Angular Velocity

This experiment explores how a falling mass influences a spinning system by converting gravitational potential energy into both linear and rotational kinetic energy. When the mass is released from rest at a constant height, it accelerates toward the ground while simultaneously causing the disk to spin as the string unwinds.

1. Data and Calculations for the 47.5 mm Disk

• Trial Times: 2.07 s, 1.75 s, 1.59 s

• Average Time:

$$t_{\text{avg}}=\frac{2.07+1.75+1.59}{3}\approx 1.80\text{s}t\_\{\backslash text\{avg\}\}\; =\; \backslash frac\{2.07\; +\; 1.75\; +\; 1.59\}\{3\}\; \backslash approx\; 1.80\; \backslash ,\; \backslash text\{s\}$$

• Linear Velocity:

$$v=\frac{0.887\text{m}}{1.80\text{s}}\approx 0.494\text{m/s}\text{<br>}\text{<br>}$$

2. Data and Calculations for the 28.6 mm Disk

• Trial Times: 2.6 s, 2.57 s

• Average Time:

$$t_{\text{avg}}=\frac{2.6+2.57}{2}\approx 2.585\text{s}t\_\{\backslash text\{avg\}\}\; =\; \backslash frac\{2.6\; +\; 2.57\}\{2\}\; \backslash approx\; 2.585\; \backslash ,\; \backslash text\{s\}$$

• Linear Velocity:

$$v=\frac{0.887\text{m}}{2.585\text{s}}\approx 0.343\text{m/s}$$

3. Data and Calculations for the 19.6 mm Disk

• Trial Times: 3.57 s, 3.88 s

• Average Time:

$$t_{\text{avg}}=\frac{3.57+3.88}{2}\approx 3.725\text{s}t\_\{\backslash text\{avg\}\}\; =\; \backslash frac\{3.57\; +\; 3.88\}\{2\}\; \backslash approx\; 3.725\; \backslash ,\; \backslash text\{s\}$$

• Linear Velocity:

$$v=\frac{0.887\text{m}}{3.725\text{s}}\approx 0.238\text{m/s}$$

Disk Diameter	Radius (m)	Average Fall Time (s)	Linear Velocity (m/s)	Angular Velocity (rad/s)
47.5 mm	0.02375	1.80	0.494	20.8
28.6 mm	0.0143	2.585	0.343	24.0
19.6 mm	0.0098	3.725	0.238	24.3

Conclusion:

The linear velocity decreases with decreasing disk diameter due to increased fall times. This

indicates that more energy is required to overcome the rotational inertia of smaller disks. Since it is taking longer to call the velocity is also smaller. The linear velocity and the angular velocity are comparable to group 3.  The uncertainty of our angular velocity is 23.0 +- 1.5. It does not fit in the uncertainty, but this could be due to human error as well as differences in measurment technique. 

Inclined track Lab

 Data gathered

Performing 3 trials, three different efficiencies were gathered to determine the uncertainty.   

The experiment output an average efficiency of 80.5% +- 5.25%. 

Conclusion

The cart had a higher velocity going down the incline and the magnets ensured it did not hit the bumper and returned the cart up the incline with a smaller velocity. The total efficiency that the cart going up and down the ramp was 80.5% +- 5.25%. 

The efficiency from group three on a leveled track was calculated to be 90.285% +- 5.09%. The velocity changed only 10% on average after the cart hit the bumper. This shows that the incline has a large effect on the carts total motion and the flat surface helps retain the motion.

The leveled track would be a better way to measure since it is easier to gather data and less factors like angles and friction becoming more of a problem. The inclined track was also harder to measure the heights and positions since we have to use trigonometry to calculate.

Analysis of Coefficient of Friction and Normal Force

Constant Height (1 & 2)

With a constant height of 0.5 m, the friction force and the normal force was calculated in excel using an input of the equations derived. The friction coefficient in this experiment with constant height was 0.94. This does not fit in the predicted range of 0.2-0.6. since the more mass was added per trial, the friction force increased higher and higher.