preparation
The diagrams above do not show the base of the apparatus, which is simply a mounting point for the axle, with a low friction ball bearing, and for the pulley which the masses are hung over.
Support the masses before the run so the system can be set up.
For the first set of runs the masses are to be hung on a line running over the pulley and attached to the first of three drive radii.
The ticker timer and guide need to be raised to the correct height (in line with rim) by being set on top of stacks of books etc.
Tape the ticker tape to the outer rim of the cylinder in line with the rim.
Feed it through the guide and on through the ticker timer.
Between the guide and ticker timer the tape needs enough distance to change orientation from ‘on edge’ to lying flat to ensure it flows through the system freely.
test run
Do a test run with the ticker timer off, during which the mass is allowed to fall and accelerate the drum, which in turn winds in a length of ticker tape. Ensure this happens freely.
data run
Data collection is carried out as follows; ensure that the ticker tape has little slack in it. Turn the ticker timer on. Release the masses and allow the system to be accelerated. Turn off the ticker timer, removethe ticker tape and analyse it to find v for each five dot period (0.1s, using five dot method).
reconfigure
Prepare the apparatus with combinations of mass 1000g, 500g and
radii = 0.0153m, 0.0252m. 0.0352m. Complete a test run and data run for each and analyse the ticker tape for v at the rim. Record your results.
analysis
The torque involved in each run is found using τ = Fr.
The Angular velocity for each time period is found using ϖ = v/r.
These are used to produce the graph Angular Velocity vs. time, the (best fit) gradient of this is equal to the average acceleration for each of the runs.
These gradients were used to produce the graph Angular Acceleration vs. Torque (Angular Acceleration on the y axis, as it is the D.V).
Analysis of Error:
The error in measurement of v was ±0.002m. This is drawn from the absolute error in measurement of the distance of the five-dot sections on the ticker tape of ±2mm.
Error in measurement of ‘r’ was negligible because it was measured with a vernier calliper accurate to less than a mm, the cylinder is likely to have larger fluctuations in radius.
Error in ‘t’ was unknown as the ticker timer was known to run at an average of exactly 50hz, but fluctuations could not be measured. So as a result error could not be reliably calculated from absolute error.
This graph shows a comparison of the value of I from the equation τ = Iα where the 500g and 1000g runs have been graphed separately.
From the gradients shown in this graph and the gradient shown in the Torque vs. Angular Acceleration graph I have found the relationship between torque and acceleration (I, rotational inertia), to be 0.135 ± 0.02kgm². This value was obtained using 1/g where g is the gradient of the graph, because I = τ/α but the graph gives a gradient equal to a/t. The error value was taken from the largest difference of these gradients from the overall average gradient 0.135.
This becomes 0.1 ± 0.02 kgm² to 1 s.f. or a percentage error of 8%.
Since error in torque was taken to be negligible, using percentage error and the formula α = τ/I I figured error in Angular acceleration to be 8%.
I have added this as y error bars in the graph Torque vs. Angular Acceleration, and the line of best fit fits through these errors, which suggests they are reasonably correct.
However, theoretically; if the mass of the cylinder is taken as 1.322 ± 0.001Kg (this was measured) and the radius taken as 0.124m, we can calculate the rotational inertia using the formula I (of a cylinder) = 1/3mr² to be 6.775 x 10ˉ³ Kgm² or 7 x 10ˉ³ Kgm².
Comparing this with 0.1 ± 0.02 kgm² there is a large unexplained error. I was unable to identify the source of this error and cannot confidently attribute it to friction.
Conclusion:
The graph Angular Velocity vs. Time was used solely to acquire several angular acceleration values for the Angular Acceleration vs. Torque graph.
However it does show the comparison of the angular acceleration caused by different masses combined with different drive radii.
The Angular Acceleration vs. Torque graph shows the important relationship; torque multiplied by 0.1 ± 0.02 kgm² is equal to angular acceleration.
This constant must be equal to I, from the equation τ = Iα where I represents rotational inertia.
The graph is a straight line graph with a positive gradient, which means Angular Acceleration is directly proportional to Torque, as stated in the hypothesis.
Evaluation:
The investigation was well designed, although error analysis was a problem.
Further investigations could include research into:
-the accuracy of ticker timers
-the accuracy of the way the ticker tape wraps around the rim of the cylinder/drum,
-the fluctuations of the cylinder’s radius.
This would enable use of absolute error to calculate derived errors for derived values such as Angular Acceleration.
The results are erroneous compared with the theoretical value of I. This is an error which can be investigated either to find the source of the discrepancy or the systematic error that caused this large difference in results.