OCR Specification focus:
‘Define centre of mass/centre of gravity; determine position experimentally for irregular laminae.’
Understanding the centre of mass and centre of gravity is crucial in analysing the stability and equilibrium of objects. These concepts explain how forces act and balance within physical systems.
The Centre of Mass
Definition and Meaning
Centre of Mass: The point within an object or system where the entire mass may be considered to act for the purpose of analysing linear motion.
Every object, regardless of shape or composition, has a centre of mass (COM). For uniform, symmetrical objects, the COM lies at the geometric centre. However, for non-uniform or irregular objects, it depends on how the mass is distributed throughout the body.
The COM is a conceptual point rather than a physical one; it helps simplify motion and stability problems by reducing an extended object to a single mass point.
Locating the Centre of Mass in Regular Objects
For simple uniform shapes (e.g. a cube, cylinder, or sphere), the COM is at the geometric centre because the mass is evenly distributed in all directions.
For composite bodies made of several uniform parts:
Identify each part’s individual COM.
Multiply each COM position by its mass.
Divide the sum of these moments by the total mass to find the overall COM position.
EQUATION
—-----------------------------------------------------------------
Centre of Mass (x̄) = (Σmᵢxᵢ) ÷ (Σmᵢ)
x̄ = position of the centre of mass (m)
mᵢ = individual mass (kg)
xᵢ = position of individual mass (m)
—-----------------------------------------------------------------
This formula applies in one dimension, but can extend to two or three dimensions using vector or coordinate methods.
The Centre of Gravity
Definition and Relation to COM
Centre of Gravity: The point through which the resultant weight of an object acts, regardless of the object’s orientation.
In a uniform gravitational field, the centre of gravity (COG) coincides with the centre of mass, as every part of the body experiences the same gravitational acceleration.
In a non-uniform gravitational field, the two points differ because different parts of the object experience slightly different gravitational forces.
Practical Importance
The COG determines balance. An object balances when supported directly below its centre of gravity.
In design and engineering, lowering the COG improves stability, reducing the likelihood of tipping over.
The COG is essential in understanding torques, moments, and rotational equilibrium.
Experimental Determination of the Centre of Mass
Method for an Irregular Lamina
A lamina is a thin, flat sheet with negligible thickness compared to its other dimensions. Irregular laminae (such as cardboard cut-outs) provide a practical way to determine COM experimentally.
Procedure:
Suspend the lamina freely from a pin or nail through one of its edges so it can rotate freely.
Allow it to hang at rest; a plumb line (a string with a small weight) is hung from the same pin.
Mark the vertical line on the lamina along the plumb line’s path.
Repeat from at least two other suspension points.
The intersection of the lines marks the centre of mass of the lamina.
Plumb-line method for an irregular lamina. The lamina is suspended from several points; a vertical line is drawn each time using a plumb line. The centre of mass lies at the intersection of these lines. Source
This simple yet effective experiment works because, in equilibrium, the weight always acts vertically downward through the centre of gravity.
Key Experimental Notes
The lamina must be free to swing without obstruction to ensure it naturally comes to rest.
Accuracy depends on marking precise plumb line paths and minimising air movement.
For a non-uniform lamina, repeating measurements from several points increases reliability.
Factors Influencing the Centre of Mass
Distribution of Mass
Changing how mass is spread affects the COM position. For example:
Adding mass to one side of a beam shifts its COM toward that side.
Removing material from one region moves the COM away from that side.
Shape and Geometry
Irregular shapes have COM locations determined by symmetry or lack thereof:
Symmetrical shapes → COM lies on the axis of symmetry.
Asymmetrical shapes → COM shifts toward the denser or larger side.
Uniform vs. Non-Uniform Bodies
Uniform density: COM depends solely on geometry.
Non-uniform density: COM depends on both geometry and mass distribution; the calculation involves integrating mass elements across the body.
Centre of Gravity and Stability
Stability Conditions
An object is stable if, when slightly displaced, it returns to its original equilibrium position. This depends on the position of the COG relative to the base of support.
Diagram of base of support and the line of action of weight through the body’s centre of gravity. Balance is maintained while the line of action passes within the BOS; tipping occurs once it passes beyond the edge. The image includes a seated figure and chair to visualise the BOS; this extra context is not required by the syllabus but clarifies the concept. Source
Types of equilibrium:
Stable equilibrium: When displaced, the COG rises, and weight creates a restoring moment.
Unstable equilibrium: When displaced, the COG falls, and weight causes further rotation away from equilibrium.
Neutral equilibrium: When displaced, the COG remains at the same height, and no restoring moment acts.
Applications
Vehicles and furniture are designed with a low COG to improve balance and safety.
Tightrope walkers use long poles to lower their combined COG, enhancing stability.
Architectural structures, such as bridges and towers, use wide bases and low centres of mass to resist toppling forces.
Gravitational Field Effects on the Centre of Gravity
In everyday conditions, the Earth’s gravitational field is nearly uniform, so COM and COG coincide. However, on larger scales—such as satellites or extended structures near planets—gravitational field variations cause the COG to shift slightly relative to the COM. This can generate torques that affect orientation, known as gravitational gradient torques.
Summary of Core Ideas
Centre of mass is a geometric and mass-distribution concept; centre of gravity concerns gravitational effects.
In a uniform field, they are identical.
The position of the COM can be found experimentally by suspension for a lamina.
Understanding COM and COG is vital for analysing equilibrium, stability, and mechanical balance in physics and engineering.
FAQ
For irregular three-dimensional objects, the centre of mass can be found using integration if the mass distribution is known.
Divide the object into infinitesimally small elements of mass (dm).
Use the equation: x̄ = (∫x dm) / (∫dm), and similarly for ȳ and z̄ coordinates.
This process effectively averages the positions of all mass elements, weighted by their mass, to give the coordinates of the centre of mass. In practice, computational methods or CAD models are often used for complex shapes.
Lowering the centre of gravity means that the line of action of weight remains within the base of support for larger tilting angles.
This increases the torque needed to topple the object, as the restoring moment (weight × perpendicular distance to pivot) becomes more effective.
Objects with a low centre of gravity, such as racing cars or furniture with wide bases, therefore resist tipping and are more stable under external disturbances.
Yes. The centre of mass does not need to be located within the material of an object; it depends solely on the mass distribution.
Examples include:
A ring or hollow cylinder, where the centre of mass lies at the empty centre.
A boomerang or horseshoe-shaped object, where the COM lies in the air between arms.
This concept is important in understanding the motion of irregular or open structures, especially when analysing rotation or balance.
When an object’s density varies, the centre of mass shifts towards the region with higher density.
For example, if one side of a rod is made denser or thicker, its COM moves closer to that end. Engineers exploit this principle when:
Balancing rotating machinery or wheels.
Designing aircraft and rockets, where internal mass distribution affects flight stability.
In calculations, variable density requires integrating over both shape and density to determine the COM position.
The centre of mass determines how the human body maintains balance, generates movement, and controls posture.
In sports:
Athletes adjust their COM to optimise stability and performance — for example, long jumpers lower it during take-off for better momentum control.
Gymnasts move their limbs to shift the COM within their base of support.
In biomechanics, analysing COM motion helps assess balance, gait, and efficiency of movement, particularly in rehabilitation and performance training.
Practice Questions
Question 2 (5 marks)
A student investigates the centre of mass of an irregular lamina made of thin card.
(a) Describe an experimental method the student could use to determine the position of the centre of mass.
(b) Explain why the method works.
Mark Scheme:
(a) (3 marks total)
1 mark: Suspend the lamina freely from a point on its edge using a pin or nail.
1 mark: Hang a plumb line from the same suspension point and mark the vertical line on the lamina.
1 mark: Repeat the process from at least two other suspension points; the intersection of the lines is the centre of mass.
(b) (2 marks total)
1 mark: When suspended, the lamina comes to rest with its centre of gravity vertically below the point of suspension.
1 mark: The intersection of the vertical lines gives the point through which the weight acts — the centre of gravity (and hence the centre of mass).
Question 1 (2 marks)
Define the term centre of gravity and state how it relates to the centre of mass for an object in a uniform gravitational field.
Mark Scheme:
1 mark for correctly defining the centre of gravity:
The point through which the resultant weight of an object acts.
1 mark for relating it to the centre of mass:
In a uniform gravitational field, the centre of gravity and centre of mass are at the same point.
