How to find the Centre of Mass of a framework

Centres of masses in Mechanics is an important topic which when using this simple method will be very easily understood.

A framework is an object where it can be described as 'hollow' where there is no inside to the object and that there is only 'wire' on the outskirts of the object.

Here is a simple method to finding the centre of masses:

1) Identify the object and split it into individual wires.

2) Identify your origin point for the shape and work out what way your axis will go (the coordinates of the origin must be decided depending on the type of framework).

3) Work out the centres of masses of the individual frames.

4) Set up a table including the mass of the frame, the centre of mass coordinates and totals for these.

5) Use the formula to calculate the centre of mass.

This will all be shown in examples below.

The formula mentioned above is rsin(a)/a. This is used to find the coordinate of the Centre of Mass Point of a semicircle. This can also be done with (2/3)(rsin(a)/a) which will find the Centre of Mass Point of a semicircular lamina.

Resolving Forces - FAST

Here is an object of 2kg lying on a slop raised from the horizontal of an angle of 'a'.
From this we know that there is a force of 2g downwards as a result of (M)x(G).
We also know that the reaction force will be perpendicular to the slope as shown.
Therefore this triangle can be drawn with the bottom line parallel to that of the slope itself.

 We can now look at this triangle in more detail. The angle joining the vertical force of gravity and the reaction force will be the same as the angle of which the slope is raised from the horizontal.

From this we can draw the triangle to the right to show the different forces acting upon the object on the slope.

 As we know the values of the downward force (gravity) and the angle a; we can derive the values of the force parallel to the slope and the value of the reaction force.

The reaction force is worked out using SOHCAHTOA.
Cos = Adjacent/Hypotenuse.
Cos a = Hypotenuse / 2g
Hypotenuse = 2g Cos a

The force parallel to the slope is worked out the same way:
Sin = Opposite / Hypotenuse
Sin a = Opposite / 2g
Opposite = 2g Sin a

The trick to resolving forces is to learn that:

• The force next to the angle will always use 'Cos'.
• The force that is not next to the angle will use 'Sin'.

Therefore we can assume the general formulae:

• 'mg cos a'

for the reaction force of an object.

• 'mg sin a'

for the force parallel to the slope.