VCE Physics

Force as a vector

A force is a push or a pull on an object, measured in newtons (N), and it has both size and direction, so it is a vector. Forces are drawn as arrows on a free-body diagram, which shows one object alone with every force acting on it and nothing else. Each arrow starts on the object and points the way the force acts. A force is always exerted by something, so if you cannot name what is exerting an arrow, it should not be there. In particular, there is no "force of motion" in the direction an object is travelling: velocity needs no force to keep going, only to change.

The forces met most often are listed below.

Mass is the amount of matter and is measured in kilograms and does not change with place. Weight is the gravitational force on the mass, so it does change: an object of mass $60\,\mathrm{kg}$ weighs about $590\,\mathrm{N}$ on Earth and about one sixth of that on the Moon.

Newton's three laws

First law

An object keeps its velocity, which includes staying at rest, unless a net force acts on it. So if the net force is zero the acceleration is zero: the object is at rest or moves in a straight line at constant speed. The law explains why a passenger is thrown forward when a bus brakes: the bus is slowed by the brakes but the passenger, with no net force on them, tends to keep the old velocity until a seat belt or seat applies one.

Second law

The acceleration of an object is proportional to the net force on it and inversely proportional to its mass, and is in the direction of the net force.

$F_{net}$ is the vector sum of all the forces on the object, in newtons, $m$ is its mass in kilograms and $a$ is its acceleration in $\mathrm{m\,s^{-2}}$. One newton is the net force that gives a $1\,\mathrm{kg}$ mass an acceleration of $1\,\mathrm{m\,s^{-2}}$. A more general form is that net force equals the rate of change of momentum, which for constant mass is the same thing.

Worked example: a car of mass $1200\,\mathrm{kg}$ is driven forward by a force of $3000\,\mathrm{N}$ from the road on its tyres while a total resistance of $600\,\mathrm{N}$ acts backwards. The net force is $3000 - 600 = 2400\,\mathrm{N}$ forward, so

Third law

When one object exerts a force on a second, the second exerts a force on the first that is equal in size and opposite in direction. The two forces are the same kind of force, act on different objects and so never cancel each other on a free-body diagram. A book on a table is pulled down by Earth's gravity and pushed up by the table. Those two are not a third-law pair, because they act on the same object. The pair of the book's weight is the gravitational pull of the book on Earth, and the pair of the table's push upwards is the book's push downwards on the table.

The third law is why a rocket accelerates: it pushes exhaust gas backwards, and the gas pushes the rocket forwards with an equal force. It is also why two colliding objects exert equal and opposite forces on each other at every instant, whatever their masses, even though their accelerations differ.

Forces along a straight line

When all forces act along one line, choose a positive direction and add the forces with signs. A lift of mass $m$ carrying a person on a scale shows how this works, because the scale reads the normal force $N$ on the person. With upward positive and upward acceleration $a$,

so $N = m(g + a)$. For a person of mass $60\,\mathrm{kg}$ in a lift accelerating upwards at $1.5\,\mathrm{m\,s^{-2}}$, $N = 60\times(9.8+1.5) = 678\,\mathrm{N}$, which is more than their weight of $588\,\mathrm{N}$. If the lift accelerates downwards at the same rate, $N = 60\times(9.8-1.5) = 498\,\mathrm{N}$. The person feels heavier or lighter because the normal force, which is what is felt, changes while the weight does not. A lift moving at constant velocity, up or down, gives $N = mg$.

Connected objects

Two objects joined by a light string that does not stretch share the same acceleration. Apply the second law to each object, or to the whole system when the string tension is an internal force and is not wanted. A truck of mass $2000\,\mathrm{kg}$ pulls a trailer of mass $500\,\mathrm{kg}$ with net forward force on the whole pair of $3000\,\mathrm{N}$. The acceleration is $3000/2500 = 1.2\,\mathrm{m\,s^{-2}}$. For the trailer alone, the tension in the coupling is the net force on it, $T = 500\times1.2 = 600\,\mathrm{N}$.

Forces in two dimensions

Forces that are not along one line are dealt with by resolving them into perpendicular components, most often horizontal and vertical or, on a slope, along and perpendicular to the slope. A force $F$ at an angle $\theta$ to the horizontal has components

The second law is then applied to each direction separately: $F_{net,x} = ma_x$ and $F_{net,y} = ma_y$. An object in equilibrium has zero net force, so the components in each direction sum to zero separately.

Worked example: a sign of mass $20\,\mathrm{kg}$ hangs from two cables, each at $40^\circ$ above the horizontal and each under the same tension $T$. Horizontally the two horizontal components cancel. Vertically the two upward components carry the weight, $20\times9.8 = 196\,\mathrm{N}$:

Each cable carries more than half the weight, and the flatter the cables, the larger the tension needed.

Inclined planes

On a slope at angle $\theta$ to the horizontal, the weight $mg$ is resolved into a component $mg\sin\theta$ down the slope and $mg\cos\theta$ into the slope. The normal force balances the second, so $N = mg\cos\theta$, and the component down the slope is what drives the motion.

Worked example: a box of mass $5.0\,\mathrm{kg}$ slides down a slope at $30^\circ$, with a friction force of $12\,\mathrm{N}$ up the slope. The component of weight down the slope is $5.0\times9.8\times\sin30^\circ = 24.5\,\mathrm{N}$. Taking down the slope as positive,

The normal force is $5.0\times9.8\times\cos30^\circ = 42\,\mathrm{N}$. With no friction the acceleration would be $g\sin\theta = 4.9\,\mathrm{m\,s^{-2}}$, independent of mass.

A force applied at an angle

Pulling a suitcase with a handle at an angle above the horizontal gives a horizontal component that accelerates it and a vertical component that lifts partly against its weight, so the normal force is reduced and so is the friction it produces. Pushing down at an angle has the opposite effect, raising the normal force and the friction.

Friction

Friction is a contact force parallel to the surfaces that resists sliding. Static friction acts when there is no sliding and matches whatever force is trying to start motion, up to a maximum. Once the object slides, the kinetic friction is roughly constant and usually a little smaller than the maximum static value, which is why a stuck object suddenly lurches. Friction between a driven tyre and the road is what pushes a car forward, so it acts in the direction of travel on the driving wheels and is a necessary force for starting, braking and turning.

Air resistance and terminal velocity

A falling object meets air resistance that grows with its speed and acts upwards. The net force $mg - F_{drag}$ therefore shrinks as the object speeds up, so the acceleration falls. When the drag equals the weight the net force is zero and the object falls at constant speed, its terminal velocity. A graph of speed against time starts with gradient $g$ and flattens as it approaches a horizontal line.