Objects Suspended in Equilibrium

What is meant by objects suspended in equilibrium?

An object can be suspended at a fixed point by a string such that it is allowed to hang freely and be in equilibrium (at rest)
- e.g. A sign hung from the ceiling of a shop by single rope (string)
- An object will either be a uniform lamina or a uniform framework
Equilibrium means
- there is zero resultant (net) force acting on the object
- there is zero resultant (net) moments about the point of suspension

Objects suspended in equilibrium have two forces acting on them
- tension in the string (acting at the fixed point)
- weight (acting at the position of the centre of mass)
These forces act through a vertical line which passes through the point of suspension and the position of the centre of mass
Problems often require the angle that is created between this vertical line and an edge of the object to be found

2-1-5-fig1-suspended-objects

A lamina could be suspended at a fixed point from a vertex (as above) or along an edge, or, suspended by a point on its surface that it would then be able to pivot (rotate) around
A lamina suspended at a pivot will have two forces acting on its
- reaction force (acting at the pivot)
- weight (acting at the position of the centre of mass)
As with the fixed-point scenario above, these forces will act through a vertical line which passes through the point of suspension and the position of the centre of mass

2-1-5-fig2-lamina-pivot

The mathematics involved in finding the angle is the same for both the fixed-point and pivot scenarios but you should be aware of the different phrasing in questions

How do I solve problems involving freely suspended laminas and frameworks?

STEP 1 Sketch a diagram or use a given one and add to it as you make progress

STEP 2 Find the position of the centre of mass of the object

Ensure you know whether you are working with a

lamina or a framework and the differences in finding the

position of the centre of mass

STEP 3 On your diagram, add the position of the centre of mass

Draw a line to represent the downward vertical

This will be a straight line connecting the point at which the body is suspended to the point of the centre of mass

Identify the angle required

STEP 4 Use trigonometry to find the required angle

Interpret the result in the context of the problem and answer the question accordingly

Worked example

As part of the set in a theatre production, a large polystyrene letter E is to be produced and will hang above the stage by a single rope attached to the ceiling.

2-1-5---we-diagram--ial-mech

The polystyrene letter is modelled as a uniform lamina and the rope is modelled as a light inextensible string.

Find the angle the rope makes with the side PQ when the sign is at rest and suspended from the point P.

2-1-5-fig3-we-solution-1

Examiner Tip

Ensure you know whether you are working with a lamina or a framework.
Once you have drawn the downward vertical line on your diagram turn your page (rather than trying to sketch it) to get an idea of how the lamina or framework would look like when suspended from the given point.

Objects Suspended in Equilibrium (Edexcel International A Level Maths: Mechanics 2)

Revision Note