Recall, describe and apply Newton's three laws of motion, including the use of free-body diagrams to identify forces acting on an object and solve problems involving weight, normal force, friction and tension

What this dot point is asking

QCAA expects you to state and apply Newton's three laws of motion, and to construct free-body diagrams to analyse the forces on a single body. The standard problem types are level-surface motion with friction, motion on an inclined plane, connected bodies (pulleys and trains of carts), and tension in a hanging string or cable.

Newton's three laws

First law (inertia). An object continues in a state of rest or uniform motion in a straight line unless a net external force acts on it. A net force of zero means constant velocity (which includes being at rest).

Second law. The net force on an object equals its mass times acceleration:

Force has SI unit newton (N): $1$ N is the force that accelerates a $1$ kg mass at $1$ m s$^{-2}$. Force is a vector. Add forces vectorially.

Third law. For every action there is an equal and opposite reaction. If body A exerts a force on body B, then body B exerts a force of equal magnitude and opposite direction on body A. The forces act on different bodies and never on the same body, which is why a third-law pair does not "cancel out" the way two opposing forces on one body do.

Free-body diagrams

Isolate one object and draw every external force acting on it as an arrow from the object's centre. Standard forces:

• Weight ($W = mg$). Straight down. Always present unless explicitly in deep space.
• Normal force ($N$). Perpendicular to the contact surface, pointing away from the surface.
• Friction ($f$). Parallel to the surface, opposing relative motion or relative motion tendency. Kinetic: $f_k = \mu_k N$. Static: $f_s \le \mu_s N$.
• Tension ($T$). Along a string or cable, pulling away from the body.
• Applied force. As stated.

Sum forces vectorially in two perpendicular directions and apply $F = ma$ in each.

Inclined planes

On a frictionless ramp at angle $\theta$:

• Weight component along the slope (down the slope): $W_\parallel = mg \sin\theta$.
• Weight component perpendicular to slope: $W_\perp = mg \cos\theta$.
• Normal force balances $W_\perp$: $N = mg \cos\theta$.
• If friction acts up the slope (object sliding down): $F_{\text{net}} = mg \sin\theta - \mu_k mg \cos\theta$.

Choose the $x$-axis along the slope and the $y$-axis perpendicular. This eliminates the need to resolve the normal force or friction.

Connected bodies

For two masses joined by a light, inextensible string over a frictionless pulley, both masses have the same magnitude of acceleration and the tension is the same throughout the string. Write $F = ma$ for each mass, then solve simultaneously.

Worked example

A $2.0$ kg block sits on a frictionless ramp inclined at $30°$. Find the block's acceleration down the ramp. Use $g = 9.8$ m s$^{-2}$.

$F_\parallel = mg \sin\theta = (2.0)(9.8)\sin 30° = 9.8$ N down the slope.

$a = F_\parallel / m = 9.8 / 2.0 = 4.9$ m s$^{-2}$ down the slope.

(Equivalent: $a = g \sin\theta = 9.8 \sin 30° = 4.9$ m s$^{-2}$.)

Common traps

Confusing action-reaction with balanced forces. A book on a table has weight (down) and normal force (up). These are not a third-law pair because both act on the book. The third-law pair to the book's weight is the book's gravitational pull on the Earth.

Forgetting to resolve weight on an incline. Weight does not act along the slope unless the slope is vertical.

Treating static friction as $\mu_s N$ at all times. Static friction is only equal to $\mu_s N$ at the threshold of slipping. Before slipping, it equals whatever force it needs to to keep the object stationary, up to $\mu_s N$.

Using $F = ma$ without summing all forces. Always tally the net force first.

In one sentence

Newton's first law says zero net force gives constant velocity, the second law $F_{\text{net}} = ma$ relates net force to acceleration, and the third law pairs equal and opposite forces on different bodies; a free-body diagram showing weight, normal force, friction and tension lets you sum forces and solve for the acceleration of a single body.