Units and Measurements

Dimensional Analysis

Dimensional analysis is a method used to check the consistency of equations and to derive relationships between physical quantities. Each physical quantity can be expressed in terms of fundamental dimensions: Length (L), Mass (M), and Time (T).

• Example: The dimension of force is .

• Formula:

• Unit conversions: Always ensure units are consistent when solving problems.

One-Dimensional Kinematics

Equations of Motion

Kinematics describes the motion of objects without considering the causes. In one dimension, the following equations are commonly used:

• Displacement:

• Velocity:

• Example: A particle moves along the x-axis according to ; its acceleration is m/s2.

Graphical Analysis

Graphs of position vs. time and velocity vs. time are useful for visualizing motion.

• Constant velocity: Straight line on position-time graph.

• Constant acceleration: Parabolic curve on position-time graph.

• Example: The slope of a velocity-time graph gives acceleration.

Vectors and Vector Operations

Vector Properties

Vectors have both magnitude and direction. Common operations include addition, subtraction, and multiplication by a scalar.

• Vector addition: Use the parallelogram or triangle method.

• Components: ,

• Example: A vector has components , ; its magnitude is .

Two-Dimensional Kinematics

Projectile Motion

Projectile motion involves two independent motions: horizontal and vertical. The horizontal motion has constant velocity, while the vertical motion has constant acceleration due to gravity.

• Horizontal displacement:

• Vertical displacement:

• Example: A ball is thrown horizontally from a height; time to hit the ground depends only on the vertical motion.

Newton's Laws of Motion

First, Second, and Third Laws

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

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

• Second Law:

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

• Example: Calculating the acceleration of a system using .

Free-Body Diagrams

Free-body diagrams help visualize all the forces acting on an object.

• Identify all forces: Gravity, normal force, friction, tension, etc.

• Sum forces: Use vector addition to find net force.

Friction and Circular Motion

Friction

Friction is a force that opposes motion between two surfaces in contact.

• Static friction: Prevents motion up to a maximum value.

• Kinetic friction: Acts during motion;

Circular Motion

Objects moving in a circle experience a centripetal acceleration directed toward the center.

• Centripetal acceleration:

• Centripetal force:

Work, Energy, and Power

Work and Kinetic Energy

Work is done when a force causes displacement. Kinetic energy is the energy of motion.

• Work:

• Kinetic energy:

Potential Energy

Potential energy is stored energy due to position.

• Gravitational potential energy:

Sample Table: Dimensional Analysis

Additional info:

• Some questions involve interpreting graphs and diagrams, such as velocity-time graphs and free-body diagrams.

• Problems cover basic kinematics, vector operations, Newton's laws, friction, and energy concepts.

• Students should be familiar with SI units and dimensional analysis for exam success.