Ch.10 Energy and Work

Forms of Energy

Energy exists in various forms, each with distinct physical meaning and applications. Understanding these forms is essential for analyzing physical systems and solving problems in mechanics.

• Kinetic energy (K): The energy of motion. For an object of mass m moving at speed v, .

• Gravitational potential energy (U_g): Energy stored due to an object's position in a gravitational field, typically above Earth's surface. , where h is the height above a reference point.

• Elastic or spring potential energy (U_s): Energy stored when a spring or other elastic object is stretched or compressed. , where k is the spring constant and x is the displacement from equilibrium.

• Thermal energy (E_{th}): The sum of the kinetic and potential energies of all the molecules in an object. Related to temperature and internal energy.

• Chemical energy (E_{chem}): Energy stored in the bonds between molecules, released or absorbed during chemical reactions.

• Nuclear energy (E_{nuclear}): Energy stored in the mass of the nucleus of an atom, released in nuclear reactions.

Example: When a ball is dropped, gravitational potential energy is converted into kinetic energy as it falls.

Physics Definition of Work

Work is a measure of energy transfer that occurs when a force acts on an object and causes displacement. The direction of the force relative to the displacement is crucial.

• Definition: Work done by a constant force F over a displacement d at angle θ is given by:

• Only the component of force in the direction of displacement does work.

• Always specify which force is doing the work.

Example: Pulling a suitcase at an angle in an airport involves calculating the work done by the pulling force.

Example: Work Done by Lifting a Book

When lifting a book at constant speed, the work done by the applied force and gravity can be calculated.

• Work by applied force:

• Work by gravity:

• The net work done on the book is zero if lifted at constant speed (work by lifting force + work by gravity).

Example: Lifting a 2.0 kg book by 1.5 m requires J.

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object. It is directly proportional to the mass and the square of the velocity.

• Formula:

• Units: Joules (J)

• Derived from kinematic equations and Newton's second law.

Example: A car towed from initial speed to final speed has a change in kinetic energy equal to the net work done on it.

Work-Kinetic Energy Theorem

The work-kinetic energy theorem relates the net work done on an object to its change in kinetic energy.

• Statement: The net work done by all forces is equal to the change in kinetic energy.

Or equivalently:

• If an object starts with initial speed, and work is done on it, its final speed will change accordingly.

Example: If a car accelerates and work is done, its final kinetic energy increases.

Kinetic Energy of Rotation

Objects can possess kinetic energy due to both translational and rotational motion. For rotating bodies, rotational kinetic energy must be considered.

• Translational kinetic energy:

• Rotational kinetic energy:

• Where I is the moment of inertia and ω is the angular velocity.

• Rolling objects (e.g., balls) have both translational and rotational kinetic energy.

Example: A rolling ball has .

Worked Example Problems

Applying the concepts of work and energy to solve practical problems.

• Example 1: Work done by a shopper pushing a cart

  • Given force, angle, and distance, calculate work using .

  • Consider friction and direction of applied force.

• Example 2: Car accelerating with given work

  • Given mass, initial speed, and work done, use the work-kinetic energy theorem to find final speed:

  • Solve for .

• Example 3: Object sliding to a stop due to friction

  • Given mass, initial speed, and friction force, use work-energy theorem to find stopping distance.

  • Work done by friction: (negative, as friction opposes motion).

  • Set and solve for .

• Example 4: Rotational kinetic energy of a rolling ball

  • Given mass, speed, and height, calculate rotational kinetic energy just before impact.

  • Use and relate and for rolling without slipping: .

Summary Table: Forms of Energy

Additional info: The notes include both typed slides and handwritten solutions, providing worked examples and step-by-step problem solving for key concepts in energy and work.