Centre of Gravity and Rigid-Body Equilibrium: Study Notes for PHY1020

Centre of Gravity and Rigid-Body Equilibrium

Introduction

This study guide covers the fundamental concepts of centre of gravity, centre of mass, and rigid-body equilibrium, as outlined in a typical college-level physics course. These topics are essential for understanding how forces and torques affect the stability and motion of objects.

Centre of Gravity

Mass vs Weight

• Mass (kg): The quantity of matter in an object. It is a measure of an object's resistance to acceleration and does not vary with location. Mass is measured using a balance scale.

• Weight (N): The force exerted on an object due to gravity. It is location-dependent and measured with a spring scale. The relationship is given by: $W = mg$ where $W$ is weight, $m$ is mass, and $g$ is the acceleration due to gravity.

Centre of Mass

The centre of mass is the point at which the mass of a system or body can be considered to be concentrated for the purposes of analyzing translational motion.

• Definition: For a system of particles, the position vector of the centre of mass from the origin (O) is: $\vec{r}_{cm} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$

• Properties:

  • The centre of mass moves like a particle when a force is applied.

  • The system may also experience torque about the centre of mass.

Centre of Gravity

The centre of gravity is the point at which the total gravitational torque on a body is zero. For most practical purposes near Earth's surface, it coincides with the centre of mass.

• Definition: The position vector for the centre of gravity is: $\vec{r}_{cg} = \frac{\sum m_i g_i \vec{r}_i}{\sum m_i g_i}$ For uniform gravity ($g_i = g$), this simplifies to: $\vec{r}_{cg} = \vec{r}_{cm}$

• Gravity Variation: On Earth's surface, $g \approx 9.81 \pm 0.02\ \mathrm{m\ s^{-2}}$.

Assumptions in Modelling

• Use the simplest model that will work, but no simpler.

• Common Assumptions:

  • Gravity ($g$) is constant.

  • Material is homogeneous.

  • Object is rigid.

  • All particles are equal in mass.

  • Spherical cow in a vacuum (idealization for simplicity).

• Whether an assumption is good depends on the situation.

Finding the Centre of Gravity

• Homogeneous, Simple Shape: Use geometric rules (e.g., centre of a rectangle).

• Small Collection of Particles: Add up positions weighted by mass and divide by total mass.

• Complex Shape: Decompose into geometric shapes, treat each as a particle, or use experimental methods (e.g., dangle from a point).

Support and Stability

• More than one support creates an area of support.

• If the centre of gravity is above the area of support, the object is stable; otherwise, it may tip due to net torque.

• Lower centre of gravity and larger support area increase stability.

Rigid Body Equilibrium

Defining a Rigid Body

• A rigid body does not deform under applied forces.

• Forces between particles are balanced and instantly propagated throughout the body.

• This is an idealization, but often a useful one in physics.

Conditions for Equilibrium

• Translational Equilibrium: The sum of all external forces is zero. $\sum \vec{F} = 0$

• Rotational Equilibrium: The sum of all external torques is zero. $\sum \vec{\tau} = 0$

• In two dimensions: $\sum F_x = \sum F_y = \sum \tau_z = 0$

Free-Body Diagrams

• Forces can be combined from any position, but are often brought together at the centre of gravity.

• Torques are always relative to a chosen point (origin). Different origins yield different torques.

• It is essential to pick one point and stick with it for consistency in calculations.

Selecting Origin for Rotations

• Choose an origin that simplifies calculations. Good options include:

  • Centre of gravity ($\vec{r}_{cg}$)

  • A fulcrum (fixed point)

  • Point where force acts through the origin (no torque produced)

Balancing the Equations

• Set up simultaneous equations for all forces and torques: $\sum F_x = \sum F_y = \sum \tau_z = 0$

• For multiple unknowns, express one variable in terms of others and substitute into the remaining equations.

• Sometimes, the solution yields limits (e.g., maximum allowable weight) rather than a specific value.

Example: Ladder Problem

• Scenario: Lancelot (800 N) climbs a 5 m ladder (180 N) at an angle of 53.1°.

• Tasks:

  • Find the normal and friction forces.

  • Determine the coefficient of friction.

  • Calculate the contact force vector.

• Application: Draw a free-body diagram, resolve forces and torques, and apply equilibrium conditions to solve for unknowns.

Limitations of Equilibrium Analysis

• Need as many independent equations as unknowns.

• Trivial equations (e.g., no force in a direction) provide no new information.

• Analysis assumes the body is rigid and in equilibrium; if not, the results do not apply.

References

• Young, H. D., Freedman, R. A., & Ford, L. A. (2020). University Physics with Modern Physics.

• Ellingboe, B. 2023, PS105 2023.

• Faller & Cook, 2024, Encyclopaedia Britannica: Gravity: Force physics and theory. [online]