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.

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Mechanics 2 Coursework

* Introduction

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.

* Assumptions

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.

. 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 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 the Kinetic Friction Law and the equation of motion. The Kinetic Friction 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 =µR

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. Tcos? and Tsin? 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:

. The weight of the block, which is mg.

2. The reaction force given by the table
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3. The vertical component of tension T, which is Tsin?.

Horizontally:

. The limiting friction between the block and table, which is µR.

2. The horizontal component of tension T, which is Tcos?.

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 ...

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