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:

• Velocity:

• Average Velocity:

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

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:

• Where: is the speed, is the radius of the circle.

Example: A ball moving at in a circle of radius has .

Quadratic Equation in Physics

Solving for Unknowns

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

• Quadratic Formula:

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:

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

Example: A spring with stretched by exerts .

Gravitational Force

Newton's Law of Universal Gravitation

Gravitational force between two masses is given by:

• Gravitational Force:

• Where: is the gravitational constant, and are masses, is the distance between centers.

Example: The force between two masses apart is .

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:

Example: For two masses at and at , .

Rotational Dynamics

Rotational Form of Newton's Second Law

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

• Rotational Newton's Second Law:

• Where: is torque, is moment of inertia, is angular acceleration.

Example: A disk with and has .

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:

• Where: is linear speed, is radius, is angular speed.

Example: A wheel of radius rotating at has .

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:

• Strain:

• Young's Modulus:

Example: A steel wire with and has .

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 .

• Center of Mass Calculation: For masses at different positions, use .

• Angular Acceleration: , where is net torque and is moment of inertia.

• Spring Stretch: , solve for given and .

• Static Friction: , where is the coefficient of static friction and is the normal force.

• Torque: , where is lever arm, is force, is angle.

Sample Table: Moon Data Comparison

The following table compares hypothetical moon data for gravitational calculations:

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:

• Angular displacement:

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:

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

Friction and Inclined Planes

• Static friction prevents sliding:

• Applications include pushing refrigerators and ladders against walls.

Summary Table: Key Equations

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.