Study Notes on Vector Quantities and Their Applications

Chapter Four: Vector Quantities & Their Applications

Key Ideas

• Vectors: Quantities that possess both magnitude and direction. Examples include forces and displacements.

  • A vector is represented by an arrow:

  • Direction: The direction of the vector is indicated by the direction the arrow points.

  • Magnitude: The length of the arrow is proportional to the vector's magnitude.

• Vector Addition: The process of combining vector quantities.

  • The result of vector addition is termed the resultant vector.

  • Vectors can be added:

  • Geometrically: By drawing to scale.

  • Mathematically: Using algebra and trigonometry.

• Resolution of Vectors: The process of breaking down a single vector into its components.

  • Simplifies problems involving two-dimensional motion, static equilibrium, and motion on inclined planes.

Key Objectives

At the conclusion of this chapter you will be able to:

• Define the terms scalar and vector, and list scalar and vector quantities.

• Represent a vector quantity by an arrow drawn to scale.

• Relate the direction of a vector to compass directions.

• Define the term resultant vector.

• Add vector quantities both graphically and algebraically.

• Relate vector subtraction to vector addition.

• Define vector resolution and resolve a vector into its x- and y-components (limited to two-dimensional analysis).

• Add vector quantities by combining their x- and y-components.

• Define the term static equilibrium.

• Solve static equilibrium problems.

• Identify and calculate the parallel and perpendicular components of an object’s weight when on an inclined plane.

• Solve problems involving motion on inclined planes and in two dimensions.

4.1 Introduction

• Scalar Quantities: Quantities described solely by their magnitudes (size) such as mass, time, and temperature.

• Vector Quantities: Must include direction in their description (e.g., velocity, acceleration, and force).

4.2 Displacement and Representation of Vector Quantities

• Displacement: Defined as a directed change in the position of an object. - Example: An indication of 30 meters [east] indicates a displacement.

• Vectors can be represented by arrows, where:

  • Arrow length represents magnitude.

  • Direction points from the tail to the tip.

  • Example scalings are provided:

  • 1 cm of arrow length represents 20 meters.

4.3 Vector Addition

• Concept: Determines the net effect of two or more vectors acting on an object.

  • Example: An airplane traveling at 300 m/s [east] enters a jet stream with a velocity of 100 m/s [north].

  • Resultant calculation performed by placing vectors head to tail:

  $OA \text{ (3.0 m [E])} + B \text{ (4.0 m [E])} = R \text{ (7.0 m [E])}$

• When vectors are in opposite directions, their magnitudes are subtracted:

  • Example: A bird flies 3.0 km [N] then 4.0 km [S], yielding a resultant of:

    $R = 1.0 km [S]$

• Vectors at right angles:

  • Can be added via the Pythagorean theorem:

    $R^2 = P^2 + J^2$

4.4 Vector Subtraction

• Definition: Subtraction considered as a special case of addition:

  • $A - B = A + (-B)$.

  • The negative vector is equal in magnitude but opposite in direction.

4.5 Vector Resolution

• Definition: The process of breaking down a single vector into its perpendicular components along specified axes (typically x and y).

  • This simplifies the analysis of vector interactions in two or three dimensions.

  • A vector $A$ can be resolved into:

    • x-component: $A_x = A \cos(\text{angle})$

    • y-component: $A_y = A \sin(\text{angle})$

      where the angle is the angle the vector makes with the positive x-axis.

4.6 Adding Vectors by Components

• Method:

  1. Resolve each vector into its x and y components.

  2. Algebraically sum all the x-components to find the resultant x-component ($R_x = \text{sum of } A_x$).

  3. Algebraically sum all the y-components to find the resultant y-component ($R_y = \text{sum of } A_y$).

  4. Calculate the magnitude of the resultant vector using the Pythagorean theorem: $R = \sqrt{R_x^2 + R_y^2}$.

  5. Determine the direction of the resultant vector using trigonometry: $\text{angle} = \arctan(R_y/R_x)$. The quadrant of the resultant must be considered based on the signs of $R_x$ and $R_y$.

4.7 Static Equilibrium

• Definition: A state where an object is at rest and the net force acting on it is zero.

  • This implies that the object is not accelerating ($a=0$).

• Conditions for Static Equilibrium:

  • The sum of all force components in the x-direction must be zero: $\text{sum of } F_x = 0$.

  • The sum of all force components in the y-direction must be zero: $\text{sum of } F_y = 0$.

• Problem Solving: Typically involves resolving all forces into components and applying these equilibrium conditions to solve for unknown forces.

4.8 Motion on Inclined Planes

• Analysis: Problems involving objects on inclined planes are simplified by resolving forces into components parallel and perpendicular to the incline.

• Weight Components: An object's weight $(Fg)$ acting vertically downwards is resolved into:

  • Parallel component: The component of weight acting down the incline: $W_{||}=Fg\sin(\text{angle})$

  • Perpendicular component: The component of weight normal to the incline: $W_{\perp}=Fg\cos(\text{angle})$

    where the angle is the angle of inclination of the plane with the horizontal.

• Forces to Consider: Normal force, friction, and any applied forces, in addition to the components of weight.

4.9 Two-Dimensional Motion

• Concept: Motion that occurs in a plane, such as projectile motion or circular motion. The x and y components of motion are generally independent but occur simultaneously.

• Independence of Components:

  • Horizontal motion (x-direction) is often characterized by constant velocity (assuming no air resistance).

  • Vertical motion (y-direction) is characterized by constant acceleration due to gravity ($g = 9.8 \text{ m/s}^2$ downwards).

• Kinematic Equations: Applied separately to the x and y components. For example:

  • $x = v_{0x}t + \frac{1}{2}a_xt^2$

  • $y = v_{0y}t + \frac{1}{2}a_yt^2$

• Projectile Motion: A common example where an object is launched into the air and follows a parabolic trajectory under the influence of gravity only.