4.5.1.1Scalar and vector quantities
Scalar quantities have magnitude only. Vector quantities have magnitude and an associated direction.
A vector quantity may be represented by an arrow. The length of the arrow represents the magnitude, and the direction of the arrow the direction of the vector quantity.
Both parts of that convention are marked: an arrow drawn twice as long must represent twice the force, and the arrow must point the way the force acts.
- Scalar quantity
- a quantity that has magnitude only, such as distance, speed or mass
- Vector quantity
- a quantity that has magnitude and an associated direction, such as displacement, velocity or force; it may be represented by an arrow whose length shows the magnitude
4.5.1.2Contact and non-contact forces
A force is a push or pull that acts on an object due to the interaction with another object. All forces between objects are either:
• contact forces — the objects are physically touching • non-contact forces — the objects are physically separated.
Examples of contact forces include friction, air resistance, tension and normal contact force. Examples of non-contact forces are gravitational force, electrostatic force and magnetic force.
Force is a vector quantity.
You should be able to describe the interaction between pairs of objects which produce a force on each object, with the forces represented as vectors.
The key idea is that forces come in pairs, and the pair acts on two different objects. A book resting on a table: the book pushes down on the table, and the table pushes up on the book with an equal force in the opposite direction. Drawn as vectors, the two arrows are the same length and point opposite ways — but each is drawn on a different object.
Common mistake
Weight is a non-contact force, even for an object sitting on the ground. The Earth pulls the object without touching it; the contact force from the ground pushing up is a separate, different force.
- Force
- a push or pull that acts on an object due to the interaction with another object
- Contact force
- a force acting where the objects are physically touching, such as friction, air resistance, tension and normal contact force
- Non-contact force
- a force acting where the objects are physically separated, such as gravitational, electrostatic and magnetic force
4.5.1.3Gravity
Weight is the force acting on an object due to gravity. The force of gravity close to the Earth is due to the gravitational field around the Earth.
The weight of an object depends on the gravitational field strength at the point where the object is.
The weight of an object can be calculated using the equation:
weight = mass × gravitational field strength W = m g
weight, W, in newtons, N; mass, m, in kilograms, kg; gravitational field strength, g, in newtons per kilogram, N/kg. In any calculation the value of g will be given.
You should be able to recall and apply this equation.
The weight of an object may be considered to act at a single point, referred to as the object's centre of mass.
The weight of an object and the mass of an object are directly proportional. You should recognise and be able to use the symbol for proportionality, ∝, so this can be written W ∝ m.
Weight is measured using a calibrated spring-balance, a newtonmeter.
Worked example
An astronaut has a mass of 80 kg. Calculate their weight on Earth (g = 9.8 N/kg) and on the Moon (g = 1.6 N/kg).
- Write the equation: W = m g.
- On Earth: W = 80 × 9.8 = 784 N.
- On the Moon: W = 80 × 1.6 = 128 N.
- Note what has changed and what has not: the mass is 80 kg in both places.
- Only the gravitational field strength differs, so only the weight differs.
Answer: 784 N on Earth and 128 N on the Moon. The mass is unchanged at 80 kg — mass is a scalar property of the object, while weight is a force that depends on where the object is.
Common mistake
Mass is in kilograms and weight is in newtons. Saying an object "weighs 80 kg" is wrong in physics — 80 kg is its mass, and its weight is the force in newtons that gravity exerts on it.
- Weight
- the force acting on an object due to gravity, calculated as mass × gravitational field strength
4.5.1.4Resultant forces
A number of forces acting on an object may be replaced by a single force that has the same effect as all the original forces acting together. This single force is called the resultant force.
You should be able to calculate the resultant of two forces that act in a straight line. Forces in the same direction add; forces in opposite directions subtract, and the resultant acts in the direction of the larger force.
Worked example
A car experiences a driving force of 3200 N forwards and a total resistive force of 1900 N backwards. Calculate the resultant force.
- The two forces act in a straight line but in opposite directions, so subtract.
- 3200 − 1900 = 1300.
- The larger force is forwards, so the resultant acts forwards.
- State both the size and the direction, because force is a vector.
Answer: 1300 N forwards
- Resultant force
- a single force that has the same effect as all the original forces acting on an object together
4.5.1.4Free body diagrams and resolving forcesHigher tier
Higher tier students should be able to describe examples of the forces acting on an isolated object or system, and use free body diagrams to describe qualitatively examples where several forces lead to a resultant force on an object, including balanced forces when the resultant force is zero.
A free body diagram shows one object on its own, with an arrow for every force acting on it and nothing else. Drawing it that way stops you mixing in forces that act on other objects — a common source of error.
When the forces are balanced the resultant is zero, and the object either stays still or continues at a constant velocity.
A single force can be resolved into two components acting at right angles to each other. The two component forces together have the same effect as the single force.
You should be able to use vector diagrams to illustrate resolution of forces, equilibrium situations, and to determine the resultant of two forces, to include both magnitude and direction — scale drawings only.
"Scale drawings only" is an important limit: you will never be asked to use trigonometry here. The method is to draw carefully and measure.
Worked example
A force of 5.0 N acts north and a force of 3.0 N acts east on the same object. Describe how to find the resultant by scale drawing.
- Choose and state a scale, for example 1 cm represents 1.0 N.
- Draw the 5.0 N force as a 5 cm arrow pointing north.
- From the head of that arrow, draw the 3.0 N force as a 3 cm arrow pointing east.
- Draw a single arrow from the tail of the first to the head of the second — this is the resultant.
- Measure its length and convert using the scale, and measure its angle from north with a protractor.
Answer: The resultant measures about 5.8 cm, so about 5.8 N, at roughly 31° east of north. Give both the magnitude and the direction, since a resultant is a vector.
Common mistake
State the scale on a vector diagram and draw the arrows head to tail. Drawing both arrows from the same point and joining the ends gives the wrong answer, and an unstated scale means the measurement cannot be marked.
Before you practise
- Scalars have magnitude only; vectors have magnitude and direction, and are drawn as arrows whose length shows the magnitude.
- Contact forces need the objects to touch — friction, air resistance, tension, normal contact force.
- Non-contact forces act at a distance — gravitational, electrostatic, magnetic.
- W = m g must be recalled; the value of g will always be given.
- Mass is in kilograms and does not change with location; weight is a force in newtons and does.
- Weight acts at the centre of mass, is directly proportional to mass, and is measured with a calibrated spring-balance.
- The resultant force is the single force with the same effect as all the forces together.
- Higher tier: free body diagrams show one object and every force on it; a force can be resolved into two components at right angles.
- Higher tier vector work is by scale drawing only — never trigonometry.