Forces

Newton's first law. An object will remain at rest or move at constant velocity in a straight line UNLESS a net (unbalanced) external force acts on it.

Newton's second law. $F = ma.$

• $F$: net force ($\text{N}$).
• $m$: mass ($\text{kg}$).
• $a$: acceleration ($\text{m/s}^{2}$).

A net force CAUSES acceleration. No net force = no acceleration (constant velocity).

Newton's third law. When body A exerts a force on body B, body B exerts an equal and opposite force on body A.

• Action and reaction are EQUAL in magnitude, OPPOSITE in direction, on DIFFERENT bodies.

Worked. A $5\,\text{kg}$ box pushed with $20\,\text{N}$, friction is $5\,\text{N}$.

• Net force: $20 - 5 = 15\,\text{N}$.
• Acceleration: $a = 15/5 = 3\,\text{m/s}^{2}$.

Worked (Newton 3). When you push a wall with $50\,\text{N}$, the wall pushes back on you with $50\,\text{N}$.

Friction. Force that opposes relative motion between two surfaces in contact. Caused by surface roughness and intermolecular bonds.

Static friction: must be overcome to start sliding. Kinetic (sliding) friction: opposes ongoing motion. Usually less than static.

On a $v$-$t$ graph for a falling object: air resistance is just friction with air. Same idea: opposes motion, increases with speed → terminal velocity.

Reducing friction.

• Lubricants (oil, grease).
• Smooth surfaces.
• Bearings (rolling > sliding).

Useful friction.

• Walking (without it you slip).
• Tyres on road (grip for braking and turning).

Hooke's law. Up to the limit of proportionality: $F = kx,$ where $F$ is the applied force, $k$ is the spring constant ($\text{N/m}$), and $x$ is the extension.

Force-extension graph.

• Linear region (start): $F = kx$, gradient = $k$.
• Limit of proportionality: the point where the graph stops being a straight line — i.e. where extension stops being proportional to force and the spring stops obeying Hooke's law.
• Elastic limit: a point slightly beyond the limit of proportionality, beyond which the spring no longer returns to its original length when the load is removed (permanent deformation). On 0625 you should know the term but a detailed understanding is not required.
• Then beyond, the spring eventually breaks.

Limit of proportionality vs elastic limit. These are NOT the same point. The limit of proportionality is the one a load-extension graph asks you to identify — it is where the line stops being straight. The elastic limit is where deformation becomes permanent.

Worked. A spring extends by $4\,\text{cm}$ when a $2\,\text{N}$ force is applied.

• $k = F/x = 2/0.04 = 50\,\text{N/m}$.
• For $5\,\text{N}$: $x = 5/50 = 0.1\,\text{m} = 10\,\text{cm}$.

Tip. Always read EXTENSION (change in length), not total length, off the graph.

Moment of a force. $\text{moment} = F \times d,$ where $F$ is the force ($\text{N}$) and $d$ is the perpendicular distance from the pivot ($\text{m}$). Units: $\text{N m}$.

Principle of moments. When an object is in equilibrium (not rotating): $\text{sum of clockwise moments} = \text{sum of anticlockwise moments}.$

Worked. A $30\,\text{N}$ force acts $0.5\,\text{m}$ from a pivot, clockwise. What force at $0.3\,\text{m}$ on the other side balances it?

• Clockwise moment: $30 \times 0.5 = 15\,\text{N m}$.
• For balance: $F \times 0.3 = 15 \Rightarrow F = 50\,\text{N}$.

Centre of gravity. The single point where the weight of an object can be considered to act. For a uniform ruler, it's at the midpoint.

Tip. Always measure $d$ PERPENDICULAR to the force. If the force is at an angle, use $d_{\perp} = d \sin\theta$.

Centre of gravity. The centre of gravity is the single point at which the entire weight of an object can be considered to act. For a uniform, symmetrical object (a ruler, a sphere, a rectangular sheet) it is at the geometric centre.

Experiment — centre of gravity of an irregular lamina. To find the centre of gravity of an irregularly shaped plane lamina (a flat sheet, e.g. stiff card):

1. Make three small holes near the edge of the lamina, well spaced apart.
2. Hang the lamina freely from a pin through one hole, so it can swing freely, and let it settle.
3. Hang a plumb line (a small mass on a thread) from the same pin.
4. When everything is at rest, the centre of gravity lies directly below the point of suspension. Mark the line of the plumb line on the card.
5. Repeat for the other two holes.
6. The centre of gravity is where the three lines cross. The third line is a check.

Stability. An object topples when the vertical line through its centre of gravity falls outside its base. While that line stays inside the base, the object's weight provides a moment that returns it upright.

Two things make an object more stable:

• a lower centre of gravity;
• a wider base.

A racing car (low, wide) is very stable; a tall, narrow object with a high centre of gravity (e.g. a loaded double-decker bus) topples with only a small tilt.

Circular motion. An object moving in a circle at constant SPEED is constantly changing DIRECTION → constantly changing velocity → ACCELERATING.

The acceleration (and net force) point toward the centre of the circle. This force is the centripetal force.

No "centrifugal force". The outward feeling is your inertia trying to keep you in a straight line. The car door pushes you inward — that's the centripetal force.

What provides the centripetal force?

• Car turning: friction between tyres and road.
• Ball on string: tension in the string.
• Satellite: gravity.

Worked qualitative. A ball on a string swung in a circle. Cut the string and the ball flies off in a STRAIGHT LINE (tangent to the circle at the moment of cut), not radially outward.