Rotational Kinematics lab

1. Use the velocity components to determine the direction of the velocity vector. Is it in the expected direction?

1. From the x-t and y-t images, the two curves show a sine and cosine relationship, indicating that the object is in uniform circular motion. Since position over time behaves as a sine and cosine function, this means that the velocity is their derivative, and the direction of the velocity is always perpendicular to the direction of the radius, so the velocity vector varies in the tangential direction, which is consistent with the theory of circular motion.

The formula used to calculate speed

2. Analyze enough different points in the same video to make a graph of the speed of a point as a function of distance from the axis of rotation. What quantity does the slope of this graph represent?

In this experiment, we investigated the relationship between radius, linear velocity and acceleration in uniform circular motion by analyzing the motion of three different points on a rotating rod. We chose three points which are 3 cm, 8 cm and 15 cm away from the center of rotation. By reading the position versus time plots (x-t and y-t plots), we measured their periods T, and then used the formula. 

1. r=0.03m，ω≈1.047rad/s，v≈0.0314m/s 

2. r=0.08m，ω≈1.653rad/s，v≈0.132m/s

3. r=0.15m，ω≈3.49rad/s，v≈0.5235m/s

3. Calculate the acceleration of each point and graph the acceleration as a function of the distance from the axis of rotation. What quantity does the slope of this graph represent?

We calculated the centripetal acceleration for each point using the formula:

For the three distances (0.03 m, 0.08 m, 0.15 m), we used the previously calculated speeds and obtained the following accelerations:

a≈0.0329m/s2 at 0.03 m

$a \approx 0.218 \, \text{m/s}^2$ at 0.08 m

$a \approx 1.827 \, \text{m/s}^2$ at 0.15 m

4.How do your results compare to your predictions?

Our results matched our predictions closely. The speed of each point increased linearly with distance from the axis, just as predicted by the equation $v = \omega r$. The acceleration also increased linearly with radius, consistent with $a = \omega^2 r$.

In addition, the velocity direction was always tangential to the circular path, which is exactly what is expected in uniform circular motion.

Overall, the experiment confirmed all theoretical predictions with clear and consistent data.