# Motion of an object as it slides down a slope at different angles to the horizontal.

Mechanics 2 Coursework        Angelina Lai

Whoosh Down the Slide

I am investigating the motion of an object as it slides down a slope at different angles to the horizontal, and falls to the ground.

I will start the investigation with an experiment. I will let an object, modelled as a particle, slide down a guttering, held at an angle with a table of a certain height from the floor. I want to know how far from the foot of the guttering would the object land for different angels the slope would make with the table and different distances down the guttering. Hence the experiment would be set up as below:

The gutter is a semi-circular plastic rigid piping with fairly smooth surface. It is held at an angle, θ, to the horizontal, the other end resting on the edge of the table. The object is a small metal cylinder. The table’s top surface is h cm from the floor. Paper is glued to the floor to be used as a marker. When the cylinder is dropped and first hit the floor, it will cut the paper. We will then measure the distance, d, of this mark to perpendicular of the table’s edge.

Modelling this situation, there are a few assumptions that I have to make first:

• Assume that the cylinder and gutter in this experiment are perfectly smooth and no friction is to be considered in the calculation – this is not going to make such big influence since they are both quite smooth. This is very important for this model since it concerns whether friction will be calculated
• Since the object is small and round with a not so big surface area, and the rate at which it drops is relatively slow, air resistance is assumed to be 0. This is also very important since otherwise the calculation will have to involve complicated steps involving the air resistance
• The object used can be modelled as a particle – hence it will adopt a random motion when no force is acting on it, and as it drops from the guttering, it will be in projectile mode. This is quite important for the modelling too since then when calculating the displacement the object travels after it leaves the edge of the table
• The gravitational acceleration, g, will be constant at 9.8ms-2
• The path of the cylinder as it slides off the gutter is straight
• The table is flat – since the angle the gutter makes with the horizontal line is calculated by measuring the distance of the gutter at various points and their corresponding perpendicular distance on the table
• As the object slides on the gutter, there is no rotation. Otherwise the surfaces in contact are not the same and the calculations will become too heavy
• The angle at which the gutter makes with the table/horizontal is constant even when the object is dropping. Since the gutter is rigid in real life situation this should not be too untrue. This is to simplify the calculation too.

In order to reduce systematic errors, I have also set a few rules about the setting up of the equipment and how to measure:

• The measurements will be made to the nearest mm. This is the most precise the experiment will get since that is what a metre ruler would give. Any more precision would not give too many definite digits in the answer
• From where the cylinder is left to slide down, the length it will travel from the starting point to the end of the gutter/edge of the table, is measured from the bottom of the cylinder before it starts, and the edge of the table. E.g. I wanted to drop the cylinder 75cm along the gutter, I will leave it like this:

• Where the cylinder will cut the paper, the cut will look like a semi-circle. The measurement is made from the floor perpendicular to the table edge to the nearest mark that is parallel to the edge of the table. This is because I have chosen to measure l from the bottom of the cylinder – measuring where it cuts the paper will be equivalent of measuring the horizontal distance where the bottom hits the floor to the edge the table. By measuring in this way I am being consistent. There will be a slight systematic error, since the angle which the cylinder makes with the horizontal when it is lying on the gutter will probably not be the same as when it hits the ground (as demonstrated below):

However, since the object is very small, the error will be very small.

Since I am modelling the cylinder as a particle, I should have really measured the distances relative to the centre of mass. However this is hard to measure during the experiment – if I did then it would be very inaccurate. What I will do is to make a systematic correction and find the distances from the centre of mass of the object when calculating the models – by taking the cosine of the angle θ. I will discuss this later when I come to the analyse section.

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

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