SPH3U Forces, Weight and Free-Body Diagrams: Notes and Practice

Key things to remember

1A force is a push or pull measured in newtons,

1 N = 1 kg \cdot m/s^{2}

, and it is always a vector.

F_{g} = m g

with

g = 9.81 m/s^{2} = 9.81 N/kg

. Mass must be in kilograms.

3Mass is matter in kilograms and does not change with location; weight is a force in newtons and does.

4Normal means perpendicular to the surface — and

F_{N}

is not automatically equal to the weight.

5Tension is always a pull along the rope, away from the object.

6A free-body diagram shows one isolated object and only the forces acting on it.

Kinematics described motion. Dynamics asks what causes it to change, and the answer is always a force — a push or a pull, measured in newtons (N), and always a vector. One newton is the force that gives a $1 kg$ mass an acceleration of $1 m/s^{2}$ , so a newton is really a shorthand for $kg \cdot m/s^{2}$ .

A force is an interaction, not a possession

An object does not 'have' a force. A force is always something else acting on it — the Earth pulling, the table pushing, a rope tugging. If you cannot name what is applying a force, it does not belong on your diagram. This is why there is no such thing as a 'force of motion' pushing a sliding puck along.

The forces you will actually meet

Fig 1Contact forces need something touching. Gravity does not.

Force	Symbol	Direction it acts	Applied by
Gravity (weight)	$F_{g}$	Straight down, toward Earth's centre	The Earth — no contact needed
Normal	$F_{N}$	Perpendicular to the surface, away from it	The surface being pressed on
Tension	$F_{T}$	Along the rope, away from the object	A rope, cable or string — always a pull
Friction	$F_{f}$	Along the surface, opposing relative sliding	The surface
Applied	$F_{a}$	Whichever way the push or pull points	A person, engine or another object
Air resistance	$F_{ai r}$	Opposite to the motion through the air	The air

Normal does not mean usual

In physics normal means perpendicular. The normal force is perpendicular to the surface, which on flat ground happens to be straight up — but on a ramp it points at right angles to the ramp, not vertically. It is also not automatically equal to the weight: press down on a book and the table pushes back harder.

Mass and weight are different quantities

Mass is the amount of matter in an object, measured in kilograms. It is a scalar and it does not change when you move. Weight is the force of gravity on that mass, measured in newtons. It is a vector, it points down, and it changes with where you are.

F_{g} = m g g = 9.81 m/s^{2} = 9.81 N/kg

Mass in kilograms gives a force in newtons. Never put grams into this equation.

Both forms of $g$ on your formula sheet say the same thing. As $9.81 m/s^{2}$ it is the acceleration of a falling object. As $9.81 N/kg$ it is the weight per kilogram — which is the more useful reading here, because it says directly that every kilogram is pulled with about $9.81 N$ .

AMass

The quantity of matter, a scalar
Measured in kilograms
Identical on Earth, the Moon and in orbit
Measured by comparison, on a balance

BWeight

The force of gravity on that mass, a vector
Measured in newtons
About one sixth as much on the Moon
Measured by a spring scale

Why astronauts float

Not because gravity has disappeared — in low orbit gravity is still around 90% of its surface strength. They float because they and the station are both in free fall around the Earth together, so there is no normal force pressing them against the floor. Weightlessness is the absence of a support force, not of gravity.

Free-body diagrams

A free-body diagram (FBD) shows one object, isolated, with every force acting on it drawn as an arrow from its centre. It is the single most useful habit in this unit, because once it is drawn the problem usually solves itself.

Fig 2Four situations, four diagrams. Only forces acting on the object are drawn.

Draw the object as a dot or a box. Nothing else — no ramp, no rope, no ground.
Draw only forces acting on that object. Forces the object exerts on other things belong on their diagrams, not this one.
Start every arrow at the object and point it the way the force acts.
Make arrow lengths roughly proportional. If the object is not accelerating vertically, the up and down arrows should look equal.
Label each arrow with its symbol — $F_{g}$ , $F_{N}$ , $F_{T}$ , $F_{f}$ , $F_{a}$ .
Mark the positive direction beside the diagram, so every sign that follows is unambiguous.

Three arrows that should never appear

A 'force of motion' in the direction something is sliding — motion is not a force. A centrifugal force flinging something outward — that is not in this course and is not a real force in this frame. And the object's own weight pushing down on the table — that force acts on the table, so it belongs on the table's diagram, not the object's.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

A student has a mass of

58 kg

. Find their weight on Earth, and state their mass and weight on the Moon where

g = 1.62 m/s^{2}

Example 2

12 kg

box sits at rest on a level floor. A person pushes horizontally with

40 N

but the box does not move. Draw the free-body diagram and state every force with its size.

Example 3

3.0 kg

lamp hangs motionless from a single vertical cord. Find the tension in the cord.

Where marks get lost

Using grams in

F_{g} = m g

. The equation needs kilograms, or the answer is out by a factor of 1000.

Calling mass 'weight' and quoting it in newtons, or quoting a weight in kilograms. Kilograms are mass; newtons are force.

Drawing a 'force of motion' arrow in the direction something is travelling. Motion is not a force and nothing is applying it.

Assuming the normal force always equals the weight. It does on level ground with nothing else pushing vertically — and not otherwise.

Putting the object's push on the table onto the object's own free-body diagram. That force acts on the table.

Drawing the ramp, the rope and the floor in the diagram. A free-body diagram shows the object alone.

Forgetting that the normal force is perpendicular to the surface, which is not vertical on a slope.

Key things to remember

Worked examples

Where marks get lost

Flashcards 9

Practice