P15: Scalar and Vector Quantities
Understanding the difference between scalar and vector quantities, forces, weight, mass, gravity, resultant forces, free body diagrams, resolving forces, and contact versus non-contact forces.
Scalars and Vectors
Scalar quantities have magnitude only. Vector quantities have both magnitude and direction.
Understanding the difference between scalars and vectors is fundamental to mechanics. Scalars are fully described by a number and unit, while vectors also require a direction to be completely specified.
| Scalar Quantities | Vector Quantities |
|---|---|
| Speed | Velocity |
| Distance | Displacement |
| Mass | Weight (force) |
| Temperature | Acceleration |
| Time | Momentum |
| Energy | Force |
| Volume | Friction |
| Density | Gravitational field strength |
Scalars can be added by simple arithmetic. Vectors must be added considering direction, often using geometry or component methods.
Contact and Non-Contact Forces
A force is a push or pull that acts on an object due to its interaction with another object. Forces are always measured in newtons (N).
| Contact Forces | Non-Contact Forces |
|---|---|
| Friction | Gravitational force |
| Air resistance | Electrostatic force |
| Tension | Magnetic force |
| Normal contact force | Nuclear force |
| Upthrust | |
| Thrust |
Contact forces require the objects to be physically touching. Non-contact forces act at a distance without physical contact between the objects.
Mass and Weight
Mass is a scalar measure of the amount of matter in an object (measured in kg). It does not change with location. Weight is a vector force due to gravity acting on an object (measured in N). It changes depending on the gravitational field strength.
Weight = mass × gravitational field strength
W = mg
W = weight (N), m = mass (kg), g = gravitational field strength (N/kg)
On Earth, g ≈ 9.8 N/kg (often rounded to 10 N/kg for calculations). On the Moon, g ≈ 1.6 N/kg. This means an object with mass 60 kg has a weight of 588 N on Earth but only 96 N on the Moon.
Worked Example
A student has a mass of 55 kg. Calculate their weight on Earth (g = 9.8 N/kg).
W = mg = 55 × 9.8 = 539 N
The weight of an object can be measured using a calibrated spring balance (newtonmeter). Mass is measured using a balance.
Gravitational Field Strength
Gravity is a non-contact force that attracts all objects with mass towards each other. The gravitational field strength at the surface of a planet depends on the planet's mass and radius.
| Location | g (N/kg) |
|---|---|
| Earth | 9.8 |
| Moon | 1.6 |
| Jupiter | 25 |
| Mars | 3.7 |
Weight is directly proportional to mass. A graph of weight against mass gives a straight line through the origin with gradient g.
Resultant Force
The resultant force is the single force that has the same effect as all the original forces acting together. It determines the overall acceleration of the object.
When forces act along the same line, the resultant is found by adding forces in one direction and subtracting forces in the opposite direction.
Worked Example
A car experiences a driving force of 5000 N forwards and drag of 1200 N backwards. Calculate the resultant force.
Resultant = 5000 - 1200 = 3800 N forwards
If the resultant force on an object is zero, the object remains at rest or continues moving at a constant velocity (Newton's first law). If the resultant force is non-zero, the object accelerates in the direction of the resultant force.
Free Body Diagrams
A free body diagram shows all the forces acting on a single object, with each force represented by an arrow. The arrow length represents magnitude and the arrow direction shows the force direction.
For an object at rest on a surface, the typical forces shown are:
- Weight acting downwards from the centre of mass
- Normal contact force acting upwards from the surface
For a falling object:
- Weight acting downwards
- Air resistance (drag) acting upwards
For an object on a slope:
- Weight acting vertically downwards
- Normal contact force perpendicular to the slope surface
- Friction acting along the slope surface (opposing motion)
Always draw force arrows from the centre of mass of the object. Make sure opposing forces are clearly shown with different lengths if they are unequal.
Resolving Forces (Higher)
A single force can be resolved into two component forces at right angles to each other. This is the reverse of finding a resultant and uses trigonometry.
For a force F at angle θ to the horizontal:
Horizontal component = F cos θ
Vertical component = F sin θ
Worked Example
A force of 80 N acts at 30° above the horizontal. Calculate its horizontal and vertical components.
Horizontal = 80 cos 30° = 80 × 0.866 = 69.3 N
Vertical = 80 sin 30° = 80 × 0.5 = 40 N
Resolving forces is particularly useful on inclined planes. The weight component parallel to the slope is mg sin θ, and the weight component perpendicular to the slope is mg cos θ, where θ is the angle of the slope.
When resolving a force on a slope, always use the angle between the slope and the horizontal as θ. The component parallel to the slope (mg sin θ) causes the object to accelerate down the slope.
Vector Addition (Higher)
When two forces are not along the same line, the resultant can be found using a scale diagram (tip-to-tail method) or by calculation using Pythagoras and trigonometry for perpendicular forces.
Worked Example
Two forces act on an object: 30 N to the east and 40 N to the north. Find the resultant force.
Resultant = √(30² + 40²) = √(900 + 1600) = √2500 = 50 N
Direction: tan θ = 40/30, θ = tan⁻¹(1.333) = 53.1° north of east
For two perpendicular forces, use Pythagoras to find the magnitude and trigonometry to find the angle. For non-perpendicular forces, draw a scale diagram with the tip of one arrow at the tail of the other.
Practice Questions
1. State whether each of the following is a scalar or a vector quantity: speed, acceleration, energy, displacement, temperature, momentum.
Speed - scalar, Acceleration - vector, Energy - scalar, Displacement - vector, Temperature - scalar, Momentum - vector.
2. Calculate the weight of a 75 kg person on Earth (g = 9.8 N/kg) and on the Moon (g = 1.6 N/kg).
Earth: W = 75 × 9.8 = 735 N. Moon: W = 75 × 1.6 = 120 N.
3. A boat has a driving force of 8000 N and experiences drag of 3500 N. Calculate the resultant force and state its direction.
Resultant = 8000 - 3500 = 4500 N in the forward direction.
4. A force of 120 N acts at 40° to the horizontal. Calculate the horizontal and vertical components.
Horizontal = 120 cos 40° = 91.9 N. Vertical = 120 sin 40° = 77.1 N.
5. Explain the difference between mass and weight and why an astronaut's mass stays the same on the Moon but their weight changes.
Mass is the amount of matter in an object and does not depend on gravitational field strength. Weight is the gravitational force on the object (W = mg), so it depends on the local value of g. The Moon has a weaker gravitational field so weight is less, but mass is unchanged.
Maths Skills
Vector Addition Using Scale Drawings
When two forces are not along the same line, find the resultant using a scale diagram: draw the first vector as an arrow to scale, then draw the second vector starting from the tip of the first (tip-to-tail method). The resultant is the arrow from the tail of the first to the tip of the second. Measure its length (for magnitude) and angle (for direction).
Worked Example
Two forces act on an object: 40 N to the east and 30 N at 60° north of east. Use a scale diagram to find the resultant. (Use a scale of 1 cm = 10 N.)
Draw 4.0 cm east, then from the tip draw 3.0 cm at 60°. Measure the closing vector: approximately 6.1 cm at 25° north of east, giving a resultant of approximately 61 N at 25° north of east.
Resolving Forces Using Trigonometry (Higher)
A single force F at angle θ to the horizontal can be split into two perpendicular components: horizontal = F cos θ and vertical = F sin θ. This is useful on inclined planes where the weight component parallel to the slope is mg sin θ and perpendicular is mg cos θ.
W = mg Calculations
Weight is calculated using W = mg. On Earth, g = 9.8 N/kg (often rounded to 10 N/kg). Remember that mass stays the same anywhere but weight changes with gravitational field strength. Always use the correct value of g for the location stated in the question.
Worked Example
A rover on Mars has a mass of 200 kg. Mars has g = 3.7 N/kg. Calculate its weight on Mars and on Earth.
Mars: W = 200 × 3.7 = 740 N. Earth: W = 200 × 9.8 = 1960 N. The mass remains 200 kg in both locations.
Common Misconceptions
Mass and Weight
Mass and weight are the same thing. Mass is a scalar measure of the amount of matter in an object, measured in kilograms (kg). It does not change with location. Weight is a vector force due to gravity, measured in newtons (N), calculated as W = mg. An astronaut's mass is the same on the Moon as on Earth, but their weight is about one-sixth because the Moon's gravitational field strength is weaker.
Forces on Stationary Objects
A stationary object has no forces acting on it. A stationary object can have multiple forces acting on it — what matters is that the forces are balanced (resultant force is zero). A book on a table has both weight (downwards) and the normal contact force (upwards) acting on it. These are equal and opposite, so the book remains stationary, but the forces are still present.
6-Mark Extended Question
Explain the difference between scalar and vector quantities. Use examples to show why the distinction matters. [6 marks]
Scalar quantities have magnitude only, while vector quantities have both magnitude and direction (1). Examples of scalars include speed, distance, mass, and temperature (1). Examples of vectors include velocity, displacement, force, and acceleration (1). The distinction matters because two objects with the same scalar speed can be moving in completely different directions — if two cars both travel at 30 m/s but one goes north and one goes south, their velocities are different (1). Similarly, an object returning to its starting point has a total distance of the full path travelled, but a displacement of zero — the scalar and vector give very different information (1). In force calculations, direction is essential: two forces of 10 N can cancel if opposite, or add to 20 N if in the same direction, so treating forces as scalars would give incorrect results (1).
AO3: Analyse and Evaluate
A free body diagram shows three forces acting on an object:
- Force A: 80 N to the right
- Force B: 60 N upwards
- Force C: 30 N to the left
(a) Calculate the resultant horizontal force. (b) Calculate the magnitude and direction of the overall resultant force. (c) The object has a mass of 25 kg. Calculate its acceleration and state the direction of acceleration.
Evaluation
(a) Horizontal resultant = 80 − 30 = 50 N to the right.
(b) The resultant has components 50 N (right) and 60 N (up). Magnitude = √(50² + 60²) = √(2500 + 3600) = √6100 = 78.1 N. Direction = tan−1(60/50) = 50.2° above the horizontal (to the right).
(c) a = F/m = 78.1/25 = 3.12 m/s² in the direction of the resultant force (50.2° above horizontal to the right).
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