Mechanics: Kinematics and Dynamics

One- and Two-Dimensional Kinematics

Kinematics describes the motion of objects without considering the causes of motion. It includes the analysis of position, velocity, and acceleration in one or more dimensions.

• Position, Velocity, and Acceleration: The position of an object as a function of time is given by , velocity by , and acceleration by .

• Equations of Motion (Constant Acceleration):

• Projectile Motion: Involves two-dimensional motion under gravity, with horizontal and vertical components analyzed separately.

Example: A satellite in elliptical orbit (see Problem 1) requires analysis of forces and velocities at different points using conservation of angular momentum and energy.

Forces and Newton's Laws

Newton's Laws describe the relationship between forces and motion.

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

• Newton's Second Law:

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

• Gravitational Force:

Example: The torsion balance experiment (see Problem 2) is used to measure the gravitational constant by analyzing the equilibrium and oscillations of a suspended system.

Energy, Work, and Power

Work and Energy

Work is the transfer of energy by a force acting over a distance. Energy can be kinetic or potential.

• Work:

• Kinetic Energy:

• Potential Energy (Gravity):

• Conservation of Mechanical Energy: (if no non-conservative forces)

Example: The rolling disk on a track (see Problem 5) involves conservation of energy and rotational motion to determine speeds and heights.

Momentum and Collisions

Linear and Angular Momentum

Momentum is a measure of an object's motion and is conserved in isolated systems.

• Linear Momentum:

• Impulse:

• Conservation of Momentum:

• Angular Momentum:

• Conservation of Angular Momentum: (if no external torque)

Example: Collisions and rotational motion are analyzed in oscillators and rolling objects (see Problems 4 and 5).

Rotational Dynamics and Inertia

Rotational Kinematics and Dynamics

Rotational motion is described by angular displacement, velocity, and acceleration, analogous to linear motion.

• Angular Displacement: (in radians)

• Angular Velocity:

• Angular Acceleration:

• Moment of Inertia: (depends on mass distribution)

• Rotational Kinetic Energy:

• Torque:

• Newton's Second Law for Rotation:

Table: Rotational Inertia for Common Shapes

Example: The rolling disk and torsion balance problems require calculation of moments of inertia and application of rotational dynamics.

Fluids and Fluid Dynamics

Fluid Statics and Dynamics

Fluids at rest and in motion are governed by principles of pressure, buoyancy, and flow.

• Pressure:

• Hydrostatic Pressure:

• Continuity Equation: (for incompressible fluids)

• Bernoulli's Equation:

Example: The manometer and flow tube (see Problem 3) require application of Bernoulli's equation and the continuity equation to determine velocities and pressure differences.

Oscillations and Waves

Simple Harmonic Motion (SHM)

SHM describes systems where the restoring force is proportional to displacement and directed toward equilibrium.

• Equation of Motion:

• Angular Frequency:

• Period:

• Mechanical Energy:

Example: The block-spring oscillator (see Problem 4) involves determining frequency, position, velocity, and energy using SHM equations.

Waves and Standing Waves

Waves transfer energy through oscillations. Standing waves form when two waves of the same frequency and amplitude travel in opposite directions and interfere.

• Wave Speed:

• Standing Wave Frequencies (Open Tube):

• Standing Wave Frequencies (Closed Tube): (odd only)

Example: The standing wave in a tube (see Problem 1, Part 2) requires analysis of pressure nodes and antinodes to determine frequency and tube configuration.

Additional Mathematical Tools

Useful Integrals and Trigonometric Identities

• Common Integrals:

• Trigonometric Identities:

• Taylor Expansion:

Summary Table: Key Equations by Topic

Additional info: This study guide is based on a final exam covering core topics in college-level introductory physics, including mechanics, fluids, oscillations, and waves. The equation sheets and problems provided are representative of standard Physics 7A (mechanics and waves) content.