Level 3 Mechanical Systems

Scalars and Vectors

• Vector: Quantity with both size and direction.
  • Examples: displacement, velocity, acceleration, force, torque.
• Scalar: Quantity with only size.
  • Examples: pressure, temperature, time, mass, energy.

Basic Motion

• Displacement: Change in position (measured in meters).
  • Displacement with direction vs. Distance (no direction).
• Velocity: Rate of displacement change (meters/second).
  • Scalar version: Speed.
  • Equation: ( \text{velocity} = \frac{\text{displacement}}{\text{duration}} ).
• Acceleration: Rate of velocity change (meters/second²).
  • Equation: ( \text{acceleration} = \frac{\text{velocity}}{\text{duration}} ).

Equating Vectors

• Vectors forming closed paths equal zero.
  • Example: A + B + C + D = 0.
• Equivalent paths: A + B = C or A + B = C + D.

Trigonometry Refresher

• Right-angled triangles.
  • Hypotenuse, adjacent, and opposite sides.
• Pythagorean Theorem: Hypotenuse² = Opposite² + Adjacent².
• SOHCAHTOA: Mnemonic for trigonometric ratios.

Vector Components

• Separation into horizontal and vertical components using SOHCAHTOA.
• Horizontal: ( v_{\text{horizontal}} = v \cos(\theta) ).
• Vertical: ( v_{\text{vertical}} = v \sin(\theta) ).

Momentum

• Momentum = Mass × Velocity (kg·m/s).
• Impulse: Longer collisions result in smaller forces.
  • Force = Change in momentum / duration.
• Conservation of Momentum: Total momentum before = total momentum after.

Center of Mass

• Weighted average position of masses.
• Balance point for two objects.
• 2D and collision center of mass remains unchanged if no external forces.

Conservation of Energy

• Energy is transformed, not created/destroyed.
• Examples: Electrical to light/heat, potential to kinetic.

Work and Power

• Work: Energy used to perform an action (Joules).
  • Equation: Work = Force × Distance.
• Power: Rate of energy consumption (Watts).
  • Equation: Power = Energy / Duration.

Force and Newton's Laws

• Force: Influence causing acceleration (Newtons).
• Newton's Laws
  • 1st: Motion changes only with force.
  • 2nd: Acceleration proportional to mass (F = ma).
  • 3rd: Every action has an equal/opposite reaction.

Circular Motion and Forces

• Terms: radius, circumference, revolution, period, frequency.
• Centripetal Acceleration: Change in direction is acceleration.
• Centripetal Force: Not a force itself but a role by others (tension, gravity).

Rotational Motion

• Angular Displacement: Change in angle (radians).
• Angular Velocity/Acceleration
  • Equations for rotational motion parallel to linear motion.
• Torque: Influence causing angular acceleration (N·m).

Rotational Inertia

• Resistance to acceleration (kg·m²).
• Equations for common shapes (ring, sphere, rod).
• Angular Momentum: Rotational inertia × Angular velocity.

Simple Harmonic Motion (SHM)

• Oscillates toward equilibrium; acceleration proportional to displacement.
• Pendulum and Spring: Period depends on length/mass, not amplitude.
• SHM Equations: Sinusoidal relationships.
• Energy of SHM: Exchanges between kinetic and potential; damping causes energy loss.