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.