One-Dimensional Kinematics

Displacement, Velocity, and Acceleration

One-dimensional kinematics deals with the motion of objects along a straight line. The primary quantities are displacement, velocity, and acceleration.

• Displacement is the change in position of an object:

• Velocity is the rate of change of displacement:

• Acceleration is the rate of change of velocity:

Example: If a car accelerates from rest at for , its final velocity is .

Graphical Analysis

• Velocity-time graphs: The slope gives acceleration; the area under the curve gives displacement.

• Position-time graphs: The slope gives velocity.

Example: A straight line on a velocity-time graph indicates constant acceleration.

Vectors and Two-Dimensional Kinematics

Vector Addition and Components

Vectors have both magnitude and direction. They can be resolved into components and added using the parallelogram or triangle method.

• Component form:

• Magnitude:

• Direction:

Example: If has components (x) and (y), .

Projectile Motion

• Horizontal and vertical motions are independent.

• Horizontal displacement:

• Vertical displacement:

Example: A stone thrown horizontally from a building will follow a parabolic path due to gravity.

Forces and Newton's Laws

Newton's Laws of Motion

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

• Second Law:

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

Example: If three forces act on a particle, the resultant is found by vector addition.

Friction and Tension

• Frictional force:

• Tension: The force transmitted through a string, rope, or cable when pulled tight.

Example: If a block slides on a rough surface, the frictional force opposes motion.

Work, Energy, and Power

Work and Kinetic Energy

• Work:

• Kinetic Energy:

• Work-Energy Theorem:

Example: A constant force applied to an object changes its kinetic energy.

Potential Energy and Conservation of Energy

• Gravitational Potential Energy:

• Elastic Potential Energy:

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

Example: A pendulum converts potential energy to kinetic energy as it swings.

Momentum and Collisions

Linear Momentum

• Momentum:

• Impulse:

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

Example: In a collision, the total momentum before equals the total momentum after.

Rotational Motion and Angular Momentum

Rotational Kinematics

• Angular displacement: (radians)

• Angular velocity:

• Angular acceleration:

• Relationship to linear quantities: ,

Example: A disk rotating with constant angular acceleration increases its angular velocity linearly with time.

Moment of Inertia and Torque

• Moment of inertia:

• Torque:

• Rotational analog of Newton's second law:

Example: A force applied tangentially to a disk produces a torque and angular acceleration.

Angular Momentum

• Angular momentum:

• Conservation of angular momentum: (if no external torque)

Example: A figure skater spins faster when pulling in their arms due to conservation of angular momentum.

Equilibrium and Statics

Conditions for Equilibrium

• The sum of all forces and the sum of all torques on a body must be zero for equilibrium.

• Translational equilibrium:

• Rotational equilibrium:

Example: A beam supported at two points with weights hanging from it can be analyzed using equilibrium conditions.

Circular Motion

Centripetal Force and Acceleration

• Centripetal acceleration:

• Centripetal force:

Example: A car turning in a circle requires friction to provide the necessary centripetal force.

Energy in Rotational Motion

Rotational Kinetic Energy

• Rotational kinetic energy:

Example: A rolling cylinder has both translational and rotational kinetic energy.

Sample Table: Comparison of Translational and Rotational Quantities

Additional info:

• This study guide is based on a cumulative exam covering introductory physics topics, including kinematics, dynamics, energy, momentum, rotational motion, equilibrium, and circular motion.

• Students should be familiar with interpreting graphs, solving vector problems, and applying conservation laws.