March 25, 2015 Centripetal Acceleration vs Angular Frequency

Objective: 
To determine the relationship between centripetal acceleration and angular speed.

The Set-Up:

Prepare an empty space, mount the accelerometer on the disk. The disk should be able to accelerate or decelerate smoothly while rotating. Set up the photogate on the side of the disk. Stick a piece of thin paper on the disk so that it can be detected when passing through the photogate.

Data Collection:
The equation for centripetal acceleration is a=rω^{2
Our goal is to calculate the radius (r) by doing an experiment to get the acceleration and angular frequency, and compare the value of r to the true value.
We measure the radius of the disk with a ruler, it is about 13.8-14 cm.
We begin the data collection by accelerating the disk using accelerometer and record the data through loggerpro.}
We have done five trials, each time with a different voltage from the accelerometer, which give us different acceleration.
We obtain the value of acceleration by looking at the acceleration vs time graph. We use the mean value as the acceleration.

Acceleration vs time graph

Next, in order to calculate omega, the angular frequency, we need the total time of rotation and the number of rotation. These information are found in another loggerpro data.

In this set of data, we are only interested in 1-16, the ones in red box because there we have the complete information. The total time is the difference between final time and starting time. In this particular data, we have 16.461-1.6716= 14.789 sec., and 8 rotations.

We input these data into excel to come up with the value of omega squared.

Acceleration vs omega squared graph

We obtain the value for omega squared. We plot the acceleration and omega squared, which gives us a graph above.The acceleration is the y axis and omega squared is x axis. The slope of this graph is the radius. We see that the slope is 0.1391 meter, which is also 13.91 centimeter. It is within our measured range of 13.8-14 cm.

Summary :
In this experiment, we seek to find the relationship between acceleration and angular frequency. We look at the equation a=rω², and set up an apparatus so that we can get both the acceleration and angular frequency. Then we calculate the value of radius and compare it to the true radius. The calculated radius is 13.91, and the true value is between 13.8-14 cm, we have achieve our goal.

Some error and uncertainty to this experiment would be we assume no friction between the accelerometer and the disk. In reality, we do not have a perfect equipment so there is friction. The friction could have slow down the the time to complete one revolution. In the same way, we assume no air resistance. Since the disk it rotating, there is air resistance involve, which slows the time to complete one revolution. There is also some human technique involve that could add more uncertainty and error to the experiment.