Physics Chapter on Vectors and Moments

Mechanics and Materials

• Motion
  • Distance is a scalar quantity, measured in meters (m).
  • Displacement is a vector quantity, measured in meters (m).
  • Speed is the rate of change of distance, measured in meters per second (ms⁻¹).
  • Velocity is the rate of change of displacement, measured in meters per second (ms⁻¹).
  • Acceleration is the rate of change of velocity, measured in meters per second squared (ms⁻²).
  • Acceleration occurs when speed changes, direction changes, or both.
  • Constant circular motion involves changing velocity, thus acceleration.

Motion Graphs

• Constant Gradient (Straight Line)
  • Gradient = (y₂ - y₁)/(x₂ - x₁)
  • Gradient = zero, if gradient is zero then gradient is horizontal
  • Gradient = infinite, if gradient is vertical then gradient is infinite
  • Constant positive gradient, if gradient is positive then it slopes upwards.
  • Constant negative gradient, if gradient is negative then it slopes downwards, or constant slope downwards.
• Varying Gradient (Curves)
  • The gradient of a curve at any given point can be determined by drawing a tangent to the curve at that point.
  • Displacement-time graph, gradient = velocity
  • Velocity -time graph, gradient = acceleration.
  • The area under the graph represents displacement and velocity; acceleration-time.

Graphs of Motion

• Object with Constant Velocity
  • Displacement graph is a straight line with a constant positive gradient.
  • Velocity graph is a horizontal line at a constant value..
  • Acceleration graph is a horizontal line with a value of zero.
• Object with Constant Acceleration
  • Displacement graph is a curve, concave upwards.
  • Velocity graph is a straight line.
  • Acceleration graph is a horizontal line at a constant value.
• Object with Constant Deceleration
  • Displacement graph is a curve, concave downwards.
  • Velocity graph is a straight line.
  • Acceleration graph is a horizontal line at a constant value.

Object Dropped and Thrown Upwards

• Dropped Object
• Assumes upward direction as positive.
• Displacement graph is a curve.
• Velocity graph has negative gradient, decreasing linearly.
• Acceleration graph is a horizontal line at a constant value -9.81 ms⁻².
• Thrown Upwards
• Assumes upward direction as positive.
• Displacement graph is a curve.
• Velocity graph is decreasing linearly with negative gradient.
• Acceleration graph is a constant value of -9.81 ms⁻².

Bouncing Ball

• Displacement, Velocity, and Acceleration graphs: The graphs of a bouncing ball display parallel lines due to constant acceleration.
• Area under Displacement Graph: the area represents total height
• Gravity is constant, negative acceleration for the upwards motion and positive upwards motion.

Vectors

• Vectors: Quantities that have both magnitude and direction.
• Addition of Vectors in a Straight Line: Add magnitudes; direction of the resultant is the direction of the original vectors.
• Vectors in Opposite Lines: Subtract magnitudes. Direction of the resultant is in the direction of the vector with the larger magnitude.
• Vectors Perpendicular to Each Other: Pythagoras' Theorem to find the resultant vector.

Parallelogram and Polygon Laws

• Parallelogram Law: If two vectors are represented as the sides of a parallelogram, the diagonal represents the resultant.
• Polygon Law: Used for adding more than two vectors arranged as the sides of a polygon; the resultant vector can be drawn as a closing side of a polygon.

Resolving Vectors

• Resolving a Vector: Finding two perpendicular components.
• Components: are the projections along the x and y axes. (x- and y- components).
• Cosine and sine are used for calculating components.
• Finding the Resultant The combination of components.

Moments

• Moment of a Force: A measure of the tendency for rotation.
• Principle of Moments: Total clockwise moments = total anticlockwise moments for equilibrium.
• Centre of Mass/Centre of Gravity (COG): The point where an object's weight appears to act.

Equations of Motion

• Equations:
  • v = u + at
  • v² = u² + 2as
  • s = ut + ½at²
  • s = ½(u + v)t
  • Where:
    • v = final velocity
    • u = initial velocity
    • a = acceleration
    • t = time
    • s = displacement

Projectile Motion

• Vertical Motion: Under the influence of gravity, initial vertical velocity and final vertical velocity at the highest point are 0.
• Horizontal Motion: Constant horizontal velocity in the absence of air resistance.

Forces

• Contact Forces: Forces acting between objects in physical contact.
• Normal Contact Force: The support force exerted by a surface on an object resting on it.
• Friction Force: The force opposing motion between surfaces in contact.
• Drag Force: The force resisting motion of an object moving through a fluid; dependent on viscosity of fluid and speed.
• Tension Force: A force transmitted through a string, rope, cable, etc.
• Gravitational force: force of attraction between objects with mass.
• Magnetic Force: Attraction force/repulsion force between magnets or magnetic materials.
• Electrostatic Force: Attractive/repulsive force between electrostatically charged objects.
• Free Body Diagrams: Diagrams showing all forces acting on an object.

Work Done

• Work Done: Product of force and displacement in the direction of the force.

Energy

• Energy: Ability to do work.
• Kinetic Energy: Energy of motion.1/2mv²
• Potential Energy: Energy of position.
• Gravitational Potential Energy: Energy due to position relative to a gravitational field.
• Conservation of Energy: Energy cannot be created or destroyed, only changed from one form to another.
• Law of Conservation of Energy: All forms of energy in a closed system remain constant unless acted upon by external forces.

Power

• Power: Rate of doing work, or rate of energy transfer.

Efficiency

• Efficiency: Ratio of useful work output to total work input.

Momentum

• Momentum: Product of mass and velocity.
• Impulse: Change in momentum.
• Conservation of Momentum: Total momentum in a closed system remains constant.

Fluids

• Density: Mass per unit volume.
• Upthrust: Upward buoyant force exerted by a fluid on an immersed body.
• Archimedes' Principle: Upthrust = weight of fluid displaced.
• Viscosity: Resistance to flow in a fluid.
• Drag Force: Force opposing motion of an object moving through a fluid, increasing with speed.
• Stokes' Law: Describes the viscous drag on a small sphere moving at low speeds through a fluid.
• Laminar flow: smooth, orderly flow in a fluid.
• Turbulent flow: Unordered, chaotic flow in a fluid.

Solids Materials

• Hooke's Law: Stress is proportional to strain in the elastic region.
• Elastic Strain Energy: energy stored in a stretched spring or material.
• Stress: Force per unit cross-sectional area.
• Strain: Change in dimension divided by original dimension.
• Young's Modulus: a material property equal to the ratio of stress to strain in the elastic region.
• Stiffness: A measure of a material's resistance to deformation.
• Malleable materials: deform under compression with little or no brittle behaviour.
• Ductile materials: deform under tension with little or no brittle behaviour.
• Brittle materials: break or fracture without much deformation first.