Conservation of Angular Momentum

We can use the rotational dynamics apparatus from the last experiment to examine the conservation of angular momentum. We will do this by allowing the two discs to spin independently, and measuring their angular momentum both before and after they collide (stick together and rotate as one disc). To perform this experiment, we will use the rotational dynamics apparatus. The apparatus should be leveled before beginning the experiment.

Name:

Lab Partners:

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

Procedure

Copy the moments of inertia of the discs from the results of the last experiment:
I_top = g cm² I_bottom = g cm²

Place the discs on the apparatus so that they rotate independently (the instructor will help you do this correctly).

Make sure the lower disc is not spinning. With the air on and the upper display selected, spin the upper disc. Do not spin it too fast, as readings over 600 are unreliable. Also, to minimize the effects of sliding during the collision, the reading should be below 200. Record the reading after it stablizes:
b_{before, 1} = bars / s
Then pull the pin (to cause the collision) and record the final reading after it stablizes:
b_{after, 1} = bars / s

Repeat step 2 nine more times:

b_{before, 2} = bars / s b_{after, 2} = bars / s
b_{before, 3} = bars / s b_{after, 3} = bars / s
b_{before, 4} = bars / s b_{after, 4} = bars / s
b_{before, 5} = bars / s b_{after, 5} = bars / s
b_{before, 6} = bars / s b_{after, 6} = bars / s
b_{before, 7} = bars / s b_{after, 7} = bars / s
b_{before, 8} = bars / s b_{after, 8} = bars / s
b_{before, 9} = bars / s b_{after, 9} = bars / s
b_{before, 10} = bars / s b_{after, 10} = bars / s

Analysis

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

w_{before, 1} = rad / s w_{after, 1} = rad / s
w_{before, 2} = rad / s w_{after, 2} = rad / s
w_{before, 3} = rad / s w_{after, 3} = rad / s
w_{before, 4} = rad / s w_{after, 4} = rad / s
w_{before, 5} = rad / s w_{after, 5} = rad / s
w_{before, 6} = rad / s w_{after, 6} = rad / s
w_{before, 7} = rad / s w_{after, 7} = rad / s
w_{before, 8} = rad / s w_{after, 8} = rad / s
w_{before, 9} = rad / s w_{after, 9} = rad / s
w_{before, 10} = rad / s w_{after, 10} = rad / s

Compute the initial angular momentum using the moment of inertia of the top disc and the final angular momentum using the total moment of inertia of both discs together:

L_{before, i} = I_top w_{before, i}
L_{after, i} = ( I_top + I_bottom ) w_{after, i}

Do this for each trial:

L_{before, 1} = g cm² / s L_{after, 1} = g cm² / s
L_{before, 2} = g cm² / s L_{after, 2} = g cm² / s
L_{before, 3} = g cm² / s L_{after, 3} = g cm² / s
L_{before, 4} = g cm² / s L_{after, 4} = g cm² / s
L_{before, 5} = g cm² / s L_{after, 5} = g cm² / s
L_{before, 6} = g cm² / s L_{after, 6} = g cm² / s
L_{before, 7} = g cm² / s L_{after, 7} = g cm² / s
L_{before, 8} = g cm² / s L_{after, 8} = g cm² / s
L_{before, 9} = g cm² / s L_{after, 9} = g cm² / s
L_{before, 10} = g cm² / s L_{after, 10} = g cm² / s

Compute the difference between the initial and final angular momenta for each trial:
DL_i = L_{after, i} - L_{before, i}

DL₁ = g cm² / s DL₆ = g cm² / s
DL₂ = g cm² / s DL₇ = g cm² / s
DL₃ = g cm² / s DL₈ = g cm² / s
DL₄ = g cm² / s DL₉ = g cm² / s
DL₅ = g cm² / s DL₁₀ = g cm² / s

Compute the average and absolute error of the quantities in the last step:
DL_avg = ( DL_{i, 1} + ... + DL_{i, 10} ) / 10
DL_avg = g cm² / s
DL_abs = Max ( DL_max - DL_avg , DL_avg - DL_min )
DL_abs = g cm² / s
Was angular momentum conserved in this experiment?

Compute the initial rotational kinetic energy using the moment of inertia of the top disc and the final rotational kinetic energy using the total moment of inertia of both discs together:

K_{before, i} = I_top w_{before, i}² / 2
K_{after, i} = ( I_top + I_bottom ) w_{after, i}² / 2

Do this for each trial:

K_{before, 1} = erg K_{after, 1} = erg
K_{before, 2} = erg K_{after, 2} = erg
K_{before, 3} = erg K_{after, 3} = erg
K_{before, 4} = erg K_{after, 4} = erg
K_{before, 5} = erg K_{after, 5} = erg
K_{before, 6} = erg K_{after, 6} = erg
K_{before, 7} = erg K_{after, 7} = erg
K_{before, 8} = erg K_{after, 8} = erg
K_{before, 9} = erg K_{after, 9} = erg
K_{before, 10} = erg K_{after, 10} = erg

Compute the difference between the initial and final rotational kinetic energy for each trial:
DK_i = K_{after, i} - K_{before, i}

DK₁ = erg DK₆ = erg
DK₂ = erg DK₇ = erg
DK₃ = erg DK₈ = erg
DK₄ = erg DK₉ = erg
DK₅ = erg DK₁₀ = erg

Compute the average and absolute error of the quantities in the last step:
DK_avg = ( DK_{i, 1} + ... + DK_{i, 10} ) / 10
DK_avg = erg
DK_abs = Max ( DK_max - DK_avg , DK_avg - DK_min )
DK_abs = erg
Was rotational kinetic energy conserved in this experiment?

Please send comments or suggestions to the author.

b_{before, 2} = bars / s				b_{after, 2} = bars / s
b_{before, 3} = bars / s				b_{after, 3} = bars / s
b_{before, 4} = bars / s				b_{after, 4} = bars / s
b_{before, 5} = bars / s				b_{after, 5} = bars / s
b_{before, 6} = bars / s				b_{after, 6} = bars / s
b_{before, 7} = bars / s				b_{after, 7} = bars / s
b_{before, 8} = bars / s				b_{after, 8} = bars / s
b_{before, 9} = bars / s				b_{after, 9} = bars / s
b_{before, 10} = bars / s				b_{after, 10} = bars / s

w_{before, 1} = rad / s				w_{after, 1} = rad / s
w_{before, 2} = rad / s				w_{after, 2} = rad / s
w_{before, 3} = rad / s				w_{after, 3} = rad / s
w_{before, 4} = rad / s				w_{after, 4} = rad / s
w_{before, 5} = rad / s				w_{after, 5} = rad / s
w_{before, 6} = rad / s				w_{after, 6} = rad / s
w_{before, 7} = rad / s				w_{after, 7} = rad / s
w_{before, 8} = rad / s				w_{after, 8} = rad / s
w_{before, 9} = rad / s				w_{after, 9} = rad / s
w_{before, 10} = rad / s				w_{after, 10} = rad / s

L_{before, 1} = g cm² / s				L_{after, 1} = g cm² / s
L_{before, 2} = g cm² / s				L_{after, 2} = g cm² / s
L_{before, 3} = g cm² / s				L_{after, 3} = g cm² / s
L_{before, 4} = g cm² / s				L_{after, 4} = g cm² / s
L_{before, 5} = g cm² / s				L_{after, 5} = g cm² / s
L_{before, 6} = g cm² / s				L_{after, 6} = g cm² / s
L_{before, 7} = g cm² / s				L_{after, 7} = g cm² / s
L_{before, 8} = g cm² / s				L_{after, 8} = g cm² / s
L_{before, 9} = g cm² / s				L_{after, 9} = g cm² / s
L_{before, 10} = g cm² / s				L_{after, 10} = g cm² / s

DL₁ = g cm² / s				DL₆ = g cm² / s
DL₂ = g cm² / s				DL₇ = g cm² / s
DL₃ = g cm² / s				DL₈ = g cm² / s
DL₄ = g cm² / s				DL₉ = g cm² / s
DL₅ = g cm² / s				DL₁₀ = g cm² / s

K_{before, 1} = erg				K_{after, 1} = erg
K_{before, 2} = erg				K_{after, 2} = erg
K_{before, 3} = erg				K_{after, 3} = erg
K_{before, 4} = erg				K_{after, 4} = erg
K_{before, 5} = erg				K_{after, 5} = erg
K_{before, 6} = erg				K_{after, 6} = erg
K_{before, 7} = erg				K_{after, 7} = erg
K_{before, 8} = erg				K_{after, 8} = erg
K_{before, 9} = erg				K_{after, 9} = erg
K_{before, 10} = erg				K_{after, 10} = erg

DK₁ = erg				DK₆ = erg
DK₂ = erg				DK₇ = erg
DK₃ = erg				DK₈ = erg
DK₄ = erg				DK₉ = erg
DK₅ = erg				DK₁₀ = erg