Statics of rigid bodies

Welcome to the World of Statics!

In this chapter, we are going to explore Statics of Rigid Bodies. If "Dynamics" is the study of things in motion, "Statics" is the study of things that are perfectly still. Think about a massive bridge, a ladder leaning against a wall, or a hanging sign outside a shop. Why don't they fall or collapse? It’s all about balance!

By the end of this unit, you’ll be able to calculate exactly where an object's "balance point" is and predict how it will behave when placed on a slope or hung from a string. Don't worry if it sounds heavy—we’ll break it down piece by piece!

1. The Centre of Mass (CoM)

Every object, no matter how weirdly shaped, has a single point where we can imagine all its weight is concentrated. This is the Centre of Mass. In Mechanics, we treat the weight of the whole object as a single force acting downwards from this specific point.

Uniform Rigid Bodies & Symmetry

A uniform body has its mass spread evenly. If a shape is uniform and has symmetry, the CoM is always on the axis of symmetry.
Example: For a uniform rectangular door, the CoM is exactly in the geometric center where the diagonals cross. For a uniform circular plate, it's the center of the circle.

Using Integration to Find CoM

Sometimes shapes aren't simple rectangles. For a uniform lamina (a flat 2D shape) bounded by a curve \(y = f(x)\), the x-axis, and the lines \(x=a\) and \(x=b\), we use integration to find the coordinates \((\bar{x}, \bar{y})\):

The x-coordinate: \(\bar{x} = \frac{\int_{a}^{b} x y \, dx}{\int_{a}^{b} y \, dx}\)

The y-coordinate: \(\bar{y} = \frac{\int_{a}^{b} \frac{1}{2} y^2 \, dx}{\int_{a}^{b} y \, dx}\)

Quick Tip: The denominator \(\int y \, dx\) is just the Area of the shape. If you already know the area, you’ve saved yourself half the work!

Composite Bodies

A composite body is just a big shape made of smaller, simpler shapes (like a "T" shape made of two rectangles). To find the CoM of the whole thing, we use a Moments Table.
1. Break the object into simple parts.
2. Find the mass (or area, if uniform) and the CoM of each part.
3. Choose a reference point (usually the origin \( (0,0) \)).
4. Use the formula: \(\bar{x} = \frac{\sum m_i x_i}{\sum m_i}\)

Step-by-Step for Composite Bodies:
1. Table it: Create columns for "Part", "Mass/Area", "x-position", and "Mass \(\times\) x".
2. Sum it: Add up the "Mass" column and the "Mass \(\times\) x" column.
3. Divide: Divide the total "Mass \(\times\) x" by the total "Mass".
4. Repeat: Do the same for the y-direction if needed.

Summary: The Centre of Mass is the "average" position of the mass. Use symmetry where possible, integration for curves, and the table method for combined shapes.

2. Equilibrium of Rigid Bodies

A rigid body is in equilibrium when it is at rest and has no tendency to start moving or spinning. For this to happen, two things must be true:

1. Resultant Force is Zero: The total force up equals the total force down, and left equals right.
2. Resultant Moment is Zero: The total clockwise moments equal the total anti-clockwise moments about any point.

Suspension from a Fixed Point

When you hang an object freely from a pivot (like a picture on a nail), it will swing until it settles. In equilibrium, the Centre of Mass must lie vertically below the point of suspension.

Analogy: Imagine holding a ruler loosely between your fingers at one end. It will always hang straight down so its middle is directly under your fingers.

Common Exam Trick: They will ask for the angle a side makes with the vertical. Simply draw a vertical line from the pivot through the CoM and use Trigonometry (usually \( \tan \theta \)) in the resulting right-angled triangle.

Rigid Bodies on Planes

When a body is placed on a horizontal or inclined plane, we look at three main forces:
- Weight (W): Acts vertically down from the CoM.
- Normal Reaction (R): Acts perpendicular to the surface.
- Friction (F): Acts parallel to the surface, opposing motion.

Inclined Planes

If a body is on a slope of angle \(\alpha\):
- Resolve weight into components: \(W \sin \alpha\) (down the slope) and \(W \cos \alpha\) (into the slope).
- At the point of sliding, \(F = \mu R\), where \(\mu\) is the coefficient of friction.

Did you know? A rigid body will topple if the vertical line through its Centre of Mass falls outside its base of support. Think of a tall bus going around a sharp, slanted corner—if the CoM shifts too far, it tips over!

Summary: Equilibrium means everything balances out. If it's hanging, the CoM is under the pivot. If it's on a slope, resolve your forces and don't forget friction!

3. Solving Statics Problems: A Survival Guide

Statics can feel overwhelming because there are so many forces to keep track of. Follow these steps to stay organized:

1. Draw a LARGE Diagram: Don't be shy with the paper! Mark the CoM, the pivot, and every force acting on the body.
2. Pick a "Smart" Point for Moments: You can take moments about any point, but choosing a point where an unknown force acts (like a hinge or a contact point) makes that force disappear from your equation because its distance is zero!
3. Resolve Forces: Pick two perpendicular directions (usually horizontal/vertical or parallel/perpendicular to a slope) and set the forces equal.
4. Check your units: Ensure mass is in kg and forces are in Newtons (\(Weight = mg\)). Use \(g = 9.8 \, ms^{-2}\) unless told otherwise.

Common Mistake to Avoid: Forgetting that the Normal Reaction (\(R\)) doesn't always act at the center of the base if the object is about to tilt. However, for most simple equilibrium problems in M3, you can assume the body is rigid and the forces act at the specific contact points described.

Key Takeaway: Statics is a puzzle where all the pieces must sum to zero. Clear diagrams and clever point selection for moments are your best friends!