Fundamental Concepts in College Physics: Kinematics, Dynamics, Work, Energy, Momentum, Rotation, and Gravitation

Kinematics

Scalars vs. Vectors

Kinematics is the study of motion without considering its causes. It involves understanding quantities such as position, displacement, velocity, and acceleration.

• Scalar: A quantity with magnitude only (e.g., speed, distance).

• Vector: A quantity with both magnitude and direction (e.g., velocity, displacemen t).

• Position (x): Location of an object in space.

• Displacement (Δx): Change in position; a vector.

• Speed vs. Velocity: Speed is scalar; velocity is vector.

• Instantaneous vs. Average: Instantaneous values are at a specific moment; average values are over a time interval.

Example: If a car moves 100 m east, its displacement is 100 m east (vector), and its distance traveled is 100 m (scalar).

Unit Conversions and Significant Figures

Physics problems require correct units and significant figures for accuracy.

• Always include units in calculations.

• Convert units as needed (e.g., cm to m).

• Use the correct number of significant figures based on measurement precision.

I 1-D and 2-D Kinematics Problem Solving

Solving kinematics problems involves identifying knowns and unknowns, choosing the correct equations, and applying them systematically.

• 1-D Kinematics equations (for constant acceleration):

$v = v_0 + at$ $x = x_0 + v_0 t + \frac{1}{2} a t^2$ $v^2 = v_0^2 + 2a(x - x_0)$

• 2-D Kinematics: Separate motion into perpendicular axes (x and y).

• Projectile motion equations:

$a_y = -9.8\,\text{m/s}^2$ $x = v_{0x} t$ $y = v_{0y} t - \frac{1}{2} g t^2$

• Relative motion: Add/subtract velocity vectors as needed.

Dynamics

Forces and Free Body Diagrams

Dynamics studies the causes of motion, primarily forces. Free body diagrams help visualize all forces acting on an object.

• Types of Forces:

  • Weight (Gravity): $F_g = mg$

  • Tension: Force in a rope or cable.

  • Normal Force: Perpendicular to a surface.

  • Friction: Opposes motion; can be kinetic or static.

• Identify all forces and their directions.

• Draw vectors to represent forces.

Newton's Laws of Motion

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

• First Law (Inertia): An object at rest stays at rest, and an object in motion stays in motion unless acted upon by a net force.

• Second Law: The net force on an object equals mass times acceleration.

$\sum \vec{F} = m \vec{a}$

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

Example: If you push a wall, the wall pushes back with equal force.

Apparent Weight and Contact Forces

Apparent weight is the normal force exerted by a surface, which can differ from actual weight due to acceleration.

• In an accelerating elevator, apparent weight changes.

Work, Energy, and Power

Work

Work is the dot product of force and displacement. It is a scalar quantity.

$W = \vec{F} \cdot \vec{d} = Fd \cos \theta$

• Work is positive when force and displacement are in the same direction.

• Work is negative when force opposes displacement.

Energy

Energy is the ability to do work. Mechanical energy can be kinetic or potential.

• Kinetic Energy: $K = \frac{1}{2} m v^2$

• Potential Energy (gravitational): $U = mgh$

• Conservation of Energy: In a closed system, total energy is conserved.

$K_i + U_i + W = K_f + U_f$

Power

Power is the rate at which work is done or energy is transferred.

$P = \frac{dW}{dt}$

• Unit: Watt (W) = 1 Joule/second

Momentum and Collisions

Linear Momentum

Momentum is the product of mass and velocity. It is a vector quantity.

$\vec{p} = m \vec{v}$

• Conservation of Momentum: In a closed system, total momentum is conserved.

$\sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}}$

Impulse

Impulse is the change in momentum caused by a force acting over a time interval.

$\vec{J} = \Delta \vec{p} = \vec{F}_{\text{avg}} \Delta t$

Collisions

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

• Inelastic Collision: Only momentum is conserved; kinetic energy is not.

Example: Two billiard balls colliding and bouncing apart (elastic); a clay ball sticking to another (inelastic).

Rotational Motion

Torque and Rotational Equilibrium

Torque is the rotational analog of force. It causes objects to rotate.

$\tau = r F \sin \theta$

• Torque is a vector; direction given by the right-hand rule.

• Rotational equilibrium: $\sum \tau = 0$

Angular Kinematics

Angular variables describe rotational motion.

• Angular displacement: $\theta$

• Angular velocity: $\omega = \frac{d\theta}{dt}$

• Angular acceleration: $\alpha = \frac{d\omega}{dt}$

Moment of Inertia

Moment of inertia is the rotational analog of mass; it depends on mass distribution.

$I = \sum m_i r_i^2$

• For a point mass: $I = m r^2$

• For extended objects, sum over all mass elements.

Rotational Kinetic Energy

Rotational kinetic energy is the energy due to rotation.

$K_{\text{rot}} = \frac{1}{2} I \omega^2$

• Conserved when no net torque acts.

Angular Momentum

Angular momentum is the rotational analog of linear momentum.

$L = I \omega$

• Conserved in the absence of external torque.

Universal Gravitation

Newton's Law of Universal Gravitation

Newton's law describes the gravitational force between two masses.

$F = G \frac{m_1 m_2}{r^2}$

• G: Universal gravitational constant ($6.67 \times 10^{-11} \text{N} \cdot \text{m}^2 / \text{kg}^2$)

• Force is attractive and acts along the line joining the masses.

Gravitational Potential Energy

Gravitational potential energy for two masses separated by distance r:

$U = -G \frac{m_1 m_2}{r}$

Orbital Motion

Objects in orbit around a mass follow circular or elliptical paths.

• Orbital velocity for circular orbit:

$v_{\text{orb}} = \sqrt{\frac{GM}{r}}$

• Escape velocity (minimum speed to leave gravitational field):

$v_{\text{esc}} = \sqrt{\frac{2GM}{R}}$

Summary Table: Types of Forces

Force Type	Symbol	Description
Weight (Gravity)	$F_g$	Downward force due to gravity; $F_g = mg$
Tension	$T$	Force in a rope or cable
Normal Force	$N$	Perpendicular to surface
Friction (Kinetic/Static)	$f_k$, $f_s$	Opposes motion; $f_k$ for moving, $f_s$ for stationary

Additional info:

• Some equations and concepts were inferred and expanded for clarity and completeness.

• Examples and applications were added to illustrate key points.

• Table reconstructed from catalog of force types mentioned in the notes. I