The true value of inertia was calculated using the equation I = 1/2 MR^2 found using mathematics and integrals. Percentage error was calculated by comparing experimental values to the accepted true values of inertia.

Photo of derivation:

Conclusion:
Angular acceleration and torque applied are related linearly where an increase in torque will cause a proportional increase in angular acceleration. This is because torque and angular acceleration are directly related through the equation t=Ia. A graph of angular acceleration vs inertia produced an inverse relationship again corresponding to the previous equation where inertia multiplied by angular acceleration will equal a constant value of torque. In comparing our experimental inertia to the true inertia of each disc, it was found that most values differed by approximately 1% while the inertia of the aluminum disc differed by approximately 4%. This larger difference may be due to its lower mass as a lower value of inertia means that small errors become more significant in calculations. However, as seen in the different values of angular acceleration while the mass moved up and down, friction was not negligible. This would explain the greater value of inertia calculated for the aluminum disc as lower angular acceleration due to friction would make it appear as if the aluminum disc has greater inertia.