Graphical Analysis 4

Determining g on an Incline

(Sensor Cart)

INTRODUCTION

The magnitude of acceleration due to gravity, g, has several applications which include but are not limited to landing of rockets, escape velocity of a certain massive body, and even weight. For instance, g is essential in determining how much the weight of an object on Earth translates to its corresponding weight in another planet. As such, several experiments have been performed to determine g. In this experiment, the acceleration of a Sensor Cart as it descends on an inclined ramp is measured. The angle of the incline is then varied to determine how it affects the acceleration of the Sensor art. Finally, the magnitude of g is extrapolated from the graph of acceleration vs. sine of the angle of incline. The main objective of the analysis is to determine whether the extrapolation of g from the aforementioned graph is valid, which can be done by comparing the experimental value of g with its literature value.

ANALYSIS

Figure 1

Figure 1, as shown above, displays the experimental setup. To vary the angle of incline, books are stacked to which one side of the ramp is propped up against. To determine the value of sin(θ), we use the mnemonic SOHCAHTOA, which provides the definition of the sine of a certain angle. The sine of an angle is given by the ratio between the side opposite the angle and the hypotenuse of the right triangle. Figure 1 shows that the side opposite the angle corresponds to the height of the stack of books, h, whereas the hypotenuse of the right triangle corresponds to the length of the ramp, x. As such, we take the sin(θ) by taking the ratio between h and x.

As the Sensor Cart rolls down the ramp, its velocity and the time it took to get to the bottom of the ramp are measured, which are then recorded as a velocity vs. time graph. From the graph, the acceleration was determined by taking the slope of the linear potion of the graph. Since three trials for each angle of incline was performed, the average acceleration is taken. Table 1 summarizes the data that will be used in determining the value of g.

Number of books	Height of books, h (m)	Length of incline, x (m)	sin(θ)	Acceleration	Average acceleration (m/s²)
				Trial 1 (m/s²)	Trial 2 (m/s²)	Trial 3 (m/s²)
1	0.044	1.164	0.038	0.335	0.331	0.330	0.332
2	0.082	1.167	0.070	0.649	0.647	0.650	0.649
3	0.128	1.170	0.109	0.990	0.997	0.995	0.994
4	0.170	1.173	0.145	1.346	1.345	1.341	1.344
5	0.210	1.180	0.178	1.631	1.638	1.639	1.636

Based on the values recorded, we see that as the angle of incline and sine of the angle of incline increases, the average acceleration increases as well. This predicts that g should be greater than the maximum recorded average acceleration, which is 1.636 m/s² at a sin(θ) equal to 0.178.

Now, we plot a graph of average acceleration vs. sin(θ) in a graph and extrapolate the value of g from the equation of the line wherein g corresponds to a sin(θ) value equal to 1. The graph is shown below:

[CHART]

Shown on the right side of the graph is the equation of the line as well as the R² value (i.e. coefficient of determination). The R² value is a measure of how well the graph fits the linear trendline; the closer the value is to 1, the more well-fitted the graph is to the linear trendline. Since the R² value is 0.997, we can say that the graph fits very well into the trendline. This means that extrapolation of values outside the graph, would be valid if the equation of the line were to be used. Substituting an x-value equal to 1, we calculate the y-value, which then corresponds to g. The calculation is shown below.

y = 9.3038(1) − 0.0145 = 9.2893

Based from the calculation above, the experimental value of g is equal to 9.2893 m/s², which is not the far from the literature value of g, which is equal to 9.80 m/s². To give a numerical value for the difference between the two values, we calculate the percent error, whose formula is given below.

$$Percent error = frac{|Experimental Value – Literature Value|}{text{Literature Value}}x 100$$

$$Percent error = frac{left| 9.2893 – 9.80 right|}{9.80} x 100 = 5.50%$$

We see that the experimental value of g has a 5.50% error when compared against the literature value of g. The low percent error further supports the validity of extrapolating of g from the graph of average acceleration vs. sin(θ)

The value of g varies at different locations around the world with different altitudes. This can be attributed to the core of the Earth acting as a magnet, wherein the closer an object is to the core, the stronger the gravitation will be. As such, the value of g would be greater in places at lower altitudes than in places at higher altitudes. Therefore, the value of g would be expected to be greater at a school at sea level than at a school in the mountains.

CONCLUSION

In this experiment, we determined that there is a linear relationship between the average acceleration of a Sensor Cart as it rolled down the incline and the angle of the incline. This linear relationship is quantified by the derived equation of the graph, which was then used to extrapolate the value of g. The experimental value of g was calculated to be 9.2893 m/s² with a 5.50% error compared against the literature value of g. This low percent error also supports the validity of extrapolating the value of g from the graph of average acceleration vs. sin(θ).