15-April-01: Lab 9 - Centripetal Force with a Motor

PURPOSE:  In a centripetal force experiment, to find a relationship between the vertical angle at which a mass revolves around a vertical shaft and the angular speed at which the mass rotates.

MATERIALS:

• The following apparatus containing various components:
• An electric motor powered the vertical and horizontal shafts to rotate at various speeds using increasing voltage powers.

• We also used a ring stand with a piece of paper sticking attached so the rubber stopper could hit it as it spun.  We used this to measure revolution time and also as a marker for the height of the rubber stopper corresponding to various angular speeds.

PROCEDURE:

• We first measured various dimensions of our apparatus:  total height from which the string hung (H), length from center to hanging string (R), length of string (L):
• H = 2 meters
• R = 0.98 meters
• L = 1.645 meters
• Performed 6 trials in which the rubber stopper revolved at 6 different angular speeds (by upping the voltage)
• For each one, we measured the time it took for the stopper to make 10 revolutions
• We used the ring stand and attached paper to measure the height at which the rubber stopper hit the paper
• Recorded the following data:

• We subtracted each height of paper (h) from the height from the horizontal rod to the floor (H), to give us the vertical length in the right triangle (shown below).  Since we also measured the length of the string L, we can use an equation to find θ: cos θ = (H-h)/L.  The formula and angle of the string for all 6 trials are shown below:

Angle of the string to vertical axis for 6 different angular speeds.

• We used the angle of the string to the vertical axis in the formula that was derived from two force equations (shown in following photo).  The force diagram on the left is of the rubber stopper with centripetal acceleration pointing towards the center.  The diagram on the right is the same but with tension (T) split into its x and y components.  We then formed equations for the sum of x components and sum of y components, substituted tension force, and then derived a final equation for angular speed;

• We used this to plug in our various angles.  However, because r (horizontal distance from the central axis to the rubber stopper) is made up of two components, we first had to solve for the horizontal component of the length of the string.  This was found using the right triangle, and we substituted L sin θ into our equation for ω:

• However, instead of calculating each equation one by one, we entered the data in excel to find angular acceleration for each trial.  Note: in column two we had to change degrees to radians.  

• In order to compare our experimental data to something, we calculated angular acceleration using the formula ω = 2π/T, taking our initial times for the 10 revolutions and dividing it by 10 to plug it back into this equation.  Our results give us the following calculated values of angular acceleration:  

To find the correlation of our two values for angular acceleration for each trial, we plotted it on the graph, with ω = sqrt((gtanθ)/(R+Lsinθ)) on the x-axis and ω = 2π/T on the y-axis.  We did a linear fit and found the slope of the line, which shows: 1.0399.

CONCLUSION:

The correlation of our two derived values of angular acceleration was not exactly 1, but this could have been due to a number of errors.  When we timed the number of seconds for 10 revolutions, the exact start and stop of our timer might have been off.  Our measured values for the height of the paper on the ring stand had an error of +/- 0.01 meters.  We also do not know exactly at what height the rubber stopper hit the paper.  In addition, we noticed that when watching the horizontal rod revolving around the axis, the rod somewhat moved up and down and was a little wobbly.

Despite our errors, what we learned from this lab was that we were able to form an equation for angular acceleration using different values for θ.  After solving for our different values, we were able to compare these results with the angular acceleration we derived from our other equation, ω = 2π/T, using only the measured period.