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.

Examples in context

Example 1. A Gladstone port crane lifts a $4000 \text{ kg}$ container at constant velocity. The free-body diagram shows weight $mg = 39200 \text{ N}$ downward balanced by cable tension $T = 39200 \text{ N}$ upward (Newton's first law). Accelerating the same container upward at $1.2 \text{ m s}^{-2}$ requires $T - mg = ma$, so $T = m(g+a) = 44000 \text{ N}$ - an extra $4800 \text{ N}$ from Newton's second law that is the design basis for the crane's safe working load margin.

Example 2. A Cairns light-rail tram traverses a $5^\circ$ incline approaching a stop. The tram (mass $45000 \text{ kg}$) experiences a component of weight along the incline $mg\sin\theta = 38500 \text{ N}$ plus rolling resistance $1800 \text{ N}$. The traction motor must apply $40300 \text{ N}$ to hold the tram stationary, by Newton's third law equally pushing the rail backward. QCAA Unit 2 EA Paper 2 incline-stem decomposition follows this template.

Try this

Q1. State Newton's third law and give one example. [2 marks]

• Cue. Forces on two interacting bodies are equal in magnitude and opposite in direction; e.g. rocket pushes gas down, gas pushes rocket up.

Q2. A $3.0 \text{ kg}$ block on a horizontal surface ($\mu_k = 0.20$) is pulled with $15 \text{ N}$ horizontal. Calculate the friction force, the net force and the acceleration. [4 marks]

• Cue. $f_k = \mu_k mg = 5.88 \text{ N}$; $F_{net} = 9.12 \text{ N}$; $a = 3.04 \text{ m s}^{-2}$.

Q3. A $1500 \text{ kg}$ car ascends a $6^\circ$ incline at constant velocity. (a) Draw the free-body diagram. (b) Calculate the driving force and the normal reaction. (c) If the car then accelerates at $1.5 \text{ m s}^{-2}$ up the slope, determine the new driving force. [3+3+2 marks; ISMG: Knowledge and conceptual understanding, Analysis and interpretation]

• Cue. (a) weight, normal, drive, opposed by gravity-component (and friction if specified); (b) $F = mg\sin 6 = 1538 \text{ N}$, $N = mg\cos 6 = 14620 \text{ N}$; (c) $F = 1538 + 2250 = 3788 \text{ N}$.