Hence there are two parts to this model. The first part being when the cylinder is sliding down the gutter, with the gravitational force acting on the cylinder and resultant force acting from the gutter to the cylinder – with the frictional force the gutter and the cylinder makes ignored and also the air resistance ignored. The second part is when the cylinder/particle leaves the gutter/edge of the table and falls to the floor in projectile motion. Therefore the forces involved in this experiment are as follows:
Sliding Section
Projectile Section
The unknown in this model is d, the horizontal distance from where the ‘particle’ cuts the paper to the edge of the table. Using the formulae of projectiles for the horizontal displacement
x = ut cosθ
θ is the angle that the particle is dropped, i.e. the angle the gutter makes with the table.
u is the initial speed when the particle is dropped from the edge of the table, which I can work out from forces in the sliding section. Ignoring friction and air resistance as assumed, the forces working on the particle while it is sliding is only its own weight:
P = mg sinθ, where m is the mass, g is the acceleration due to gravity, taken as 9.8
According to Newton’s Second Law
P = ma, where a is acceleration of the object
Hence
ma = mg sinθ
a = g sinθ
According to one of the suvat equations (constant acceleration formulae), since the object is modelled as a particle so that it is assumed to take a linear motion,
v2 = u2 + 2as, where u is the initial speed and v the end speed, s is the displacement which the particle has travelled
v is the speed the particle assumes when it leaves the gutter, i.e. the initial speed when it starts projectile motion. The object started from rest, hence the initial speed is 0. a is expressed in terms of g sinθ. s is denoted by l, the length that the object has travelled in the gutter. Therefore
v2 = 2gl sinθ
Hence the horizontal displacement in this case is
sx = d = t cosθ
The t in the projectile equation is time, which is an unknown, but since the vertical displacement is known in this case, t can be put into the equations and be cancelled out. Hence rearranging the above equation into an equation with t being the subject
t =
The projectile formula for the y distance is:
y = ut sin θ - ½ gt2
The angle, θ, that the particle makes with the horizontal is really negative, therefore in this situation the vertical displacement the particle has travelled in its projectile motion when it reaches the floor is
sy = - ut sinθ - ½ gt2 = -h (where h is the height of the table. It is negative because where the particle started to be dropped is taken as the 0 point, hence the vertical displacement it has travelled when it reaches the floor would be negative in reference to the starting place)
sy = ½ gt2 + ut sinθ - h = 0
Substitute t in terms of sx
+ - h = 0
+ - h = 0
+ d tanθ - h = 0
This is now in the form a quadratic equation in terms of d. Using the formula
Hence in a controlled experiment, with the values for the variables l, θ and h, I can hopefully obtain the value of d.
At the mean time I have done a controlled experiment in a group of five.
The experiment has been set up in the way it was described in modelling the situation. The semi-circular gutter is of length 1m. It is held by a clamp stand so that the angle can be kept – as opposed to a human holding the gutter. The metal cylinder has base diameter 25mm and height 27mm – hence quite a small object. We have not measured the mass since my equation told me that the mass of the object is not needed. The table’s top surface is 70cm from the floor, i.e. h = 70cm.
Actions have been carried out to ensure that the model that we make is as accurate as possible:
-
When the cylinder cuts the paper which is glued to the floor (so that it wouldn’t move and ruin the measurement), the measurement is taken from the nearest mark that is parallel to the edge of the table (since the mark will be a semicircle) to perpendicular the edge of the table. This is demonstrated in the diagram below:
- When we are measuring the distance on the table we are assuming that the table is flat, i.e. parallel to the horizontal – because the measurements are made to calculate the angles and if the table is not flat then the calculation would not be accurate. Hence the table is as horizontal to the floor as we can make it.
-
The gutter shouldn’t move when the cylinder is dropped, otherwise it destroys the model and the assumptions of a set length l and a set angle. Our solution is to blue-tag the end of the gutter to the edge of the table
- The cylinder’s bottom end has been marked so that if there is any inaccuracy as to the coefficient of friction for the surface of the cylinder or centre of mass, then in a way that is accounted for
Hence in summary, the useful measurements are:
Table height (h): 70 cm
Cylinder measurements:
Base diameter: 25 mm
Height: 27 mm
From this, we have experimented with two different angles. In order to find what these two angles are, we have taken measurements along the gutter and its corresponding horizontal distance from the edge of the table.
Knowing these measurements I can then work out the angle of friction, α, using cosine since
, as demonstrated below:
In order to reduce random error, we have taken three measurements and taken the average values (in a spreadsheet):
Angle α:
Angle β:
With each angle, we have also had controlled l, the lengths from where we drop the particle – starting 15cm from the edge of table along the gutter, taking every 15cm up the gutter for 5 times, until 75cm from the edge of the table. For each value of l we have dropped the particle 5 times and taken the average distance, d. This is in order to allow for random variations in the experiment and also the measurements.
Hence the results are as follows:
Angle α:
Angle β:
As mentioned before, there would be a systematic error in the measurement of d, since the distance should really be measured to the centre of mass rather than the corner of the cylinder that cuts the paper:
In order to find the horizontal length from the corner A (where it cuts the paper – i.e. where d is measure to) and centre of the mass, C, I need to know the angle the cylinder makes with the horizontal, angle δ and then use trigonometry. However, angle δ changes all the time and it is very hard to measure it. This means there will be systematic error persisting all the time.
Closely inspecting this diagram it is easy to see that the horizontal length from A to C is not really a lot. Because the error will exist anyway, and because it will be very small, I will not make any systematic corrections to the error in this measurement.
Therefore putting the figures onto graphs:
I have attempted to put a trendline onto the values of d. As the graphs have shown, this has been an unsuccessful attempt since the line doesn’t actually pass through any of the points on both graphs. Hence I know that d and l do not have a linear relationship. What the graphs do show though, is that l and d have a positive correlation relationship, i.e. as l increases, d increases. Putting this back to the model, it means that the longer the cylinder had to slide, the longer horizontally it would travel after it left the edge of the table. This is expected since the more the object slid, the more g, the gravitational force would come into effect, the more speed the particle will ‘gather’ before it leaves the gutter and starts to drop in projectile motion.
To find the actual relationships between l and d, I have rearranged the information to show the average values and minimum and maximum values, so that I can show the information on graphs with error bars:
Angle α:
Angle β:
For angle α, it is easy to see that the error bars have a tendency to reduce in size as l increases. This arguably can be expected for any angles since if the absolute errors stay constant or near constant, then the longer the d, the less relative errors there would be. However, the opposite can also be argued to be true, i.e. that if it is the relative errors that stay constant, then the absolute errors will increase as d increases. In the case of our experiment with angle α however, the error bound has reduced as d increased.
Angle β was not too much bigger than α, hence not too much difference in results. However, the graph looks different to the angle α graph in the way that the error bar sizes are quite different. But when looking closer I can see that the trend would be similar if the minimum and maximum values for l = 15cm were more apart. It may just be that we were lucky in this particular part of the experiment that this error bar is so small.
In general I am quite satisfied with our experiment results, since although there are variations, they are quite small. Variations in these results are mainly due to random errors.
For the actual graph of d, I expect it to be somewhere between the error bar lines, which accounts for the random variations, as explained above.
In the meanwhile, using my model,
, where h = 70cm
Before I put in all the numbers I need to make a systematic correction in order to match the model results to the experimental results. I have identified that there is systematic error in the measuring of l and d, therefore there should be some systematic corrections taking into account.
Consider the following diagram:
The centre of mass, C, will be halfway of the cylinder, as labelled on the diagram. The actual length l should be from the centre of mass to the edge of the table rather than the bottom of the cylinder. The reason we didn’t measure l in the first place is because that it would be very inaccurate to attempt to measure the centre of the mass. In comparison just taking the measurements from the bottom of the cylinder is much easier and more accurate – and now I can just add on the length from the bottom of the cylinder to the centre of mass to the values of l. I have already measure that the cylinder is of length 27mm. Because it is symmetrical, the centre of mass for the cylinder will be at the midpoint (as shown on the diagram) – i.e. half of the length of the cylinder, 27/2 = 13.5 mm. Hence the values of l in calculations would be:
l = the length from the bottom of the cylinder to the edge of the table + 13.5mm
Again using spreadsheet, I have worked out the theoretical values of d:
Although the model
would give 2 values of d (since there are normally two roots to a quadratic equation), for each of the cases above one value of d has come to negative – in all cases, it was the calculation of
Because in a real life situation it is not possible to have a negative value of d, I have hence ignored them all together.
At first sight these figures do not match our experiment results. But this can be expected because there has been assumptions made in prior to the model – hence some forms of variation in the results are expected.
Putting the figures onto graphs:
The shapes of the graphs look similar to the ones I have obtained in the experiment. In order to compare by how far the variations were, I have put the experiment results (average, min. and max. distances) and the model figures onto the same graphs:
It is confirmed that the shapes are quite identical, taking into accounts of random and systematic errors that could have occurred in the experiment. The differences between the model and experiment results though, are quite large. I have shown this in the form of absolute and relative errors, based on the model results:
Absolute error = |Experimental results – Model results|
Relative error =
since I have assumed the model figures are the actual values and the experimental results variations from the ‘actual’ value.
Hence:
Angle α:
Angle β:
The errors are quite large, with the absolute error ranging from 4.6 to 13.1 – as the graph has shown too, that the error increases as l increases. This means a relatively consistent relative error from around 0.14 to 0.22. These variations are mainly due to the systematic variations which I have identified before. Also as a major part they are due to the assumptions that I have made in order to put the situation to a theoretical model.
I feel that the match has been good, since the shapes of the graphs were quite similar – confirming that the relationship between l and d that I have found in my model is close to the real life relationship of the two variables. However I feel that the relatively large error between the model and the experiment results was not quite satisfactory and maybe one of the assumptions of the model can be taken out to make the model more realistic. I believe that friction would have been a very good factor to be considered in the model.
Putting friction into the big picture will only affect one thing in the whole model – the speed when the cylinder leaves the gutter, i.e. the forces of the sliding section will be different. Although this does not sound like a very big influence to the final results, I believe it will be, and it can be illustrated by the new model below. One assumption that needs to be made for this new model though, is that the friction is constant throughout. All other assumptions stand still.
Going back to the force diagram, now added friction:
According to the force diagram, the forces acting parallel to the plane can be represented by:
P = mg sinθ - F, where P represents the forces acting on the cylinder parallel to the gutter, and F is friction
According to the Colomb’s Law, when there is limiting equilibrium or when sliding occurs,
F = µR, where R is the resultant force acting on the object
R can be found by resolving forces perpendicular to the gutter:
R = mg cosθ
Therefore:
P = mg sinθ - µR = mg sinθ - µ mg cosθ
Since P = ma
∴ mg sinθ - µ mg cosθ = ma
The mass of the object, m, can be conveniently cancelled out
a = g sinθ - µg cosθ
Putting this into the suvat equation
v2 = u2 + 2as = u2 + 2s (g sinθ - µg cosθ) = u2 + 2gl (sinθ - µ cosθ)
Since u = 0,
v2 = 2gl (sinθ - µ cosθ)
Hence the horizontal displacement is
sx = d = t cosθ
⇒ t =
And sy = ½ gt2 + ut sinθ - h = 0
+ - h = 0
+ - h = 0
+ d tanθ - h = 0
∴
Before putting the values of the variables into the equation, I need to know µ. µ is the coefficient of friction, as proposed in the Colomb’s Law.
According to this law, I know that when limiting equilibrium occurs
F = µR = the opposite force
Limiting equilibrium occurs when the forces acting on the object are balanced. Therefore, if I have found the angle at which the limiting equilibrium occurs (which we call the angle of friction), then I can find the value of µ in this situation.
We have hence experimented to find the angle of friction. After gradually moving the clamps holding the gutter up and down, we have found a position where the object would slide with one slight touch but otherwise remains where it is. We have decided that the angle that the gutter is making with the table is the angle of friction.
Therefore:
Since the measurements of the lengths can only be taken to the nearest 0.1cm (as explained earlier), I am taking the value of angles to the nearest 0.1° too, as that is the digit that I can be most sure about.
The measurements look quite accurate to me, apart from the first one – but I consider it as an anonymous result, since the distances are shorter and hence a greater error margin. Because the 2nd and 3rd measurements actually agreed on each other to the nearest 0.1°, I have decided to take their values of α and ignore the 1st measurement.
Therefore the angle of friction is 12.9°.
Use this to find the value of µ:
F = µR = µ mg cosθ = mg sinθ
µ = = tanθ = tan 12.9° ≈ 0.229
Putting all the values into the new equation (still taking l as
l = the length from the bottom of the cylinder to the edge of the table + 13.5mm
Immediately it is easy to see that the new model results are very similar to my actual experiment results. Show the match in error form:
Angle α:
Angle β:
This can be illustrated even better with graphs
This shows the definite match between this model and the experiment results. Hence my belief that friction being the largest component of the imperfections of the first model is proved. Other assumptions made in the models, such as no air resistance, no rotation, etc. do make the model imperfect, however they are of little significance in these cases.