Fundamental Concepts and Problem Solving in College Physics

Motion in One Dimension

Kinematic Equations and Concepts

Motion in one dimension describes the movement of objects along a straight line, characterized by position, velocity, and acceleration. The following equations are fundamental for analyzing such motion:

• Displacement: $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• Velocity: $v = v_0 + a t$

• Average Velocity: $\bar{v} = \frac{x - x_0}{t}$

Example: If a car starts from rest and accelerates at $2\,\mathrm{m/s^2}$ for $5$seconds, its final velocity is$v = 0 + 2 \times 5 = 10\,\mathrm{m/s}$.

Circular Motion and Centripetal Acceleration

Uniform Circular Motion

Objects moving in a circle at constant speed experience a centripetal acceleration directed toward the center of the circle. The magnitude of this acceleration is given by:

• Centripetal Acceleration: $a_c = \frac{v^2}{r}$

• Where: $v$ is the speed, $r$ is the radius of the circle.

Example: A ball moving at $4\,\mathrm{m/s}$ in a circle of radius $2\,\mathrm{m}$ has $a_c = \frac{16}{2} = 8\,\mathrm{m/s^2}$.

Quadratic Equation in Physics

Solving for Unknowns

The quadratic equation is often used to solve for time or position in kinematic problems:

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Used to find the time when an object reaches a certain position under constant acceleration.

Spring Force and Hooke's Law

Elasticity and Oscillations

Springs obey Hooke's Law, which relates the force exerted by a spring to its displacement:

• Hooke's Law: $F = -kx$

• Where: $k$ is the spring constant, $x$ is the displacement from equilibrium.

Example: A spring with $k = 200\,\mathrm{N/m}$ stretched by $0.1\,\mathrm{m}$ exerts $F = -20\,\mathrm{N}$.

Gravitational Force

Newton's Law of Universal Gravitation

Gravitational force between two masses is given by:

• Gravitational Force: $F = G \frac{m_1 m_2}{r^2}$

• Where: $G$ is the gravitational constant, $m_1$ and $m_2$ are masses, $r$ is the distance between centers.

Example: The force between two $1\,\mathrm{kg}$ masses $1\,\mathrm{m}$ apart is $F = 6.67 \times 10^{-11}\,\mathrm{N}$.

Center of Mass

Definition and Calculation

The center of mass of a system is the weighted average of the positions of all the objects in the system:

• Center of Mass: $x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$

Example: For two masses $m_1$ at $x_1$ and $m_2$ at $x_2$, $x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$.

Rotational Dynamics

Rotational Form of Newton's Second Law

For rotational motion, Newton's second law is expressed as:

• Rotational Newton's Second Law: $\tau = I \alpha$

• Where: $\tau$ is torque, $I$ is moment of inertia, $\alpha$ is angular acceleration.

Example: A disk with $I = 0.5\,\mathrm{kg\cdot m^2}$ and $\tau = 1\,\mathrm{N\cdot m}$ has $\alpha = 2\,\mathrm{rad/s^2}$.

Rolling Without Slipping

Relationship Between Linear and Angular Speed

For rolling objects, the point of contact does not slip, and the following relationship holds:

• Rolling Condition: $v = r \omega$

• Where: $v$ is linear speed, $r$ is radius, $\omega$ is angular speed.

Example: A wheel of radius $0.5\,\mathrm{m}$ rotating at $2\,\mathrm{rad/s}$ has $v = 1\,\mathrm{m/s}$.

Tensile and Compressive Stress

Stress and Strain in Materials

Stress is the force per unit area applied to a material, and strain is the resulting deformation:

• Stress: $\sigma = \frac{F}{A}$

• Strain: $\epsilon = \frac{\Delta L}{L_0}$

• Young's Modulus: $Y = \frac{\sigma}{\epsilon}$

Example: A steel wire with $A = 1\,\mathrm{mm^2}$ and $F = 100\,\mathrm{N}$ has $\sigma = 100\,\mathrm{MPa}$.

Problem Solving: Sample Questions and Applications

Multiple Choice and Calculation Problems

The following sample problems illustrate the application of the above concepts:

• Centripetal Force Direction: Always directed toward the center of the circular path.

• Gravitational Force Change: If the distance between two masses is doubled and one mass is doubled, the new force is $F' = \frac{2F}{4} = \frac{F}{2}$.

• Center of Mass Calculation: For masses at different positions, use $x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$.

• Angular Acceleration: $\alpha = \frac{\tau}{I}$, where $\tau$ is net torque and $I$ is moment of inertia.

• Spring Stretch: $F = kx$, solve for $x$ given $F$ and $k$.

• Static Friction: $f_s = \mu_s N$, where $\mu_s$ is the coefficient of static friction and $N$ is the normal force.

• Torque: $\tau = r F \sin \theta$, where $r$ is lever arm, $F$ is force, $\theta$ is angle.

Sample Table: Moon Data Comparison

The following table compares hypothetical moon data for gravitational calculations:

Moon	Mass	Radius	Orbital Radius	Orbital Period
Moon A	$4.0 \times 10^{20}$ kg	unknown	$2.0 \times 10^8$ m	$4.0 \times 10^5$ s
Moon B	$1.5 \times 10^{20}$ kg	$2.0 \times 10^5$ m	$3.0 \times 10^8$ m	unknown

Main Purpose: This table is used to compare the properties of two moons for gravitational and orbital calculations.

Additional Problem Types

Rotational Kinematics

• Angular speed and acceleration: $\omega = \omega_0 + \alpha t$

• Angular displacement: $\theta = \omega_0 t + \frac{1}{2} \alpha t^2$

Statics and Equilibrium

• Sum of forces and torques must be zero for equilibrium.

• Applications include ladder problems, signs suspended by wires, and playground teeter-totters.

Stress and Strain in Wires

• Young's modulus relates stress and strain: $Y = \frac{F/A}{\Delta L / L_0}$

• Used to calculate tension in wires and deformation under load.

Friction and Inclined Planes

• Static friction prevents sliding: $f_s \leq \mu_s N$

• Applications include pushing refrigerators and ladders against walls.

Summary Table: Key Equations

Concept	Equation
Kinematics	$x = x_0 + v_0 t + \frac{1}{2} a t^2$
Centripetal Acceleration	$a_c = \frac{v^2}{r}$
Gravitational Force	$F = G \frac{m_1 m_2}{r^2}$
Hooke's Law	$F = -kx$
Torque	$\tau = r F \sin \theta$
Rotational Newton's Law	$\tau = I \alpha$
Stress	$\sigma = \frac{F}{A}$
Young's Modulus	$Y = \frac{\sigma}{\epsilon}$

Additional info: These notes expand on the original questions and diagrams by providing definitions, formulas, and context for each concept, ensuring a self-contained study guide for exam preparation.