Angular Motion and Torque

By applying a known torque to a freely rotatable disc and measuring the resultant angular acceleration, we can compute the moment of inertia of the disc. The known torque will result from a falling mass attached to the disc with a pulley.

Since the known torque is

t = m g r

and the torque is related to the angular acceleration by

t = I a,

the moment of inertia is

I = m g r / a.

To perform this experiment, we will use a rotational dynamics apparatus, a caliper and a balance. The apparatus should be levelled before beginning the experiment.

Name:

Lab Partners:

When entering numeric data, use exponentials: ie., 1.6 * 10^-19 = 1.6E-19.

Procedure

Measure the mass of each disc (1 is the top disc and 2 is the bottom):
m₁ = g m₂ = g

their inner and outer radii:
r_{1, in} = cm r_{2, in} = cm
r_{1, out} = cm r_{2, out} = cm

the radius of the pulley (use the small pulley):
r_p = cm

and the mass of the weight hanger:
m_h = g

Place the discs and pulley on the apparatus so that all three rotate together (use the pin to make sure the bottom disc spins freely, and use the open screw so both discs spin together; the instructor will help you do this correctly).

Attach a 20 g weight to the end of the string and, with the air on at approximately 9 PSI, rotate the discs until the weight is near the air pulley on the edge of the apparatus. With the upper display selected, wait until the display reads zero. Then remove your hand from the discs and record the numbers on the display until the weight hits the floor:

b_{1, 1} = bars / s b_{1, 5} = bars / s
b_{1, 2} = bars / s b_{1, 6} = bars / s
b_{1, 3} = bars / s b_{1, 7} = bars / s
b_{1, 4} = bars / s b_{1, 8} = bars / s

Ignore the first nonzero reading and any readings after the weight hits the floor.

Repeat the last step four more times:

b_{2, 1} = bars / s b_{2, 5} = bars / s
b_{2, 2} = bars / s b_{2, 6} = bars / s
b_{2, 3} = bars / s b_{2, 7} = bars / s
b_{2, 4} = bars / s b_{2, 8} = bars / s

b_{3, 1} = bars / s b_{3, 5} = bars / s
b_{3, 2} = bars / s b_{3, 6} = bars / s
b_{3, 3} = bars / s b_{3, 7} = bars / s
b_{3, 4} = bars / s b_{3, 8} = bars / s

b_{4, 1} = bars / s b_{4, 5} = bars / s
b_{4, 2} = bars / s b_{4, 6} = bars / s
b_{4, 3} = bars / s b_{4, 7} = bars / s
b_{4, 4} = bars / s b_{4, 8} = bars / s

b_{5, 1} = bars / s b_{5, 5} = bars / s
b_{5, 2} = bars / s b_{5, 6} = bars / s
b_{5, 3} = bars / s b_{5, 7} = bars / s
b_{5, 4} = bars / s b_{5, 8} = bars / s

Analysis

Multiply each reading by p / 100 to convert from bars per second to radians per seconds:

w_{1, 1} = rad / s w_{1, 5} = rad / s
w_{1, 2} = rad / s w_{1, 6} = rad / s
w_{1, 3} = rad / s w_{1, 7} = rad / s
w_{1, 4} = rad / s w_{1, 8} = rad / s

w_{2, 1} = rad / s w_{2, 5} = rad / s
w_{2, 2} = rad / s w_{2, 6} = rad / s
w_{2, 3} = rad / s w_{2, 7} = rad / s
w_{2, 4} = rad / s w_{2, 8} = rad / s

w_{3, 1} = rad / s w_{3, 5} = rad / s
w_{3, 2} = rad / s w_{3, 6} = rad / s
w_{3, 3} = rad / s w_{3, 7} = rad / s
w_{3, 4} = rad / s w_{3, 8} = rad / s

w_{4, 1} = rad / s w_{4, 5} = rad / s
w_{4, 2} = rad / s w_{4, 6} = rad / s
w_{4, 3} = rad / s w_{4, 7} = rad / s
w_{4, 4} = rad / s w_{4, 8} = rad / s

w_{5, 1} = rad / s w_{5, 5} = rad / s
w_{5, 2} = rad / s w_{5, 6} = rad / s
w_{5, 3} = rad / s w_{5, 7} = rad / s
w_{5, 4} = rad / s w_{5, 8} = rad / s

How many bars are around each disc?

The time between readings is two seconds. Compute the angular acceleration between each pair of readings by dividing their difference by two:

a_{1, i} = ( w_{1, i+1} - w_{1, i} ) / 2

a_{1, 1} = rad / s² a_{1, 5} = rad / s²
a_{1, 2} = rad / s² a_{1, 6} = rad / s²
a_{1, 3} = rad / s² a_{1, 7} = rad / s²
a_{1, 4} = rad / s²

a_{2, 1} = rad / s² a_{2, 5} = rad / s²
a_{2, 2} = rad / s² a_{2, 6} = rad / s²
a_{2, 3} = rad / s² a_{2, 7} = rad / s²
a_{2, 4} = rad / s²

a_{3, 1} = rad / s² a_{3, 5} = rad / s²
a_{3, 2} = rad / s² a_{3, 6} = rad / s²
a_{3, 3} = rad / s² a_{3, 7} = rad / s²
a_{3, 4} = rad / s²

a_{4, 1} = rad / s² a_{4, 5} = rad / s²
a_{4, 2} = rad / s² a_{4, 6} = rad / s²
a_{4, 3} = rad / s² a_{4, 7} = rad / s²
a_{4, 4} = rad / s²

a_{5, 1} = rad / s² a_{5, 5} = rad / s²
a_{5, 2} = rad / s² a_{5, 6} = rad / s²
a_{5, 3} = rad / s² a_{5, 7} = rad / s²
a_{5, 4} = rad / s²

Compute the average angular acceleration (the sum of all a's divided by 35 ):

a = rad / s²
and the absolute and relative error:

a_abs = Max ( a_max - a , a - a_min )
a_abs = rad / s²

a_rel = a_abs / a * 100 %
a_rel = %

The moment of inertia of each disc is

I_i = m_i (r_{i, in} ² + r_{i, out} ²) / 2.
Compute the expected total moment of inertia as the sum of that for each disc:

I_expected = I₁ + I₂
I₁ = g cm² I₂ = g cm²

I_expected = g cm²

Then compute the moment of inertia of the discs together using the formula
I = ( m_h + 20 ) g r_p / a
I = g cm²

Compute the relative error between the expected and measured moments of inertia:
I_rel = | I - I_expected | / I_expected * 100 %
I_rel = %

Please send comments or suggestions to the author.

b_{1, 1} = bars / s				b_{1, 5} = bars / s
b_{1, 2} = bars / s				b_{1, 6} = bars / s
b_{1, 3} = bars / s				b_{1, 7} = bars / s
b_{1, 4} = bars / s				b_{1, 8} = bars / s

b_{2, 1} = bars / s				b_{2, 5} = bars / s
b_{2, 2} = bars / s				b_{2, 6} = bars / s
b_{2, 3} = bars / s				b_{2, 7} = bars / s
b_{2, 4} = bars / s				b_{2, 8} = bars / s

b_{3, 1} = bars / s				b_{3, 5} = bars / s
b_{3, 2} = bars / s				b_{3, 6} = bars / s
b_{3, 3} = bars / s				b_{3, 7} = bars / s
b_{3, 4} = bars / s				b_{3, 8} = bars / s

b_{4, 1} = bars / s				b_{4, 5} = bars / s
b_{4, 2} = bars / s				b_{4, 6} = bars / s
b_{4, 3} = bars / s				b_{4, 7} = bars / s
b_{4, 4} = bars / s				b_{4, 8} = bars / s

b_{5, 1} = bars / s				b_{5, 5} = bars / s
b_{5, 2} = bars / s				b_{5, 6} = bars / s
b_{5, 3} = bars / s				b_{5, 7} = bars / s
b_{5, 4} = bars / s				b_{5, 8} = bars / s

w_{1, 1} = rad / s				w_{1, 5} = rad / s
w_{1, 2} = rad / s				w_{1, 6} = rad / s
w_{1, 3} = rad / s				w_{1, 7} = rad / s
w_{1, 4} = rad / s				w_{1, 8} = rad / s

w_{2, 1} = rad / s				w_{2, 5} = rad / s
w_{2, 2} = rad / s				w_{2, 6} = rad / s
w_{2, 3} = rad / s				w_{2, 7} = rad / s
w_{2, 4} = rad / s				w_{2, 8} = rad / s

w_{3, 1} = rad / s				w_{3, 5} = rad / s
w_{3, 2} = rad / s				w_{3, 6} = rad / s
w_{3, 3} = rad / s				w_{3, 7} = rad / s
w_{3, 4} = rad / s				w_{3, 8} = rad / s

w_{4, 1} = rad / s				w_{4, 5} = rad / s
w_{4, 2} = rad / s				w_{4, 6} = rad / s
w_{4, 3} = rad / s				w_{4, 7} = rad / s
w_{4, 4} = rad / s				w_{4, 8} = rad / s

w_{5, 1} = rad / s				w_{5, 5} = rad / s
w_{5, 2} = rad / s				w_{5, 6} = rad / s
w_{5, 3} = rad / s				w_{5, 7} = rad / s
w_{5, 4} = rad / s				w_{5, 8} = rad / s

m₁ = g				m₂ = g

r_{1, in} = cm				r_{2, in} = cm
r_{1, out} = cm				r_{2, out} = cm

a_{1, 1} = rad / s²				a_{1, 5} = rad / s²
a_{1, 2} = rad / s²				a_{1, 6} = rad / s²
a_{1, 3} = rad / s²				a_{1, 7} = rad / s²
a_{1, 4} = rad / s²

a_{2, 1} = rad / s²				a_{2, 5} = rad / s²
a_{2, 2} = rad / s²				a_{2, 6} = rad / s²
a_{2, 3} = rad / s²				a_{2, 7} = rad / s²
a_{2, 4} = rad / s²

a_{3, 1} = rad / s²				a_{3, 5} = rad / s²
a_{3, 2} = rad / s²				a_{3, 6} = rad / s²
a_{3, 3} = rad / s²				a_{3, 7} = rad / s²
a_{3, 4} = rad / s²

a_{4, 1} = rad / s²				a_{4, 5} = rad / s²
a_{4, 2} = rad / s²				a_{4, 6} = rad / s²
a_{4, 3} = rad / s²				a_{4, 7} = rad / s²
a_{4, 4} = rad / s²

a_{5, 1} = rad / s²				a_{5, 5} = rad / s²
a_{5, 2} = rad / s²				a_{5, 6} = rad / s²
a_{5, 3} = rad / s²				a_{5, 7} = rad / s²
a_{5, 4} = rad / s²