Work and Kinetic Energy

Introduction to Energy

Energy is a central concept in physics, representing the ability to cause change or do work. In this chapter, we explore how energy is transferred and transformed, focusing on the relationship between work and kinetic energy.

• System and Environment: A system is the object or group of objects under study, while the environment is everything else. Energy can be transferred between the system and the environment or transformed within the system.

• Forms of Energy:

  • Kinetic Energy (K): Energy of motion.

  • Potential Energy (U): Energy associated with position.

  • Thermal Energy (E_{th}): Energy due to random motion of atoms.

• Units: Energy is measured in joules (J).

Key Definitions

• Kinetic Energy (K): The energy an object possesses due to its motion. For a particle of mass m moving at speed v:

• Potential Energy (U): Stored energy due to position, such as gravitational potential energy.

• Thermal Energy (E_{th}): The sum of microscopic kinetic and potential energies of all atoms and bonds in an object.

• Work (W): The process of energy transfer to or from a system by mechanical means (pushing or pulling). Work is done when a force causes displacement.

• Power (P): The rate at which energy is transferred or transformed.

The Energy Principle

Statement of the Principle

The energy principle states that the change in a system's energy equals the work done on it by external forces:

• For kinetic energy:

• For systems with friction:

Energy can be transformed within a system (e.g., potential to kinetic) or transferred between system and environment (work or heat).

Work and Kinetic Energy for a Single Particle

Work Done by a Constant Force

When a constant force acts parallel to the direction of motion:

• (if force and displacement are parallel)

• For an angle between force and displacement:

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

Signs of Work

• Positive Work: Force and displacement in the same direction (object speeds up).

• Negative Work: Force and displacement in opposite directions (object slows down).

• Zero Work: Force is perpendicular to displacement (e.g., uniform circular motion).

Work Done by a Variable Force

If the force varies in magnitude or direction, work is calculated as:

• Graphically, this is the area under the force vs. displacement curve.

Dot Product and Work

• The dot product of two vectors and is: where is the angle between the vectors.

• In component form:

• Work is the dot product of force and displacement vectors.

Examples and Applications

Example: Pulling a Suitcase

• A 20 N force pulls a suitcase 100 m at a 45° angle.

• Work done: J

Example: Launching a Rocket

• Rocket mass: 150,000 kg; thrust: N; height: 500 m.

• Work by thrust:

• Work by gravity:

• Net work changes the rocket's kinetic energy.

Example: Skier on a Slope

• 70 kg skier, 50 m slope at 10°, frictionless.

• Work by gravity: (angle between force and displacement must be considered).

Example: Car Pulled by Variable Force

• 1500 kg car towed 200 m; tension varies with distance.

• Work found by area under force vs. distance graph.

• Final speed from (since ).

Restoring Forces and Hooke's Law

Hooke's Law

A restoring force returns a system to equilibrium. For springs:

• is the spring constant; is displacement from equilibrium.

Work Done by a Spring

• Work done as the spring moves from to :

Stick-Slip Motion

• Occurs when friction alternately prevents and allows motion (e.g., earthquakes).

• Elastic energy builds up until it overcomes friction, causing sudden movement.

Dissipative Forces and Thermal Energy

Dissipative Forces

• Friction and drag convert macroscopic kinetic energy into thermal energy.

• Thermal energy always increases due to dissipative forces.

Example: Crate Pulled Across Floor

• 10 kg crate, 30 N tension, 3.0 m displacement, .

• Friction force:

• Increase in thermal energy:

• Work by tension:

• Change in kinetic energy:

Power

Definition and Units

• Power is the rate of energy transfer or transformation.

• SI unit: watt (W), where .

• English unit: horsepower (hp), .

Calculating Power

• For constant force and velocity: (if force and velocity are parallel)

Example: Lifting a Motor

• 35 kg motor lifted 3.0 m in 8.0 s.

• Work:

• Power: or

Summary Table: Key Equations and Concepts

Additional info:

• Thermal energy is only defined for extended objects, not for point particles.

• In all examples, air resistance is neglected unless otherwise stated.

• For all work calculations, ensure the correct angle between force and displacement is used.