(1.1) This experiment is about determining the effect of angle of inclination in the acceleration of an object. The independent variable is the angle of inclination of the slope of motion; the dependent variable is the acceleration of the object.

## Hypothesis

(1.2) The angle of inclination of the slope is directly proportional to the acceleration of the object. Without initial applied force, the only acceleration of the object would only be due to gravity. Thus, the relationship will follow the equation:

a=gsinθ

Where a is the acceleration of the object, g is acceleration due to gravity, and θ is the angle of inclination. Thus, the determining factor is the term with the angle: sinθ.

## Background

(1.3) In an inclined plane, an object has the tendency to move down the surface of the plane because of the effect of gravity. The forces affecting the object on an inclined plane is illustrated by the following figure :

Figure 1. Forces affecting objects on an Inclined Plane

Fg is the force due to gravity, FN is the force normal to the surface, and θ is the angle of inclination. The force due to gravity is:

Fg=m∙g

Where in m is the mass, and g is the acceleration due to gravity. The component Fgcosθ is the one that is related to friction. The effective net force is the other component which is Fgsinθ. Now, with this term at hand, with the same mass, the acceleration of the object becomes:

m∙a=m∙g∙sinθ→a=gsinθ

(1.4) In this experiment, the effects of friction and air resistance are neglected. Moreover, human error could also introduce deviations in the results. The imperfect, manual synchronization of the timer and cart release, the errors in angle measurements, and even slight instabilities with the ramp set-up could all contribute to inaccuracies of the measured results. It is therefore necessary to minimize these errors as much as possible.

The other formula to be concerned about is the displacement formula, since the experimental set-up does not automatically measure acceleration. The displacement formula is :

x=x0+v0t+12a∙t2

Where x-x0 is the displacement, v0 is the initial velocity, a is the acceleration, and t is the time elapsed. The first term in the equation is neglected because no initial velocity is introduced in the experiment. Therefore, the displacement (S) formula becomes:

S=12a∙t2

## This experiment assumes no initial force applied, and the effect of friction on the object’s motion is insignificant.

Method

(2.1) The following materials are used for the experimental set up:

power supply, cart, weight to equal 2190.55 g, gate, foam, vector stand, busman clamp, protector, timer, ruler, pencil, camera, band.

## The materials are set up as follows:

- Set up apparatus as shown in the picture, choosing the first angle of incline.

- The weight of the cart is 782.60g. Also, we have attached 2 weights which the weight of the first one is 1001.55 g and the weight of the second one is 1006.40g. The total weight is 2190.55g and it is constant during the whole experiment.

- Measure the base length and height to the bottom of the ramp.

- Plug in power supply.

- Place the cart at the top of the ramp.

- Begin the first run on experiment and release both the cart and the timer button at the same time.

- The cart is stopped once it has travelled down the ramp and hit the gate to stop the timer.

- Stop the run on experiment.

- Repeat steps 1-7 using 5 different heights for the ramp for 3 times for more accuracy.

- Using the measurements, calculate the angle of incline and sin angle of incline for each trial.

- Using the Data results, calculate the rate of acceleration for each trial.

(2.2) The experimental set-up looks like the following figure:

Figure 2. Experimental Set-up

(2.3) The height h is controlled by changing the angle of the slope. The length L and the mass of the moving car are kept constant. The length L is maintained at this value because of two reasons: this one ramp is used in the whole of this experiment, and the cart is assumed to travel this distance at all the runs of the experiment. The friction on the ramp is assumed negligible because the surface of the board is significantly smooth.

(2.4) The angle of the slope of inclination of the plane is measured carefully to minimize measurement errors; after all, this entity is the one being studied in this experiment. Additionally, since real-world friction is always present, neglecting this factor would always introduce errors in the experimental results. Moreover, at each run of the experiment, the cart is carefully initialized at the top of the ramp such that no unnecessary change of direction may occur that may disrupt the experiment. Also, the gate is maintained at a functioning state such that each cart release is accurately measured. Some images of the set-up are presented in the following figure:

Figure 3. Images of the Set-up

## Results

(3.1, 3.2) The measured quantities are tabulated in the following table. The values in the “Formula 1 Acceleration” column are calculated using the following formula derived from the displacement equation:

a=2St2

## Wherein S is the displacement, a is acceleration, and t is the time elapsed.

(3.4) The values of acceleration show an increasing trend as angle increases. Conversely, the time values are decreasing (which obviously translate to higher acceleration values). The following graph shows the relationship between the angle of inclination and the acceleration of the moving car:

Figure 4. Effect of Angle of Slope On Acceleration

## For both formulas, the trend is generally increasing (almost linearly).

Analysis

(3.4) The results show that there is a direct proportionality relationship between the angle of inclination and the acceleration of the moving car. The plots generated display a linear relationship between the two variables. This behavior provides significant evidence to prove the hypothesis. The values of formula 1 are all less than the corresponding values of formula 2. This can be attributed to the effect of friction, which would basically reduce the acceleration experienced across the surface of the plane.

Some additional errors may be due to the accuracy of measurements on the angle of inclination, the timer, and the direction travelled by the car on the plane surface.

(3.3) The time measurements among the trials display precision because the values are not so very far apart. This observation adds to the validity of the experimental results. The linear similarities between the two formulas (with an almost fixed offset) translate to high accuracy of the measurements done. This fundamentally shows that there is a fixed friction force present on the surface of the inclined plane (the friction is represented by the fixed offset).

## Conclusion

(4.1) A direct linear relationship was established between the two variables in the experiment: the angle of inclination, and the acceleration of the moving car. Following the formula a=gsinθ

Since acceleration due to gravity g is constant, the term increasing with the angle is sinθ, which can be considered as the factor of proportionality.

Since precision and accuracy were established, it can be concluded that the results are valid. Therefore, the hypothesis was proven true. Even though friction was not included in the calculations, the effect of friction was observed in the results, which further validated the accuracy of the measurements.

(4.2) These results can be applied to real-life situations. For example, a person driving a vehicle on a mountainous terrain would control the vehicle’s acceleration based on how steep the path is. If the slope is steep, minimal or even negative acceleration might be needed to avoid undesirable speeds. If the slope is not very steep, normal driving can be applied.

(4.3) This experiment can be further improved by incorporating friction in the measurements and calculations. Moreover, a more accurate method of measuring angles can be introduced. Since it was discussed beforehand, this experiment could be further improved by reducing manual involvement. Thus, a mechanized synchronization of timer and cart release can be incorporated. The gate can also be optimized in terms of sensitivity.

(4.4) In conclusion, there is a direct relationship between the angle of inclination and the acceleration of an object on an inclined plane. Experimental results show that as the angle of inclination increases, the acceleration of the moving car along the inclined plane also increases. Based on the plots, this relationship is linear in nature. The factor of proportionality is the sine of the slope angle θ, or sinθ. This relationship was established clearly for slope angles from 100 to 300.

## References

Golwala, S 2007, 'Lecture Notes on Classical Mechanics for Physics 106ab', Lecture Notes, California Institute of Technology. (This is lecture notes used in a physics class of a renowned university: California Institue of Technology. The year is also relatively recent, 2007, making the reference still very relevant.)

Young, HD & Freedman, RA 2012, 'Motion Along a Straight Line', in University Physics with Modern Physics, 13th edn, Addison-Wesley, San Francisco. (This physics textbook by Young and Freedman is a reliable reference used by several educational institutions. The year of edition. 2012, is also very recent, making it very relevant today.)