Patterns of Motion and Equilibrium: Foundations of Classical Mechanics

Patterns of Motion and Equilibrium

Aristotle on Motion

Early understandings of motion were shaped by Aristotle, who classified motion into two types: natural motion (straight up or down, such as falling objects) and violent motion (motion caused by external forces, such as pushing or pulling).

• Natural motion: Objects move toward their natural place (e.g., stones fall, smoke rises).

• Violent motion: Requires an external agent to maintain movement.

Galileo’s Concept of Inertia

Galileo challenged Aristotle’s views, introducing the concept of inertia. He demonstrated that, in the absence of friction, objects in motion remain in motion unless acted upon by an external force.

• Inertia: The tendency of an object to resist changes in its state of motion.

• Galileo’s experiments with inclined planes showed that objects roll back to their original height, and on a frictionless surface, would continue moving indefinitely.

Key Point: Inertia is a property of matter, not a cause of motion.

Mass—A Measure of Inertia

Mass quantifies the amount of matter in an object and is a direct measure of its inertia. The greater the mass, the greater the inertia and the more force required to change its motion.

• Mass: Quantity of matter; measured in kilograms (kg).

• Weight: The gravitational force acting on an object; varies with location (e.g., less on the Moon).

• Volume: The amount of space an object occupies.

• Density: Mass per unit volume. $\text{Density} = \frac{\text{mass}}{\text{volume}}$

Example: On Earth, 1 kg weighs about 10 N; on the Moon, it weighs less, but its mass remains unchanged.

Net Force

A force is a push or pull. The net force is the vector sum of all forces acting on an object. It determines the object's acceleration and change in motion.

• Net force: $\sum \vec{F}$

• If net force is zero, the object’s motion does not change (equilibrium).

The Equilibrium Rule

The equilibrium rule states that an object is in equilibrium when the vector sum of all forces acting on it is zero:

• $\sum \vec{F} = 0$

• Applies to both objects at rest (static equilibrium) and objects moving at constant velocity (dynamic equilibrium).

Dynamic Equilibrium

An object in dynamic equilibrium moves at constant velocity because all forces acting on it cancel out. Both rest and constant velocity are forms of equilibrium.

Support Force (Normal Force)

The support force (or normal force) is the upward force that balances the weight of an object on a surface. When standing on two scales, each scale supports half your weight if distributed evenly.

The Force of Friction

Friction is a resistive force that opposes motion between two surfaces or through a fluid. It depends on the nature of the surfaces and how hard they are pressed together.

• Acts opposite to the direction of motion.

• Exists for objects at rest and in motion.

Speed and Velocity

Speed is the distance traveled per unit time. Velocity is speed in a given direction. Average speed is total distance divided by total time, while instantaneous speed is the speed at a specific moment.

• Speed: $\text{Speed} = \frac{\text{distance}}{\text{time}}$

• Velocity: $\vec{v} = \frac{\Delta \vec{x}}{\Delta t}$

Motion is Relative

All motion is measured relative to a chosen reference frame. For example, a car’s speed is measured relative to the ground, and the Earth moves relative to the Sun.

Acceleration

Acceleration is the rate at which velocity changes with time. It can involve changes in speed, direction, or both. Galileo’s experiments with inclined planes led to the concept of acceleration.

• Acceleration: $a = \frac{\Delta v}{\Delta t}$

• Units: meters per second squared (m/s2)

Free Fall

When gravity is the only force acting on an object, it is in free fall. The acceleration due to gravity near Earth’s surface is approximately $9.8\ \text{m/s}^2$ (often rounded to $10\ \text{m/s}^2$ for estimation).

• Velocity increases by about $10\ \text{m/s}$ each second during free fall.

• Distance fallen increases with the square of the time: $d = \frac{1}{2}gt^2$

Summary Table: Free-Fall Motion

Time (s)	Velocity (m/s)	Distance Fallen (m)
0	0	0
1	10	5
2	20	20
3	30	45
4	40	80
5	50	125

Additional info: Table values inferred from $v = gt$ and $d = \frac{1}{2}gt^2$ with $g = 10\ \text{m/s}^2$.