Work and Kinetic Energy

Introduction

This chapter introduces the fundamental concepts of work, kinetic energy, and power in physics. These concepts are essential for understanding how forces transfer energy to objects and how energy is conserved in physical systems. The chapter also explains the limitations of Newton's laws and the necessity of energy-based approaches for certain problems.

• Work is the process by which a force causes displacement of an object.

• Kinetic energy is the energy an object possesses due to its motion.

• Power is the rate at which work is done.

Work

Work is done by a force when it causes an object to move. The amount of work depends on the magnitude of the force, the displacement of the object, and the angle between the force and displacement vectors.

• Definition: Work is done when a force causes displacement of an object.

• Formula for constant force in the direction of displacement: where W is work, F is the constant force, and s is the displacement.

• General formula (force at an angle): where is the angle between the force and displacement vectors.

• Vector notation (dot product):

• Units: The SI unit of work is the joule (J), where .

Types of Work

• Positive Work: Force component is in the direction of displacement.

• Negative Work: Force component is opposite to the direction of displacement.

• Zero Work: Force is perpendicular to displacement, or there is no displacement.

Example:

• Pushing a car: People exert a force on a car, causing it to move, thus doing work.

• Weightlifter holding a barbell stationary: No work is done since there is no displacement.

Units of Work

The SI unit of work is the joule (J), named after James Prescott Joule. Work is calculated as force times distance.

• 1 joule = 1 newton × 1 meter

• If you lift a 1 N object by 1 m, you do 1 J of work.

Kinetic Energy

Kinetic energy is the energy of motion. It depends on the mass and speed of the object, and is always a scalar quantity.

• Formula: where K is kinetic energy, m is mass, and v is speed.

• Kinetic energy is always non-negative and is zero only when the object is at rest.

• The SI unit of kinetic energy is the joule (J).

• Kinetic energy does not depend on the direction of motion, only on speed and mass.

Examples:

• Two objects with the same mass and speed but different directions have the same kinetic energy.

• Doubling the mass at constant speed doubles the kinetic energy.

• Doubling the speed at constant mass quadruples the kinetic energy.

The Work-Energy Theorem

The work-energy theorem states that the net work done on a particle equals the change in its kinetic energy.

• Formula: where is final kinetic energy and is initial kinetic energy.

• If , the particle speeds up; if , it slows down; if , its speed remains constant.

Work and Energy with Varying Forces

When forces are not constant, or the path is curved, work is calculated using integrals.

• Work by a varying force along the x-axis: where is the component of force along the x-axis, and , are initial and final positions.

• The area under the force vs. position graph represents the work done.

• For a spring: where is the spring constant and is the displacement.

Work-Energy Theorem for Curved Paths

For motion along a curved path under a varying force, work is calculated using a line integral:

• Formula: where is an infinitesimal displacement along the path.

Power

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

• Average power: where is work done over time interval .

• Instantaneous power:

• Power in terms of force and velocity: where is the velocity of the object.

• The SI unit of power is the watt (W), where .

• Another common unit is horsepower (hp), where .

Examples:

• Lifting a box slowly (over 5 s): if of work is done.

• Lifting the same box quickly (over 1 s): for the same work.

Summary Table: Work, Kinetic Energy, and Power

Additional info: The notes include expanded explanations and formulas for work with varying forces, work-energy theorem for curved paths, and practical examples for power calculations.