Welcome to Vectors in Mechanics!
In this chapter, we are going to bridge the gap between pure math and the physical world. While a simple number (like "5 kg") tells us how heavy something is, it doesn't tell us which way a car is driving or where a force is pushing. That is where vectors come in! Vectors are essentially arrows that tell us two things at once: How much? and Which way?
By the end of these notes, you’ll be able to track a ship's position, calculate the speed of a plane in the wind, and work out exactly where a particle will end up after being pushed by multiple forces. Don't worry if it seems like a lot to take in—we’ll break it down step-by-step!
1. What is a Vector?
In Mechanics, we deal with two types of quantities:
- Scalars: These only have magnitude (size). Examples: Mass, time, distance, and speed.
- Vectors: These have both magnitude AND direction. Examples: Displacement, velocity, acceleration, and force.
Vector Notation
We usually write vectors in two ways:
- Unit Vector Form: Using \( \mathbf{i} \) (one unit to the right) and \( \mathbf{j} \) (one unit up). For example: \( \mathbf{v} = 3\mathbf{i} + 4\mathbf{j} \).
- Column Vectors: Written as \( \begin{pmatrix} x \\ y \end{pmatrix} \). So, \( 3\mathbf{i} + 4\mathbf{j} \) becomes \( \begin{pmatrix} 3 \\ 4 \end{pmatrix} \).
Quick Review: Think of \( \mathbf{i} \) as "steps East" and \( \mathbf{j} \) as "steps North". A vector of \( 5\mathbf{i} - 2\mathbf{j} \) means 5 steps right and 2 steps down.
2. Magnitude and Direction
Sometimes a question will give you a vector like \( 3\mathbf{i} + 4\mathbf{j} \) and ask for its magnitude (how long is the arrow?) or its direction (what is the angle?).
Calculating Magnitude
Because the \( \mathbf{i} \) and \( \mathbf{j} \) components are at right angles, we can use Pythagoras' Theorem. For a vector \( \mathbf{a} = x\mathbf{i} + y\mathbf{j} \):
\( |\mathbf{a}| = \sqrt{x^2 + y^2} \)
Example: The magnitude of \( 3\mathbf{i} - 4\mathbf{j} \) is \( \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = 5 \).
Calculating Direction
We use trigonometry (SOH CAH TOA) to find the angle \( \theta \). Usually, we find the angle with the positive \( x \)-axis (the \( \mathbf{i} \) direction):
\( \tan \theta = \frac{y}{x} \)
Common Mistake: Always draw a quick sketch! If your vector is \( -3\mathbf{i} + 2\mathbf{j} \), it is in the top-left quadrant. A calculator might give you a negative angle, but your sketch will help you find the correct bearing or angle from the North.
Key Takeaway: Magnitude is the "size" (always positive), and direction is the "angle." Together, they define a vector.
3. Resultant Vectors and Resolving
In Mechanics, we often have multiple forces acting on one object. The Resultant Vector is just the single vector you get when you add all the individual vectors together.
Adding Vectors
To find the resultant, just add the \( \mathbf{i} \) parts together and the \( \mathbf{j} \) parts together.
If \( \mathbf{F}_1 = 2\mathbf{i} + 5\mathbf{j} \) and \( \mathbf{F}_2 = 4\mathbf{i} - 1\mathbf{j} \):
\( \text{Resultant } \mathbf{R} = (2+4)\mathbf{i} + (5-1)\mathbf{j} = 6\mathbf{i} + 4\mathbf{j} \).
Resolving a Vector
This is the opposite of finding magnitude. If you are given a magnitude \( R \) and an angle \( \theta \), you can split it into components:
- Horizontal component (\( \mathbf{i} \)): \( R \cos \theta \)
- Vertical component (\( \mathbf{j} \)): \( R \sin \sin \theta \) (if \( \theta \) is measured from the horizontal)
Memory Aid: "Cos is Close to the angle." If the angle is touching the axis you are looking for, use Cosine.
4. Vectors in Kinematics
This is where we apply vectors to moving objects. We use the following symbols:
- \( \mathbf{r} \): Position Vector (where the object is relative to the origin).
- \( \mathbf{v} \): Velocity Vector (speed and direction).
- \( \mathbf{a} \): Acceleration Vector.
Constant Velocity Formula
If an object moves at a constant velocity, its position at time \( t \) is given by:
\( \mathbf{r} = \mathbf{r}_0 + \mathbf{v}t \)
Where:
\( \mathbf{r}_0 \) is the starting position (at \( t=0 \))
\( \mathbf{v} \) is the constant velocity
\( t \) is the time elapsed
Did you know? Speed is just the magnitude of the velocity vector. If your velocity is \( 3\mathbf{i} + 4\mathbf{j} \), your speed is \( 5 \text{ m/s} \).
Constant Acceleration
If the acceleration is constant, we can use the vector versions of the SUVAT equations:
- \( \mathbf{v} = \mathbf{u} + \mathbf{a}t \)
- \( \mathbf{s} = \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2 \)
Key Takeaway: When solving kinematics problems, treat \( \mathbf{i} \) and \( \mathbf{j} \) separately. They don't interfere with each other!
5. Vectors in Dynamics (Forces)
Newton's Second Law (\( F = ma \)) works perfectly with vectors. If several forces act on a particle, their resultant force causes the acceleration.
\( \Sigma \mathbf{F} = m\mathbf{a} \)
Step-by-Step Process for Force Problems:
- List all the force vectors given (e.g., \( \mathbf{F}_1, \mathbf{F}_2 \)).
- Add them up to find the Resultant Force (\( \mathbf{R} \)).
- Set \( \mathbf{R} = \text{mass} \times \text{acceleration vector} \).
- Solve for the missing values by comparing the \( \mathbf{i} \) and \( \mathbf{j} \) components.
Example: A mass of 2 kg is acted on by forces \( (3\mathbf{i} + \mathbf{j}) \) and \( (\mathbf{i} + 5\mathbf{j}) \).
Resultant \( \mathbf{F} = 4\mathbf{i} + 6\mathbf{j} \).
Using \( \mathbf{F} = m\mathbf{a} \): \( 4\mathbf{i} + 6\mathbf{j} = 2\mathbf{a} \).
So, acceleration \( \mathbf{a} = 2\mathbf{i} + 3\mathbf{j} \text{ m/s}^2 \).
Key Takeaway: If a particle is in equilibrium or moving with constant velocity, the resultant force must be zero (\( 0\mathbf{i} + 0\mathbf{j} \)).
Summary Checklist
Before moving on, make sure you are comfortable with these "Quick Review" points:
- Can you find the magnitude using \( \sqrt{x^2 + y^2} \)?
- Do you remember that speed is the magnitude of velocity?
- Can you use the formula \( \mathbf{r} = \mathbf{r}_0 + \mathbf{v}t \) for constant velocity?
- Can you add force vectors to find the resultant and use \( \mathbf{F} = m\mathbf{a} \)?
Don't worry if this seems tricky at first! Vectors are just a way of doing two math problems at the same time (one for \( \mathbf{i} \) and one for \( \mathbf{j} \)). Practice drawing the diagrams, and the patterns will start to show!