There is a very large packing case, far too heavy to lift, and I want to drag it along the floor by a piece of rope. I want to find that how does the force I have to use vary with the angle that the rope makes with the horizontal. And also I want to find the angle which minimises the force required.

First I will work out the angle theoretically, in other words, I will make a model of this situation. Then I will do a simple experiment to find what the degree of the special angle is practically. Finally I would like to compare the experiment results with the prediction of the model.

* Assumption

As do an experiment, I cannot make the environment to be perfect and when I make the model, it is impossible for me to consider every factory in this case. Therefore I need make some assumptions.

1. The string used in this experiment is light and inextensible.

The rope used in this particular experiment is thin and about 1.5m long. So for this experiment, the mass of the rope is extremely small and it can be ignored. The string itself will extend when there are several forces acting on it. Hence it will create a tension on the string. However it is very small as well. Therefore I will ignore it when I model the experiment. The tension on the string would have the same magnitude as the weight has been put in the end of the string.

2. The pulley used in the experiment is smooth

In this case, I assume that the coefficient of friction of this pulley is zero. So there is no friction force between pulley and string.

3. There is no air resistance.

In such situation, the packing case would be actually moving very slowly, so the air resistance will be very small; and also it’s very complicated to model the air resistance for this object. Therefore it’s worth to assume that the air resistance in this situation is negligible.

4. Assume the wood block I used in the experiment is a particle rather than a body.

In this situation the block is only used for providing a force. So it doesn’t matter whether it is a body or not, while if consider the block is a body, I need consider toppling and sliding in this experiment. However it dos not improve anything. Therefore I assume the block is a particle.

5. Assume that the block is on the exactly horizontal table

Assume that the wood block is on the horizontal ground, so the friction acting on the block would have the same magnitude as the horizontal component of the tension on the string.

* Manipulating the Model

I am going to use some mechanics theories to set up several equations to manipulate this model. And I also will use these equations to predict the results of the experiment.

The main theories I will use here are Coulomp’s Law and the equation of motion. The Coulomp’s Law says if the object is on the point of sliding or sliding, the frictional force between the object and the surface is given by:

F =ï¿½R

And the equation of motion shows that resultant force acting on an object equals to the mass of this object multiply the acceleration of this object. The equation is given by:

F=ma

The diagram below shows the actual experiment I have done.

According the assumption I made, T should equal to m’g. TcosA and TsinA are the horizontal and vertical components. g is the acceleration of gravity, which is 9.8 N/kg. Followed is the list of all the force which is act on the block.

Vertically:

1. The weight of the block, which is mg.

2. The reaction force given by the table

3. The vertical component of tension T, which is TsinA.

Horizontally:

1. The limiting friction between the block and table, which is ï¿½R.

2. The horizontal component of tension T, which is TcosA.

If the block at rest, just move or sliding at a constant speed, I can say it is in equilibrium. So the final force on the block should be zero. In this particular case, we only interested in the equilibrium on horizontal. Because the block is putting on the table, vertically it should be always equilibrium. For horizontal, friction force should equal the horizontal component of T. Therefore we can obtain an equation ï¿½R=TcosA.

Because R=mg-TsinA, ï¿½(mg-TsinA)=TcosA.

ï¿½mg-ï¿½TsinA=TcosA

T(cosA+ï¿½sinA)= ï¿½mg

T=ï¿½mg/(cosA+ï¿½sinA)

Since there is one string link the block and the weights, the tension which is acting on the block is the same as it acting on the weights. Therefore the tension T is the same as the weight of these weights, which is m’g. Hence the equation above can be written as follows.

T=ï¿½mg/(cosA+ï¿½sinA)=m’g

By this equation, I will make the prediction of the results of this experiment. The purpose of the experiment is that find the minimum T through change the degree of A. ï¿½mg in this equation is constant, so I should make the (cosA+ï¿½sinA) as big as possible to decrease T as much as possible.

Suppose y=(cosA+ï¿½sinA)

Use calculus dy/dA= ï¿½cosA-sinA

When dy/dA=0 ï¿½cosA – sinA = 0

ï¿½cosA = sinA

ï¿½ = tanA

Now I work out an equation that is ï¿½ = tanA. Therefore I know that when tanA is equal to the coefficient of friction for the pair of surface it requires less tension to move the block. In other words, it requires less force to lift the case.

* The experiment

Here is a list for the apparatus will be used in this experiment.

1. a wood block

2. a pulley

3. a string

4. a table

5. a stand

6. a rule

7. an electronic scale

8. some weights

The diagram on the next page shows how this experiment works.

First I measured the mass of the block by using the electronic scale. And the mass of the block is 356g that is 0.356kg.

Then I made the sinA equal 0 that means the height from the top of the block to the pulley is zero, and the degree of angle A is zero as well. The reason why I use sin not the degree is it is very hard the measure degree when I did the experiment. And I cannot get an exactly degree of an angle. As compare with using degree, sin is more accurate. When the block is equilibrium, if A is zero degree, the tension on the string is the same as the friction force between the block and the table. So the friction force is the same as the weight of the weights. Then I can use the equation F=ï¿½R to calculate the coefficient of friction ï¿½.

At this situation the weight of the weights is 92.56g (to 2 decimal place). Through calculation, the coefficient of friction ï¿½ is 0.26 (to 2 decimal place).

As I know theï¿½, I can give the exactly value of the angle which minimises the force required theoretically. tanA=0.26 so A=14.57ï¿½

The graph below shows the theoretical curve.

Then I have tried several angles by changing sinA. When I changed sinA, I did not change the height from the top of block to the pulley, but I changed the length of string between pulley and block, because in this case, change the height of the pulley was more different and would produce more errors.

I measured the force for each angle I have taken three times and then take the average force. The reason why I did this is also reduce the error.

sin(angle)

Mass(g)

Mass(g)

Mass(g)

Average mass(g)

0.0714541

101.3

106.4

101.4

103.03

0.0769231

101.4

101.3

101.4

101.37

0.0833333

96.5

96.5

96.4

96.47

0.0909091

91.3

91.5

96.3

93.03

0.1

91.5

91.4

91.4

91.43

0.1111111

86.6

91.5

86.4

88.17

0.125

86.6

86.4

86.3

86.43

0.1428571

81.4

81.4

91.4

81.40

0.1666667

81.5

81.7

81.4

81.53

0.2

86.6

81.6

86.3

84.83

0.25

86.5

86.4

86.4

86.43

0.3333333

91.4

91.4

91.6

91.47

0.5012531

106.8

101.3

101.3

103.13

The smallest mass of my weight is about 0.1g, so it is the limitation of the accuracy of this experiment. And there is an out line data appeared which is 91.4. So when I calculate the average mass, I avoid this data. By the way, the average mass is to 2 decimal place. Tension is the mass multiply by g.

Followed is the graph of these data.

Through this experiment, I found the angle which can minimise the force required is about 8.21ï¿½. The result is not quite close to the theoretical one which is about 14.57ï¿½. I think this is because there are too many variation in the experiment and model as well.

* Discussion of variation in the experimental results

The graph of the experimental results is not smooth; some points are a little bit up or down. I think it is because of the errors.

1. Measurement

When I was doing the length measurement, the veracity of my reading was dependent on the rule I used. Even if I took my reading three times, I couldn’t avoid all the errors of the measurements. Also when I measured the weight of my weights, I didn’t measure the suttle of the weight. Because the electronic scale was very sensitive, even some dust would influence the measurement.

2. The change of the situation

Every time when I wanted to regress the block, I couldn’t do that is exactly as it was before, in other words, the situation of this experiment had a tiny change every time.

There is a graph to show the effects of these variations.

* Comparison between the experimental results and the prediction of the model

In my prediction, the angle which can minimise the force required is about 14.57ï¿½. However through the experiment, I gained the result which the angle is about 8.21ï¿½. It is far away from the theoretical angle. So the two results do not match each other very well. The graph below shows this clearly.

The experimental line is more curved than the theoretical one. That means it is more changeable. I consider the reason of this is the variations which I mentioned above. Take the wrong reading and change the situation make the difference between each value bigger than it should be.

But the basic sharps of two graphs are the same. And in fact, 6 degree is not a very big angle. Therefore I can say the match of the theoretical results and the experimental results is not too bad.

* Revision of the process

The other reason why these two results are difference is the assumption I made at the beginning. There are some factors I did not consider when I manipulate the model and did the experiment. The biggest two errors I think are the friction between string and pulley and the extension of the string.

Since the friction between string and pulley dos exist, I need heaver weights to move the block. The reason is I need create extra force to cancel the friction. The extension of the string also required the bigger tension, because the extension of the string itself produced the extra resistant force.

In order to improve this experiment, I consider the easiest point is that make the measurements more accurate. I can use a better rule to take the distance measurement, and use a more sensitive scale. Also I can improve the environment of the experiment to reduce the effect to the weight measurement.

In order to improve the model, I think the best thing to do is considering more factors than I did in this course. For example, if I did not assume the pulley is smooth, the theoretical result should be more close to the real one, because the pulley does not smooth in practice. And I can consider the extension of the string as well. However that will make the calculation much more complicated than I have done in this coursework. If the mentioned calculations have been done, I can avoid these two biggest factors which make the prediction imprecise, and the prediction should be much better.