Mechanics and Energy Study Notes

Definition: Branch of physics dealing with the motion of objects and the forces acting upon them.
Key Concepts:
- Kinematics: Describes motion (displacement, velocity, acceleration) without considering forces.
  - Equations of Motion:
    - ( v = u + at )
    - ( s = ut + \frac{1}{2}at^2 )
    - ( v^2 = u^2 + 2as )
- Dynamics: Examines forces and their effect on motion.
  - Newton's Laws of Motion:
    1. An object at rest stays at rest; an object in motion stays in motion unless acted upon.
    2. ( F = ma ) (Force equals mass times acceleration).
    3. For every action, there is an equal and opposite reaction.
- Types of Motion:
  - Linear, rotational, oscillatory.

Definition: The capacity to do work; it exists in various forms and can be converted from one form to another.
Forms of Energy:
- Kinetic Energy (KE): Energy of motion.
  - Formula: ( KE = \frac{1}{2}mv^2 )
- Potential Energy (PE): Stored energy based on position or configuration.
  - Gravitational Potential Energy: ( PE = mgh )
  - Elastic Potential Energy: ( PE = \frac{1}{2}kx^2 ) (for springs, where ( k ) is the spring constant and ( x ) is the displacement).
- Mechanical Energy: The sum of kinetic and potential energy in a system.

Principle: Energy cannot be created or destroyed, only transformed from one form to another.
Mechanical Energy Conservation: In a closed system with no external forces, the total mechanical energy remains constant.

Definition: The process of energy transfer when a force is applied to an object causing displacement.
Formula: ( W = Fd \cos(\theta) )
- ( W ): Work done
- ( F ): Force applied
- ( d ): Displacement of the object
- ( \theta ): Angle between force and displacement direction
Units: Joules (J)

Definition: The rate at which work is done or energy is transferred.
Formula: ( P = \frac{W}{t} )
- ( P ): Power
- ( W ): Work done
- ( t ): Time taken
Units: Watts (W), where ( 1 \text{ W} = 1 \text{ J/s} )

Mechanics involves kinematics and dynamics, governed by Newton's laws.
Energy exists in kinetic and potential forms, and is governed by the conservation principle.
Work and power are key concepts in understanding energy transfer and efficiency in mechanical systems.

Mechanics focuses on object motion and the forces impacting them.
Kinematics analyzes motion parameters: displacement, velocity, and acceleration, excluding forces.
Equations of Motion provide foundational formulas for kinematics:
- ( v = u + at ): Final velocity calculation based on initial velocity, acceleration, and time.
- ( s = ut + \frac{1}{2}at^2 ): Distance calculation incorporating initial velocity and acceleration over time.
- ( v^2 = u^2 + 2as ): Relates final and initial velocities with acceleration and distance.
Dynamics studies forces and their influence on motion.
Newton's Laws of Motion:
- First Law: Objects remain at rest or in uniform motion unless acted upon.
- Second Law: ( F = ma ) connects force, mass, and acceleration.
- Third Law: Every action has an equal and opposite reaction.
Recognizable motion types include linear, rotational, and oscillatory movement.

Energy is defined as a system's capacity to perform work, available in various forms, transformable across types.
Kinetic Energy (KE) represents the energy an object possesses due to its motion.
- Calculated using ( KE = \frac{1}{2}mv^2 ), where ( m ) is mass and ( v ) is velocity.
Potential Energy (PE) is energy stored based on an object's position or configuration.
- Gravitational Potential Energy uses ( PE = mgh ) (mass, gravitational acceleration, height).
- Elastic Potential Energy for springs is given by ( PE = \frac{1}{2}kx^2 ) (where ( k ) is spring constant and ( x ) is displacement).
Mechanical Energy is the cumulative energy in a system, summing kinetic and potential energy.

The conservation principle asserts that energy is neither created nor destroyed but can change forms.
In a closed system without external forces, total mechanical energy remains unchanged over time.

Work describes energy transfer via force applied across displacement.
Work formula is ( W = Fd \cos(\theta) ), where:
- ( W ): Work done,
- ( F ): Applied force,
- ( d ): Displacement,
- ( \theta ): Angle between the force and displacement direction.
Work is measured in Joules (J).

Power quantifies the rate at which work is performed or energy is transferred.
Power is calculated with ( P = \frac{W}{t} ), where:
- ( P ): Power,
- ( W ): Work done,
- ( t ): Time taken.
Power is expressed in Watts (W), where ( 1 \text{ W} = 1 \text{ J/s} ).