Angle of a Ramp and the Acceleration of a Cart

Objective:

The objective of the lab is to determine a relationship between the acceleration of an object on an incline and the angle of the incline.

Click here for information on using the TI-83 calculator instead of the Logger Pro software. The explanation of the data has been expanded in this section.

Setup: Equipment Pasco cart and track Picket fence for cart Photogate LabPro LoggerPro software Meter stick Calculator with trig fucntions. The lab may also be done with a CBL and TI-83 calculator. The Logger pro software should be setup for photogate timing with the appropriate width of the picket fence. We are using the 1cm picket fence with 13 slots.

Data Collection

Measure the length of the ramp and the height of the raised end of the ramp. Calculate the angle of the ramp in radians and record the values.

Position the photogate along the ramp and record the distance vs time data for a number of trials. From the distance vs. time data, use a quadratric regression to determine the acceleration. More on this can be found here.

Nineteen trials are shown on the graph below. As the ramp is raised, the acceleration increases and is indicated by the curve rising more rapidly than the previous curve. A curve rising faster than a previous curve indicates the increase in acceleration.

Reducing the number of graphs for demonstration purposes results in a set of graphs with regression equations shown below. A complete data set is shown in chart form. The Logger Pro uses a quadratric regression in the form of A + Bx + Cx^2 which may be confusing since the TI series calculators use ax^2 + bx + c as is the standard in most math text books. Here the value of C is 1/2 of the acceleration. Deriviation of this is shown on a separate activity.

Several things are important to note.

• The cart was released the same distance from the photogate and in the same manner each trial.
• The values of A (the y-intercept) are very close to zero meters. This would be expected because the timing began when the first section of the picket fence passed through the photogate.
• The values of B increase as the ramp angle is increased. The B values are the initial velocity (m/sec) or velocity when time is zero which occurs when the fence first passes through the photogate.
• The values of C are 1/2 of the acceleration of the object down the ramp in m/sec^2. As the angle of the ramp increases, the acceleration increases.

Complete Data Set

Angle (radians) Acceleration (m/s^2) 0.044 0.448 0.069 0.672 0.085 0.822 0.099 1.046 0.136 1.328 0.160 1.558 0.187 1.89 0.238 2.29 0.277 2.632 0.316 2.978 0.386 3.578 0.438 4.004 0.513 4.732 0.561 5.168 0.611 5.582 0.652 5.92 0.741 6.506 0.876 7.386 0.938 7.896

Evaluation of the Data

At first glance, the data set appears to be linear. Finding a linear regression, as shown at the right indicates the data is below the line at the ends and above the line in the middle, hence the model has some type of curve to it. The vertical angle would be 90 degrees or /2 radians. The linear regression shown is y = 8.460x +0.230. The model would be Acc(m/sec^2) = 8.460 (m/sec^2*rad)* x radians + 0.230 m/sec^2. The model implies that the acceleration of an object with an angle of 0 radians is 0.230 m/sec^2 when it should be zero. Secondly, by dropping a picket fence vertically through the photogate, we found the acceleration to be 9.8 m/sec^2. If we use an angle of /2 radians in this model, the predicted acceleration is 13.52 m/sec^2 which is an error of 38%.

If we consider what happens when the angle passes 90 degrees or /2 radians, the object still rolls down the ramp toward the origin. The acceleration is still in the same direction. If the reference or base angle in the second quadrant is the same as that of the first, the acceleration should be the same.

As the ramp angle goes towards 180 degrees or radians, the acceleration will decrease to zero again. The angle in radians for the reference triangle in the second quadrant is - 0 radians. Since the base angle is the same, the acceleration will remain the same. The graph below, shows the data from the first and second quadrants.

The data from the first and second quadrants may appear to quadratic. However, recall that a quadratic model continues infinitely infinitely at its endpoints. This would mean that as the angle continued to increase, the acceleration would continue negatively. This means we must consider what happens for angles in the 3rd and 4th quadrants.

Consider what happens in quadrants III and IV. Here, the cart will no longer accelerate toward the origin but will accelerate away from the origin. Hence, the acceleration for the base angle will have the same value but the change of direction causes a change of sign. Now the acceleration will be negative. The angle in the quadrant III can be found by adding + 0. The cart in quadrant IV will also move away from the origin and will be negative. The angle in this quandrant is found by subtracting the angle from 2, or 2 - 0. Once the data for all four quadrants is graphed, a graph will appear similar to the one below. It appears to be a sinusoid.

		Fitting a sinusoidal model to the data, one gets Acceleration (m/sec^2) = 0.00 m/sec^2 + 9.673 m/sec^2 * sin(1.000(0 radians + 0.001 radians)).
One would expect the k value or midline to be 0 because the acceleration at 0 radians would be 0 m/sec^2 and the acceleration should increase the same amount positively and negatively resulting in a midline value of 0. The amplitude would be expected to be 9.8 m/sec^2 because the acceleration of gravity directly vertical is 9.8 m/sec^2. This is an error or (9.8 - 9.673)/9.8 * 100 or 1.296% which is acceptable for the equipment used considering there is a small amount of friction in the cart and track. The D value is the horizontal stretch which relates to the number of periods in the normal period of 2. Since we made one complete revolution around the circle, this value should be 1. Finally, the horizontal shift (H) value should be 0 because the start of the period will occur when the acceleration is 0 which was at the angle of 0 radians. The model simplifies to Acceleration (m/sec^2)= 9.67 m/sec^2)* sin(0 ) where acceleration toward the origin is positive. The expected model would be Acceleration (m/sec^2)= 9.8 m/sec^2)* sin(0 ) which is an error of less than 1.3%.

Click here for information on using the TI-83 calculator instead of the Logger Pro software.