Centre of gravity Revision Notes

The centre of gravity of an object is the point at which the entire weight of the object can be considered to act. Weight is a force measured in newtons (N), calculated by W = mg, where m is the mass in kilograms (kg) and g is the acceleration due to gravity (9.8 m/s² near Earth's surface).

This point is a conceptual location that simplifies analysis of forces and motion, especially when considering balance and stability. It is important to note that the centre of gravity depends on the distribution of mass within the object and the gravitational field acting on it.

• Weight acts through the centre of gravity.
• Weight is a force measured in newtons (N).
• Centre of gravity simplifies force analysis.

For symmetrical objects with uniform density, such as a uniform rod or a regular solid, the centre of gravity coincides with the geometric centre. For example, the centre of gravity of a uniform rectangular block is at its midpoint.

In irregularly shaped or non-uniform objects, the centre of gravity may not be at the geometric centre and can even lie outside the physical material of the object. This occurs because the mass distribution is uneven, shifting the point where the weight effectively acts.

Understanding the position of the centre of gravity is essential for predicting how an object will balance or topple.

• Symmetrical objects: centre of gravity at geometric centre.
• Irregular objects: centre of gravity may be offset or outside the object.
• Mass distribution determines centre of gravity location.

The stability of an object depends on the position of its centre of gravity relative to its base of support. The base of support is the area beneath an object that supports its weight.

If the vertical line through the centre of gravity falls within the base of support, the object is stable and will not topple easily.

If this vertical line falls outside the base, the object becomes unstable and may topple under small disturbances or forces. This principle is used in designing stable structures and vehicles, and in sports to improve balance.

• Stability requires centre of gravity above base of support.
• Centre of gravity outside base causes toppling.
• Used in engineering and design for balance.

Although this objective does not require practical methods, it is useful to understand that the centre of gravity can be found experimentally by suspending an object from different points and drawing vertical lines downwards from the suspension points. The intersection of these lines locates the centre of gravity.

Mathematically, for composite objects, the centre of gravity can be calculated by dividing the object into simpler shapes, finding each shape's centre of gravity, and then using weighted averages based on their masses.

This calculation involves the formula: x_cg = (Σ m_i x_i) / Σ m_i, where x_cg is the horizontal coordinate of the centre of gravity, m_i is the mass of part i, and x_i is the position of the centre of gravity of part i. A similar formula applies for the vertical coordinate.

• Experimental method: suspend and draw vertical lines; intersection is centre of gravity.
• Calculation method: weighted average of component centres of gravity.
• Formula: x_cg = (Σ m_i x_i) / Σ m_i.

To draw a diagram showing the centre of gravity of a flat object:

1. Sketch the outline of the object to scale on graph paper.

2. Identify or estimate the geometric centre if the object is symmetrical; mark this point as the centre of gravity for uniform objects. For irregular shapes, divide the shape into simpler parts (rectangles, triangles). Mark the centre of gravity of each part (usually at the geometric centre). Use the weighted average method to find the overall centre of gravity and mark it on the diagram.

• Draw object outline to scale.
• Mark centres of gravity of component shapes.
• Calculate and mark overall centre of gravity.

The centre of gravity (CG) of a body is the point at which the entire weight of the body can be considered to act. For uniform shapes, the CG is at the geometric centre, but for non-uniform flat shapes (laminae), the CG may lie away from the geometric centre due to uneven mass distribution.

Determining the CG experimentally is important when the shape is irregular or made from materials of varying density. The method involves suspending the shape and using the direction of the weight force (vertical line) to locate the CG.

• Centre of gravity is the point where weight acts.
• Non-uniform shapes have CG not at geometric centre.
• Experimental method uses suspension and vertical plumb lines.

The apparatus required includes: a non-uniform flat lamina (e.g., cardboard cut-out), a thin strong string, a stand with a clamp or a fixed support, a plumb line (a small weight suspended on a string), a pencil or marker, and a ruler or set square for drawing vertical lines.

The lamina is suspended from a chosen point on its edge by tying the string securely. The plumb line is used to mark the vertical line from the suspension point downwards on the lamina.

• Non-uniform flat lamina (cardboard or plastic)
• Thin string for suspension
• Clamp stand or fixed support
• Plumb line (weight on string)
• Pencil and ruler for marking

1. Tie the lamina securely to the string at a point on its edge and suspend it from the clamp stand so it can swing freely and come to rest.

2. Once the lamina is stationary, hang the plumb line from the same suspension point so that it hangs freely alongside the lamina.

3. Use a pencil to mark a line along the plumb line on the lamina. This line represents the vertical line of action of the weight from the suspension point (line of gravity). This is the dependent measurement line for that suspension point, as it depends on the lamina's orientation under gravity and mass distribution. 4. Repeat steps 1 to 3 by suspending the lamina from at least one other different point on its edge, marking the vertical line each time. 5. The centre of gravity is located at the intersection of the vertical lines drawn from the different suspension points.

• Suspend lamina from a point on edge and let it settle.
• Hang plumb line from same point and mark vertical line on lamina.
• Repeat suspension and marking from at least one other point.
• Locate CG at intersection of vertical lines.

Independent variable: The suspension point on the lamina's edge (location where the lamina is hung).

Dependent variable: The vertical line marked on the lamina from the plumb line.

Control variables: The lamina itself, the plumb line length, environmental conditions (avoid drafts). Measurements involve accurately drawing the vertical lines using the plumb line and ruler. Repeating the suspension from multiple points improves accuracy by providing multiple intersecting lines.

• Independent variable: suspension point location
• Dependent variable: vertical plumb line position
• Control variables: lamina, plumb line, environment
• Repeat suspensions for accuracy

The centre of gravity is found by locating the intersection of the vertical lines drawn from different suspension points. If the lines do not intersect at a single point due to experimental errors, the approximate CG is taken near the region where the lines come closest.

To improve accuracy: Use a fine pencil and sharp ruler for marking lines. Suspend from at least three different points to get multiple lines and better intersection estimation. Ensure the lamina is free to swing and comes to complete rest before marking. Avoid air currents or vibrations that may cause movement. Use a long plumb line to reduce angular error in vertical line marking. Repeat the experiment and average the intersection points if possible.

Safety considerations include handling the lamina and strings carefully to avoid cuts or entanglement, and ensuring the stand is stable to prevent falling equipment.

• Find CG at intersection of vertical lines.
• Use multiple suspension points for better accuracy.
• Ensure lamina is at rest before marking.
• Avoid drafts and vibrations.
• Use fine marking tools.
• Handle apparatus safely.

The centre of gravity of an object is the single point at which the entire weight of the object can be considered to act. It is the average location of the weight distribution of the object.

Weight is a force measured in newtons (N) and is calculated by the equation W = mg, where m is the mass in kilograms (kg) and g is the acceleration due to gravity (9.8 m/s² near Earth's surface).

The centre of gravity depends on the shape and mass distribution of the object. For uniform objects, it is at the geometric centre; for non-uniform objects, it may be closer to the heavier parts.

• Centre of gravity is the balance point of an object.
• Weight acts vertically downwards through the centre of gravity.
• Measured in newtons (N).

An object is stable if it tends to return to its original position after being slightly tilted. Stability depends on the location of the centre of gravity relative to the base of support.

If the centre of gravity is low and lies vertically above the base area, the object is more stable because the weight acts within the base, creating a restoring moment when tilted.

If the centre of gravity is high or moves outside the base when tilted, the object is less stable and more likely to topple because the weight creates an overturning moment.

• Low centre of gravity increases stability.
• Centre of gravity must lie above the base for stability.
• High centre of gravity decreases stability and increases risk of toppling.

The base of support is the area beneath an object that includes all points of contact with the supporting surface.

A wider base increases stability because the centre of gravity can move further before it passes outside the base, preventing toppling.

For example, a pyramid shape with a wide base and low centre of gravity is very stable, whereas a tall, narrow object with a small base is unstable.

• Base of support is the contact area with the ground.
• Wider base increases stability.
• Narrow base decreases stability.

Vehicles are designed with a low centre of gravity and wide wheelbase to reduce the risk of overturning during turns.

Tall buildings have wide foundations and structural designs to keep the centre of gravity low and within the base to maintain stability against wind forces.

Sports like gymnastics use a low centre of gravity and wide stance to improve balance and stability.

• Cars have low centre of gravity and wide base for stability.
• Buildings use wide foundations to keep centre of gravity low.
• Athletes adjust posture to lower centre of gravity for balance.

To draw a stability diagram: first, sketch the object and mark its base of support on the surface.

Locate and mark the centre of gravity as a point inside the object.

Draw a vertical line (weight line) downwards from the centre of gravity to the base plane. If this line falls within the base, the object is stable; if outside, it is unstable and likely to topple.

• Draw object and base on horizontal line.
• Mark centre of gravity inside object.
• Draw vertical weight line from centre of gravity to base.
• Check if weight line lies within base for stability.