Cart+on+Incline+-+Nathan+Devan,+Sneha+Mittal,+&+Michael+Gary

__Title of Lab: __ Cart on Incline

__Researchers: __ Nathan Devan, Sneha Mittal, Michael Gary

__Research Question: __ How does the angle of inclination affect the force parallel to the incline that is required to keep a cart in equilibrium?

__Research: __ Consider the triangle depicted. Using trigonometry, one can conclude that mgx, a component of the force mg, equals mg (sin θ).



 mgx = mg(sin θ)

As θ tends to 90, sin θ becomes larger and mg (sin θ), becomes closer and closer to mg, the weight of the cart. According to Newton’s second law, the sum of the forces equals mass times acceleration. In the case of equilibrium, this statement can be simplified to sum of the forces equals zero; thus, the upward forces must equal the downward forces and the rightward forces must equal the leftward forces.

 Σ F = ma  Σ F = 0

Hence, as sin θ increases with the angle of inclination, the tension of the string attached to the block must also increase to maintain equilibrium.

__Hypothesis: __ Since both the tension, T, and sin θ increase with angle of inclination, the two are directly proportional according to the following equation:

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;"> T = mg (sin θ)

__<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Procedure: __

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Materials: board, protractor, string, cart, spring scale, triple beam balance

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">1. All instruments were calibrated. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">2. A board was positioned with one end on the edge of a table and the other on the floor. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">3. A protractor was used to find x, the angle the board made against the table. Using complementary angles, the angle of inclination was found. An angle of 30 degrees was tested first. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">4. One end of a piece of string was attached to the spring scale; the other end was attached to the cart. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">5. The cart was placed on the incline and the tension of the string was noted in the spring scale. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">6. After three trials, Steps 3-5 were repeated with angles (in degrees) 0, 50, 60, 70, and 90. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">7. The mass of the string and the cart were measured using a triple-beam balance. __<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Data: __

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">The following values are the result of the experiment:
 * || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Tension (N) Trial 1 || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Tension (N) Trial 2 || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Tension (N) Trial 3 ||
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Angle of Inclination (°) ||  ||   ||   ||
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">0° || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">0 N || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">0 N || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">0 N ||
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">30° || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">2.1 N || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">2.2 N || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">2.1 N ||
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">50° || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">3.6 N || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">3.5 N || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">3.5 N ||
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">60° || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">3.9 N || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">4.0 N || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">4.0 N ||
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">70° || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">4.5 N || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">4.5 N || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">4.4 N ||
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">90° || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">4.7 N || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">4.8 N || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">4.8 N ||

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Based on the precision of the data, the following table's values will be used in the graph:

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">The mass of the cart and string came out to be 490 g or 0.49 kg. To determine the weight of the cart, 0.49 kg and the acceleration of gravity (9.81 m/s2) are multiplied to produce 4.8 N, to the correct number of significant figures.
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Angle of Inclination (°) || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Tension (N) ||
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">0° || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">0 N ||
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">30° || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">2.1 N ||
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">50° || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">3.5 N ||
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">60° || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">4.0 N ||
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">70° || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">4.5 N ||
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">90° || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">4.8 N ||

__<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Data Analysis: __

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">The experimental data was plotted: <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">When the sine of the angles were plotted with their experimental outcome, the best-fit line is linear. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">The graph of the experimental data closely resembles the graph of the theoretical data. Additionally, using the linear regression calculation on Microsoft Excel, the "Tension vs. sin θ" graph depicts a constant rate of change of 4.7 N, which represents the theoretical value of the weight. Considering the hypothesized equation T = mg (sin θ) where mg equals 4.8 N, the percent accuracy is calculated to be 98%.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">__Conclusion__: <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Based on the percent accuracy, one can conclude that the experimental results supported the proposed hypothesis. The 2% difference could be a factor of instrumental error; in future experiments, the procedure will be revised as to include the calibration of the spring scale each time the angle of inclination is adjusted to help further diminish the percent error.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Experimentation and data analysis lead to a proper understanding of basic physical forces that enables technological innovation; engineers, architects, and many others apply these findings in every day life. Thus, physics is a central science needed for the betterment of society and life as whole.