01 April 2015- Centripetal force with a motor

Lab 7: Establishing a Relationship between Angular Speed and Angle of Rotation


Purpose: The purpose of the lab is to find a relationship between angular velocity(w=mv^2) and the angle that the object is rotating at represented by theta (ø).

Procedure/Material: In this experiment, a mechanism was created by attaching a long metal pole to a large tripod which controlled a wooden stick that spun in a circular motion at the very top. This wooden stick ran parallel to the ground. A string was attached to one end of the stick which was sent into a circular motion around and around as it was powered by an electric generator which allowed a constant speed for the conical pendulum to rotate. We did this over again with different wattage to give a large speed which we would calculate theoretically and experimentally to prove our relationship.

Before we start with the experiment and run several trials changing the angular speed, we must first measure the total heigh of the mechanism which is taken to be 2 meters tall. We also needed to know the length of the string because it stays constant at 1.654 meters. There is also a rubber stopper at the end of the string just for safety purposes.

This is the mechanism that we were observing and analyzing.

This is the diagram of what the mechanism looks like
and also what we helped use to solve for the different
variables in order to find a relationship.

Once we got the electric generator to turn on we began watching the top of the mechanism spin allowing the rubber stopper attached to the string to begin spinning, as it reached equilibrium and was stable we began recording the times on our phones and the amount of revolutions (complete circle) done with each different wattage of the generator. In order to find the height from the ground of the spinning rubber stopper, we used a metal rod which had a paper taped at the end. We would higher and lower the height of the metal pole in order to get the best height, we knew how high to put it when the rubber stopper just skimmed the paper. This height was then recorded and represented the distance of h. We repeated this process eight times with eight different heights which we found when the professor began to higher the voltage of the generator allowing the wooden stick with the string and rubber stopper attached to rotate faster. This data will help us to solve mathematically for the angular velocity.

This is the tool used to measure the different heights from the ground of the rubber stopper.

After we worked out the data was used to find the results by using an excel spread sheet. We also used some white boards to show the work that we did too. We first used the first diagram that helped us to solve for the angular velocity (W). By using the force diagram we were able to use trig to find the angle at which the string and rubber stopper swung at (ø) by first finding the distance from the pendulum to the rubber stopper vertically and then using the length of the string to derive a trig equation which we took the cosine inverse of to solve for the angle .

This is the work which solved for the angular velocity that we first started with.

After we solved for theta to find the angular speed we had to convert the answer into radians. We then used the proven physics equation to solve for the angular velocity. We used the physics equation  W= 2(pi)/T and compared it to the equation we derived using our mechanism 

W^2= (gsinø)/cosø(d+lsinø). 

This is the work we used to solve for theta (ø) which is a function we derived.

The equation derived will be compared to a physics law that has been proven which is W(angular speed)= 2(pi)/T. Which is shown above.

Now we plotted an equation for the lows and highs of angular velocity using the derived equation and compared them to the angular velocity of the given W=2(pi)/T. In doing so we found below that the slope of the line is .971 which is the period equation (W= 2(pi)/T).

We found that when we solved for the lows and highs of the equation as well, that the results were almost identical. And the slopes of the lines were almost completely equal. 

This the slope of the high equation meaning that the highest error value it could have had. Which we found the slope to be .9824 

This is the slope of the low equation which is the lowest error value it could have had. We found the slope to be .9696.

The slopes show that we were off by only 2 percent from the from the hypothetical value since the slopes are not completely identical. But as each trial went on the graph shows that the angular velocity did increase. The small error we had could have been from measuring the height of the mechanism as it rotated or when we measured the period which involved several decimals which may have cause the error because of rounding. In conclusion, the experiment was successful in finding a relationship between omega(W-angular speed) and theta(ø-angle). As the angular speed increased the angle also increased showing that the large the angle the higher the angular speed of the object and in this case the rubber stopper.