INTRODUCTION
Everything covered in this lesson, in one place - useful for revision or printing.
Mechanics Vectors and Scalars Magnitude, direction and the resultant
Some quantities only need a size. Others also need a direction. In this lesson you will learn to:
time mass distance speed force displacement velocity weight
A physical quantity is a property that can be measured. Every physical quantity has a number and a unit: 55 s, 2 kg, 35 N.
Physical quantities fall into two groups: vectors and scalars.
Vector Scalar Displacement Distance Velocity Speed Acceleration Time Force Mass Weight Temperature Energy
vector A vector is a physical quantity with magnitude and direction.
scalar A scalar is a physical quantity with magnitude only.
Ask: does the quantity need a direction to be complete? Yes: vector. No: scalar.
Thato walked 5 m scalar Thato walked 5 m east vector
"Thato walked 5 m" tells us how much ground was covered, but not which way: a scalar (distance).
"Thato walked 5 m east" gives the magnitude (5 m) AND the direction (east): a vector (displacement).
A distance of 2 m scalar A force of 35 N acting 30° W of N vector A velocity of 3 m·s⁻¹ on a bearing of 115° vector 55 s scalar 4 m·s⁻¹ scalar 165 m north vector
Look for a direction word: east, north, a bearing, an angle, up or down. If it is there, the quantity is a vector.
4 m·s⁻¹ on its own is a speed (scalar); 3 m·s⁻¹ on a bearing of 115° is a velocity (vector).
Distance: the total path length travelled (scalar, never negative).
Displacement: the difference in position, from start to end (vector).
You walk 5 m east and then 3 m west. Take east as positive:
\[ \begin{gathered} \text{speed} = \dfrac{\text{distance}}{\text{time}} \\ \text{velocity} = \dfrac{\text{displacement}}{\text{time}} \end{gathered} \]
Speed tells us how fast something moves: a scalar. Velocity is a speed in a specific direction: a vector.
A car travels 100 m in 20 s: \(v = \tfrac{100}{20} = 5\ \text{m}\cdot\text{s}^{-1}\).
A car moves 100 m east in 20 s: \(v = 5\ \text{m}\cdot\text{s}^{-1}\ \text{east}\).
Find the distance and the displacement for the journey A to B to C to D to E (it starts by going east).
The length of the arrow represents the magnitude; the arrowhead shows the direction. A longer arrow means a larger magnitude.
Scale: 1 cm represents 2 m·s⁻¹. So \(10\ \text{m}\cdot\text{s}^{-1} = \tfrac{10}{2} = 5\ \text{cm}\) east, and 8 m·s⁻¹ west is 4 cm.
\[ \begin{gathered} 1\ \text{cm} : 2\ \text{N} \;\Rightarrow\; \tfrac{19}{2} = 9{,}5\ \text{cm} \\ 1\ \text{cm} : 3\ \text{N} \;\Rightarrow\; \tfrac{19}{3} = 6{,}33\ \text{cm} \\ 1\ \text{cm} : 5\ \text{N} \;\Rightarrow\; \tfrac{19}{5} = 3{,}8\ \text{cm} \end{gathered} \]
How long should a line be to represent 19 N? Divide the magnitude by the value that 1 cm represents.
A bigger scale value gives a shorter arrow. Always write down the scale you used.
\[ \begin{gathered} \vec{F} \qquad \vec{v} \qquad \vec{a} \end{gathered} \]
A vector symbol has a small arrow above it: the arrow shows that the quantity has a direction.
The magnitude on its own is written without the arrow: \(F = 35\ \text{N}\). A vector answer always needs both the magnitude and the direction: 35 N east.
Equal vectors: They have the same magnitude and the same direction.
Negative vector: The negative of a vector has the same magnitude but the opposite direction.
A vector may be moved, as long as its length and direction stay the same.
The resultant is the single vector having the same effect as two or more vectors together.
2 N right and 7 N right point the same way, so the magnitudes add: \(F_R = 2 + 7 = 9\ \text{N right}\).
4 N right and 12 N left. First choose a positive direction: right is positive, so left is negative.
\(F_R = 4 + (-12) = -8\ \text{N}\). The minus sign means left: the resultant is 8 N left.
\[ \begin{gathered} F_R = 3 + (-6) + 15 + (-8) \\ F_R = 4\ \text{N right} \end{gathered} \]
Right is positive: every force to the left gets a minus sign. Add them all; the sign of the answer gives the direction.
The drawing and the calculation must agree: both give 4 N to the right.
\[ \begin{gathered} 5.1\quad 2 + 7 = 9\ \text{N} \\ 5.2\quad 4 + (-12) = -8\ \text{N} \\ 5.3\quad (-12) + (-16) = -28\ \text{N} \\ 5.5\quad (-16) + 24 + (-13) = -5\ \text{N} \end{gathered} \]
Take right as positive and find the resultant:
No direction in a vector answer Adding opposite forces without a sign Arrows not drawn to scale Resultant drawn from head to tail Distance given for displacement
8 N is not a resultant; 8 N left is. Opposite directions need a negative sign. Draw every arrow with the same scale, and draw the resultant from the first tail to the last head.
\[ \begin{gathered} \text{vector: magnitude + direction} \\ \text{scalar: magnitude only} \\ F_R = F_1 + F_2 + F_3 + F_4 \end{gathered} \]