Physics (9630) | Scalars and vectors

Introduction: Why Directions Matter!

Welcome to the foundational chapter of Mechanics! Before we can calculate how fast things move or how much force is needed to stop a train, we need to understand a very simple, yet crucial, distinction: Does the direction matter?

In physics, some quantities are just numbers (like mass or time), but most quantities involved in movement and forces depend completely on the way they are pointed. Mastering this difference between scalars and vectors is the key to successfully solving almost every problem in Mechanics.

Don't worry if vector addition looks like complicated geometry—we'll break it down into simple, manageable steps, both graphically and mathematically!

1. Defining Scalars and Vectors (3.2.1)

Physical quantities are categorized based on whether they require a specified direction to be fully described.

1.1 Scalars: Magnitude Only

A scalar quantity is fully defined by its magnitude (size) alone. Direction is irrelevant.

• Key Feature: They are added and subtracted just like regular numbers (arithmetic).
• Analogy: Think about baking a cake. If the recipe calls for 200g of flour, the direction you pour the flour doesn't change the amount of flour you have!

Examples of Scalar Quantities (Memorize these!)

• Mass (How much 'stuff' is there)
• Time (Duration)
• Distance (Total path length travelled)
• Speed (Rate of change of distance)
• Energy, Power, Density, Temperature, Volume.

1.2 Vectors: Magnitude and Direction

A vector quantity requires both a magnitude (size) and a specific direction to be fully described.

• Key Feature: The direction is absolutely critical. If two forces have the same size but act in opposite directions, their effect is zero!
• Analogy: Imagine pushing a heavy box. Pushing with 100 N of force is useless if you are pushing in the wrong direction (like pushing towards the wall instead of away from it).

Memory Aid: Vector has Direction.

Examples of Vector Quantities (Memorize these!)

• Displacement (Change in position from start to end point)
• Velocity (Rate of change of displacement)
• Force (A push or pull)
• Weight (The force of gravity, always downwards)
• Acceleration (Rate of change of velocity)
• Momentum, Impulse.

Key Takeaway from Section 1

Scalars are simple quantities (like mass or time). Vectors require a direction (like force or velocity), which completely changes how we add them up.

2. Adding Vectors: The Resultant Vector

When multiple vectors (like forces or displacements) act on an object, we need to find the single vector that represents their total effect. This is called the resultant vector (or net vector).

2.1 Method 1: Vector Addition by Scale Drawing

Scale drawings are excellent for visualising vector addition and can be used to find the resultant vector when calculations are tricky.

Step-by-Step Guide (The Tip-to-Tail Method)

1. Choose a Scale: Decide on a clear scale (e.g., 1 cm = 10 N).
2. Draw the First Vector (A): Draw the vector starting at a specific point, making sure the length matches the magnitude (using your scale) and the angle is correct.
3. Draw the Second Vector (B): Draw the second vector starting from the tip (arrowhead) of the first vector. Ensure its length and angle are correct relative to the tail of the first vector.
4. Find the Resultant (R): Draw a line from the tail (start point) of the first vector to the tip (end point) of the last vector. This line is the resultant (R).
5. Measure Magnitude and Direction: Measure the length of R and convert it back to the actual magnitude using your scale. Use a protractor to measure the angle (direction) of R.

Example: Adding a force of 30 N East and 40 N North. You would draw a line 3 cm East, then a line 4 cm North from the tip of the first line. The resultant connects the start point to the end point.

2.2 Method 2: Vector Resolution and Calculation

When vectors are not along the same line or perpendicular, we use trigonometry to break them down into their basic components (usually horizontal, \(x\), and vertical, \(y\)). This is called vector resolution.

Step 1: Resolve All Vectors into Components

For any vector \(F\) acting at an angle \(\theta\) to the horizontal:

Horizontal Component (\(F_x\)): \(F_x = F \cos \theta\)
Vertical Component (\(F_y\)): \(F_y = F \sin \theta\)

Tip: The component adjacent to (touching) the angle \(\theta\) usually uses cosine.

Step 2: Sum the Components

Add up all the horizontal components to get the total net horizontal force, \(\sum F_x\). Remember to treat left and right as opposite (+/-).

Add up all the vertical components to get the total net vertical force, \(\sum F_y\). Remember to treat up and down as opposite (+/-).

Step 3: Calculate the Resultant Magnitude

Since \(\sum F_x\) and \(\sum F_y\) are now perpendicular, we use Pythagoras’ theorem to find the resultant magnitude, \(R\):

\[ R = \sqrt{ (\sum F_x)^2 + (\sum F_y)^2 } \]

Step 4: Calculate the Resultant Direction

Use the tangent function to find the angle \(\alpha\) of the resultant relative to the horizontal:

\[ \tan \alpha = \frac{\sum F_y}{\sum F_x} \]

\[ \alpha = \arctan \left( \frac{\sum F_y}{\sum F_x} \right) \]

Don't forget to state the direction clearly (e.g., 25° above the horizontal or 40° North of East).

3. Conditions for Equilibrium (3.2.1)

In the context of Mechanics, equilibrium is one of the most important concepts. If an object is in equilibrium, it means its state of motion is not changing.

3.1 Defining Equilibrium

An object is in translational equilibrium if the net force acting on it is zero.

The Key Condition for Equilibrium

For two or three coplanar forces (forces acting on the same flat 2D plane) acting at a point, the condition for equilibrium is:

The sum of all vectors acting on the object must be zero.

This means the resultant force \(R\) is zero, or:

\[ \sum F = 0 \]

3.2 The Two Ways to Achieve Equilibrium

Since force is a vector, \(\sum F = 0\) means that the forces must cancel out in all directions. Using components, this translates to:

1. Horizontal Equilibrium: The sum of all forces in the horizontal (x) direction is zero.
\[ \sum F_x = 0 \]

2. Vertical Equilibrium: The sum of all forces in the vertical (y) direction is zero.
\[ \sum F_y = 0 \]

3.3 What Equilibrium Means for Motion

If an object is in equilibrium, according to Newton’s First Law, it must be doing one of two things:

• At Rest: The object is stationary. (Static Equilibrium)
• Constant Velocity: The object is moving at a constant speed in a straight line (zero acceleration). (Dynamic Equilibrium)

Did you know? A satellite orbiting the Earth at a constant speed is not in equilibrium, even though its speed is constant! This is because its direction is constantly changing, meaning it has an acceleration (centripetal acceleration), and therefore a net force (the gravitational force).

3.4 Solving Equilibrium Problems

When solving problems involving an object held stationary by three coplanar forces (like a weight hanging from two ropes), use the equilibrium conditions:

Step-by-Step for Solving Equilibrium

1. Draw a Free-Body Diagram: Sketch the object and draw arrows representing all forces acting on it, labeling magnitudes and angles.
2. Resolve Forces: Break down any forces that are acting diagonally into their horizontal (x) and vertical (y) components.
3. Apply Equilibrium Conditions:
   • Set "Forces Up" = "Forces Down" (\(\sum F_y = 0\)).
   • Set "Forces Left" = "Forces Right" (\(\sum F_x = 0\)).
4. Solve Simultaneous Equations: Use the equations derived in Step 3 to find any unknown forces or angles.

Key Takeaway from Section 3

Equilibrium means the total force acting on an object is zero, allowing it to remain at rest or move at constant velocity. This requires that the components of forces cancel out in all directions (\(\sum F_x = 0\) and \(\sum F_y = 0\)).