GROUP MEMBERS: Xavier L., Billy J.
PURPOSE:
To analyze the conservation of angular momentum about a point that is external to a rolling ball.
THEORY:
A steel ball travelling down a friction-less incline has some linear momentum (p = mv). If the ball were to hit a free, rotating arm a distance away from the pivot, then we say that the system now has angular momentum (L = Iω). If there are no net forces acting on the system, then both linear and angular momentum are conserved.
APPARATUS:
Note: The apparatus was assembled beforehand for this particular lab.
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| Apparatus fully assembled |
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| Look familiar? |
We used kinematics to find the initial velocity of the steel ball at the bottom of the ramp (just like in the Ballistic Pendulum Lab).
After the ball hit the rotating arm, we used the Pasco rotational sensor to measure the ω and α of the system (refer to Angular Acceleration Lab). This allowed us to calculate the moment of inertia of the system.
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| Moment of Inertia Calculation |
DATA/GRAPHS:
Mball = 28.9 +/- 0.1 g
Mhanging = 24.6 +/- 0.1 g
diameter of ball = 19.0 +/- 0.1 mm
diameter of pulley = 49.9 +/- 0.1 mm
Trial 1:
R ball to axis = 7.6 +/- 0.1 cm
ω (experimental) = 2.269 rad/s
ω (theoretical) = 2.4304 rad/s
% difference = 6.6%
Trial 2:
R ball to axis = 4.2 +/- 0.1 cm
ω (experimental) = 1.353 rad/s
ω (theoretical) = 1.487 rad/s
% difference = 9.0%
ANALYSIS:
After measuring the omega of our system at two different radii from the pivot of the rotating arm, we found that angular speed of the ball increased as the radius increased. For example, in the first experiment, we acquired an experimental value of ω = 2.269 rad/s while for the second experiment (a smaller radius), we measured ω = 1.353 rad/s. This is consistent with our theory because the angular momentum of the system is given by the equation L = Iω = p x r (where r is the length of the moment arm). This means that since linear momentum is proportional angular momentum by a factor of r, we should expect the angular speed of the system to increase as the radius from the pivot increases. Furthermore, the linear momentum of the system should remain constant at the bottom of the ramp because we dropped the ball at the same height for both trials.
CONCLUSION:
Our lab was a success. We were able to competently identify and mathematically demonstrate the relationship between angular momentum and the radius at which the ball was caught. On the other hand, this does not hide the fact that there were some inherent uncertainties in our theoretical calculations for the omega of the system. On average, our trials had approximately 8% error between the two. This error can most likely be attributed to certain assumptions we made about the experiment. For instance, we did not take into account any frictional torque, air resistance, kinetic friction or other external forces on the system. Additionally, we could have made errors in taking the measurements of crucial components of the system, such as mass, radius, and relative distances.
Mhanging = 24.6 +/- 0.1 g
diameter of ball = 19.0 +/- 0.1 mm
diameter of pulley = 49.9 +/- 0.1 mm
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| ω and α of the rotating arm system |
α up = 5.279 rad/s^2
α down = -5.982 rad/s^2
α avg = 5.6305 rad/s^2
Trial 1:
R ball to axis = 7.6 +/- 0.1 cm
ω (experimental) = 2.269 rad/s
ω (theoretical) = 2.4304 rad/s
% difference = 6.6%
Trial 2:
R ball to axis = 4.2 +/- 0.1 cm
ω (experimental) = 1.353 rad/s
ω (theoretical) = 1.487 rad/s
% difference = 9.0%
ANALYSIS:
After measuring the omega of our system at two different radii from the pivot of the rotating arm, we found that angular speed of the ball increased as the radius increased. For example, in the first experiment, we acquired an experimental value of ω = 2.269 rad/s while for the second experiment (a smaller radius), we measured ω = 1.353 rad/s. This is consistent with our theory because the angular momentum of the system is given by the equation L = Iω = p x r (where r is the length of the moment arm). This means that since linear momentum is proportional angular momentum by a factor of r, we should expect the angular speed of the system to increase as the radius from the pivot increases. Furthermore, the linear momentum of the system should remain constant at the bottom of the ramp because we dropped the ball at the same height for both trials.
CONCLUSION:
Our lab was a success. We were able to competently identify and mathematically demonstrate the relationship between angular momentum and the radius at which the ball was caught. On the other hand, this does not hide the fact that there were some inherent uncertainties in our theoretical calculations for the omega of the system. On average, our trials had approximately 8% error between the two. This error can most likely be attributed to certain assumptions we made about the experiment. For instance, we did not take into account any frictional torque, air resistance, kinetic friction or other external forces on the system. Additionally, we could have made errors in taking the measurements of crucial components of the system, such as mass, radius, and relative distances.
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