GROUP MEMBERS:
Christian R.and Haokun Z.PURPOSE:
To calculate the time it takes for a cart to travel down a ramp when its acceleration is inhibited by frictional torque.
THEORY:
Consider a metal disk rotating about a central shaft. If we find the moment of inertia of this object as well as its angular deceleration, then we can find the frictional torque (-Tfriction = Iα) acting upon the disk. We can then use Newton's second law to create an expression for the acceleration of a cart attached to the central shaft of the disk. Utilizing the acceleration we just calculated, we can then use kinematics to find the time it takes for the cart to travel one meter down the ramp.
APPARATUS:
Our apparatus consists of two main parts. The first section is a large metal disk which rotates about a central shaft. The second section is a friction-less ramp. Notice that a string is tied to the central shaft of the disk such that it can be wound and released for the cart to travel down the ramp.
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| Inertial Disk |
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| Disk + Cart + Ramp Setup fully assembled |
First, we found the moment of inertia of the system by treating the metal disk as three distinct cylinders and summing up their individual moments of inertia. Next, we spun the metal disk in order to measure its angular displacement over a period of time. This allowed us to use kinematics to find the angular deceleration of the disk. With this information, we were able to find the frictional torque of the system. Then, we we attached the cart to the metal disk so we could measure the time it took for the cart to travel one meter down the ramp.
DATA/GRAPHS:
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| Measurements |
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| Mass and Volume Calculations |
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| Moment of Inertia Calculations |
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| Frictional Torque Calculation |
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| Time to roll down ramp calculation |
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| Calculated Results |
ANALYSIS:
By finding the moment of inertia and the angular deceleration of the system, we were able to calculate the frictional torque that impeded the acceleration of the small cart moving down a ramp. After using Newton's second law to find the linear acceleration of the cart, we were able to calculate the theoretical time it would take for the cart to move one meter down the ramp. For our experimental approach, we simply used a stopwatch to measure how long it took for the cart to run the same distance.
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| Final Lab Results |
CONCLUSION:
In this lab, we were not as successful as we had initially hoped. Our theoretical lab results for the time ended up being over 140% different from what we experimentally measured. There are several factors that could have attributed to this uncertainty. For example, our group could have erroneously calculated the volume of the cylinders. This is a crucial calculation because the mass of each component of the inertial disk relies on the accurate calculation of their volumes. Another factor to consider is the calculation of the frictional torque. This is because the frictional torque of the system is dependent upon many measurable parts. For instance, the moment of inertia relies upon mass and radius while the angular acceleration relies upon the measurement of angular displacement and duration. The latter two components are the most likely culprits because our group "eyeballed" both the angular displacement and the time it took for the inertial disk to complete it.
There is an error in the sign of your frictional torque in your "time to roll down the ramp" equation. If you change the sign, the predicted time turns out to be 12.64 seconds, very close to your experimental result.
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