Kinematics

Acceleration and Types of Accelerated Motion

Acceleration is a fundamental concept in kinematics, describing how the velocity of an object changes with time. Understanding the different types of accelerated motion is essential for analyzing physical systems.

• Definition of Acceleration: Acceleration is the rate of change of velocity with respect to time. It is a vector quantity.

• Formula:

• Types of Accelerated Motion: Includes uniform acceleration (constant), non-uniform acceleration (variable), and centripetal acceleration (in circular motion).

• Projectile Motion: In 2-D projectile motion, the horizontal component of velocity remains constant (if air resistance is neglected), while the vertical component changes due to gravity.

• Velocity Components: For an initial speed at angle :

  • Horizontal: (constant)

  • Vertical: (changes with time)

• Acceleration Components: Only the vertical component is affected by gravity:

Example: A ball thrown at 30° above the horizontal will have a constant horizontal velocity and a vertical velocity that decreases due to gravity until it reaches its peak, then increases in the negative direction as it falls.

Dynamics & Friction

Newton's Laws and Applications

Newton's laws of motion are the foundation for understanding dynamics. Friction is a force that opposes motion between surfaces.

• Newton's Laws:

  • First Law: An object remains at rest or in uniform motion unless acted upon by a net force.

  • Second Law:

  • Third Law: For every action, there is an equal and opposite reaction.

• Static Friction: The force that prevents an object from moving when a force is applied. Maximum static friction:

• Application: To prevent vertical motion (sliding) of an object pressed against a vertical surface, calculate the horizontal force using the coefficient of static friction.

• Elevator Problem: The normal force in an accelerating elevator is if accelerating upward, if downward.

Example: A 5 kg object pressed against a wall with requires a horizontal force such that to prevent sliding.

Conservation of Mechanical Energy & Work

Energy Transformations and Work

Mechanical energy is conserved in the absence of non-conservative forces. Work is the process of energy transfer by force.

• Kinetic Energy:

• Potential Energy (Gravity):

• Conservation of Energy: remains constant if only conservative forces act.

• Work-Energy Theorem:

• Work by Non-Conservative Forces:

• Air Resistance: The work done by air resistance can be found from the change in mechanical energy.

• Friction: Work done by friction:

Example: An object thrown upward loses kinetic energy and gains potential energy until it reaches its maximum height.

Conservation of Momentum & Collisions

Types of Collisions and Momentum Conservation

Momentum is conserved in isolated systems. Collisions are classified as elastic or inelastic.

• Momentum:

• Conservation of Momentum:

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Completely Inelastic Collision: Objects stick together after collision; only momentum is conserved.

• Final Velocity (Inelastic):

Example: Two carts collide and stick together; their combined velocity is found using conservation of momentum.

Rotational Motion, Torque & Gravity

Rotational Dynamics and Gravitational Forces

Rotational motion involves angular quantities. Torque causes rotation, and gravity acts between masses.

• Torque:

• Moment of Inertia: depends on mass distribution.

• Angular Momentum:

• Conservation of Angular Momentum:

• Gravitational Force:

• Balancing Gravitational Forces: To find a point where forces cancel, set and solve for distance.

Example: If a figure skater pulls in her arms, her moment of inertia decreases and her angular speed increases.

Simple Harmonic Motion & Sound

Oscillations and Wave Properties

Simple harmonic motion describes systems like springs and pendulums. Waves transfer energy through oscillations.

• SHM Force:

• Acceleration:

• Kinetic Energy:

• Wave Speed:

• Standing Waves: Nodes and antinodes form at fixed points; speed can be calculated from frequency and wavelength.

Example: A string fixed at both ends vibrates at its fundamental frequency; the speed of the wave is .

Equilibrium, Elasticity & Fluids

Statics, Elasticity, and Fluid Mechanics

Equilibrium occurs when net forces and torques are zero. Elasticity describes how materials deform, and fluid mechanics involves buoyancy and density.

• Torque Equilibrium:

• Balance Condition:

• Density:

• Buoyant Force:

• Apparent Weight: Difference between weight in air and in water gives buoyant force, used to find density.

Example: A see-saw is balanced when the product of mass and distance from the pivot is equal for both sides.

Thermodynamics

First Law and Ideal Gas Law

Thermodynamics studies energy transfer. The first law relates internal energy, heat, and work. The ideal gas law describes the behavior of gases.

• First Law:

• Ideal Gas Law:

• Constant Pressure Process:

• Internal Energy Change: For a monatomic ideal gas,

Example: If a gas expands at constant pressure, the heat added equals the increase in internal energy plus the work done by the gas.

Additional info: Academic context and formulas have been expanded for completeness and clarity. This guide covers the main topics listed in the study guide and provides essential equations and examples for exam preparation.