# Investigating the forces acting on a trolley on a ramp

Physics coursework

Investigating the forces acting on a trolley on a ramp

Contents

Page 3                 ->                 Method

Page 4                 ->                 Theory

Page 7                ->                 Results

Page 9                 ->                 Error

Page 18                 ->                 Appendixes

Method

The aim of the investigation was to investigate the forces acting on a trolley as it rolled down a ramp, and also to investigate the factors which may contribute to the results. To do this, a trolley and a ramp set at a variety of angles of incline were used, and then, using a light gate, the speed at which the trolley was moving when it passed through the light gate was calculated. The variables were the starting distance of the trolley in relation to the light gate and the angle of the ramp.

Firstly, the equipment was set up as in fig. 1. The trolley was then run down the ramp with a piece of card attached to the side. This card was of a known length and could hence be used to calculate the velocity at which the trolley was moving. While the light gate did actually calculate the velocity, it only gave the answer to 2 decimal places, whereas it gave the time to 2 decimal places. Furthermore, the light gate calculated the velocity with the assumption that the card was exactly 100mm, whereas when the card was actually measured, this was a value closer to 102mm (±0.5mm). Next, after the trolley had passed through the light gate, the information from that ‘run’ appeared on the LCD of the light gate, and this was noted down.  Three readings were taken for each distance from the light gate so that an appropriate average could be taken. These were taken across a range of seven distances between 250mm and 1750mm inclusive. Finally, the angle of the ramp was altered and the measurements taken again for three different angles.

By measuring the vertical height at the point where the light gate was and a fixed distance from this, it was possible to calculate the exact angle at which the ramp rested by using simple trigonometry.

Theory

There were two ways to approach the experiment. It is possible to calculate gravity (g) by considering the ‘conservation of energy’, which calculates the gravitational potential energy of the trolley and uses this information to find the acceleration of the trolley. In this experiment, the variable is the vertical component of the ramp. Alternatively, it is possible to roughly calculate the effect of gravity using the angle of the ramp, the final velocity and the distance from the light gate. A more accurate calculation can be made if friction between the trolley and the ramp is taken into account and this was the approach taken.  Friction has a greater effect on the results the smaller the angle becomes, and so because the angles that were worked with were all less than 12˚, it is especially noticeable in the results and is be the single most prominent source of systematic error.

In order to successfully calculate the acceleration, it is important to have a good understanding of all the forces acting on the trolley at any one point. This is better displayed as a force diagram (fig. 2)

Figure 2 – Diagram to show the forces acting on an object on a slope where m is the mass of the object, g is gravity, Fr is the force of friction (equal to μN), N is the normal force (mg cos Ɵ) and Ɵ is the angle of the ramp. Because the block is neither floating off the ramp nor sinking into the ramp, N = mg cos Ɵ.

When the block is stationary;

Fr ≥ mg sin Ɵ.

However, because the block is moving;

mg sin Ɵ > Fr

Hence, the resultant force is;

mg sin Ɵ - Fr, which is equal to ma.

Thus, we have the formula;

ma = mg sin Ɵ – μN

Rewriting N as mg cos Ɵ gives us the formula;

ma = mg sin Ɵ – μ mg cos Ɵ

Hence, because m is a constant in all parts of the equation, we can divide through by m to give;

a = g sin Ɵ – μ g cos Ɵ

Simplified;

a = g (sin Ɵ – μ cos Ɵ)

Therefore, to calculate g taking friction into account;

g = a / (sin Ɵ – μ cos Ɵ)

Alternatively, it is possible to take an approximate value for g without taking friction into account. This is done in the following way;

In g = a / (sin Ɵ – μ cos Ɵ), μ cos Ɵ is equal to the frictional force Fr, hence, if we want to discount Fr, we remove this from the equation. This gives the formula;

g = a / sin Ɵ

However, to take a more accurate result, it is necessary to incorporate friction into the results and take steps to eliminate it. In order to find μ, the coefficient of friction, it is necessary to find at what maximum angle of the ramp friction will exactly counteract the pull of gravity. At this point the trolley will be stationary. This is shown in the force diagram below (fig. 3).

Figure 3 – Diagram showing the forces acting on the block when the angle of the ramp balances the force of friction.

In this circumstance, Fr = mg sin Ɵ because there is no acceleration, i.e. the forces are balanced. Since Fr can be rewritten as μ mg cos Ɵ, it is possible to find μ using the data known.

As Fr = mg sin Ɵ, and Fr = μ mg cos Ɵ we can say that;

mg sin Ɵ = μ mg cos Ɵ

Hence, to find μ;

mg sin Ɵ = μ mg cos Ɵ

Divide by mg on both sides;

sin Ɵ = μ cos Ɵ

Therefore;

μ = sin Ɵ / cos Ɵ

Because of trigonometric identities;

sin Ɵ / cos Ɵ = tan Ɵ

So finally;

μ = tan Ɵ

Results

The results from the experiment were recorded in a table with two columns; distance from light gate measured in millimetres and time taken to pass through the light gate, measured in milliseconds. Without taking friction into account, it was possible to take a rough estimate of acceleration due to gravity and hence it was possible to prove that friction has a greater effect at gentler slopes than at steeper angles. This observation holds true until the angle of the ramp is 90˚, when the trolley doesn’t suffer from friction due to contact with the surface because it is in free fall; there is no contact between the surfaces.

The results from the experiment were tabulated and a value for v2 was produced for every time reading. These were then used to find an average value for v2 for every distance. This average value for v2 was then plotted against the distance, and this meant that a graph of v2 over 2s was produced. Using to the equation of motion; v2 = u2 – 2as (where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the distance), it was possible to find the acceleration along the slope by rearranging the formula and plotting a graph. Thus, because in this experiment u was 0, the resulting rearranged formula was;

a = v2 / 2s

Hence, it was possible to find the acceleration along the slope by taking the gradient of the resulting regression lines plotted through the data. This could then be used to find the acceleration due to gravity.  However, due to the number of possible sources of error, a perfectly accurate result would be tricky to obtain.

The resulting graph is found in ‘Appendix 1 – Graph of results’. From this, the gradients of the regression lines through the data were taken. These are shown below;

When Ɵ = 10.8˚, ...

#### Here's what a star student thought of this essay

Their spelling and grammar is very good throughout. The report is very clearly presented, with subtitles and a contents page Ã¢â‚¬â€œ this makes it much easier both for a teacher to mark it and someone to read. However, usually a report should be presented fairly chronologically, to make it clear to the reader the steps and thought processes involved Ã¢â‚¬â€œ but in places the author has discussed things a bit early, for example Ã¢â‚¬Å“because the angles that were worked with were all less than 12Ã‹Å¡, it is especially noticeable in the results and is be the single most prominent source of systematic error.Ã¢â‚¬Â was in the theory section (near the beginning) when I would not have mentioned it until the sources of error section. However, this is a fairly minor issue, and probably unlikely to have an effect on the marks. They have used diagrams and graphs throughout, which helps to illustrate their method and workings-out, although we were advised to include a Ã¢â‚¬Ëœworking graphÃ¢â‚¬â„¢ with gradient lines drawn on (in pencil) in order to gain more marks.

The author has carried out the experiment well and produced some good results. They have then used graphs and maths to calculate the value of the acceleration due to gravity on the trolley. S/he has evaluated all the uncertainties in their measurements, and suggested improvements to the experiment. However, they have not carried out the improved experiment, which may be necessary for some exam boards to gain full marks as you can compare results and discuss whether the improvements actually worked. They have shown their workings throughout, explaining in detail the thought behind all of the derived formulae, which is essential to gain good marks (it shows you actually understand what youÃ¢â‚¬â„¢re doing rather than copying some formulae out of a textbook).

The authorÃ¢â‚¬â„¢s response to the question is very good Ã¢â‚¬â€œ they have explained the experiment clearly, discussed their results, and evaluated the uncertainties. They have calculated a range of possible values for the acceleration due to gravity on the trolley, which is close to the true value of g. They have shown an excellent understanding of the physics and a good mathematical ability. They have provided a detailed evaluation of the experiment and a conclusion, in which they summarised their experiment. However, I would have worked out an average calculated value for the acceleration due to gravity and stated this in the conclusion, mentioning how close it is to the accepted value of 9.81 as this clearly shows you have found an answer to the original question.