Energy and Work

Forms of Energy

Energy exists in various forms, each with distinct physical significance and applications in physics. Understanding these forms is essential for analyzing physical systems and solving problems related to energy conservation and transfer.

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

• Gravitational potential energy (U_g): Energy stored due to an object's position in a gravitational field, typically above the ground: 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.

• Chemical energy (E_{chem}): Energy stored in the bonds between molecules.

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

Physics Definition of Work

Work is a measure of energy transfer that occurs when a force acts upon an object to cause displacement. The amount of work done depends on the magnitude of the force, the displacement, and the angle between the force and displacement vectors.

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

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

• Units: Joules (J), where 1 J = 1 N·m.

• Example: Pulling a suitcase at an angle in an airport; only the horizontal component of force contributes to work in the direction of motion.

Example: Work Done by Lifting a Book

When lifting a book vertically at constant speed, two forces do work: the applied force and gravity.

• Work by applied force:

• Work by gravity: (since the force of gravity acts downward, opposite to displacement)

• Net work: The net work done on the book is zero if lifted at constant speed, as the work by the applied force is balanced by the work done by gravity.

• Example calculation: Lifting a 2.0 kg book by 1.5 m:

Example Problem: Work Done by a Shopper (Serway 7.1)

This problem involves calculating the work done by a force applied at an angle, considering friction and horizontal displacement.

• Given: Force of 35.0 N at 25.0° below horizontal, displacement of 50.0 m, friction balances applied force.

• Work calculation:

• Free-body diagram: Used to analyze forces acting on the cart.

• Friction: If friction force remains constant, the applied force does not change.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. It is directly proportional to the mass and the square of the velocity.

• Formula:

• Units: Joules (J)

• Derivation: From kinematic equations and Newton's second law, the work done on an object results in a change in its kinetic energy.

• Example: A car towed from initial speed to final speed ; the work done equals the change in kinetic energy.

Work-Kinetic Energy Theorem

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

• Statement:

• Alternate form:

• Implication: If an object starts with initial kinetic energy and net work is done on it, its final kinetic energy changes accordingly.

Example Problem: Car Accelerating

Calculating the final speed of a car given the work done and initial speed.

• Given: kg, m/s, J

• Solution: Use the work-kinetic energy theorem: Solve for :

• Example calculation: m/s

Example Problem: Friction and Stopping Distance

Finding the distance an object slides before stopping due to kinetic friction.

• Given: kg, N, m/s,

• Solution: Work done by friction equals the change in kinetic energy: Solve for :

• Example calculation: m

Kinetic Energy of Rotation

Rotational motion involves kinetic energy due to rotation, in addition to translational kinetic energy for rolling objects.

• Translational kinetic energy:

• Rotational kinetic energy: where I is the moment of inertia and ω is the angular velocity.

• Rolling objects: Both forms of kinetic energy are present.

Example Problem: Rotational Kinetic Energy of a Ball

Calculating the rotational kinetic energy of a rolling ball just before it hits the floor.

• Given: g kg, m/s, m

• For a solid sphere:

• Rotational kinetic energy: For rolling without slipping,

• Example calculation: Substitute values to find just before impact.

Summary Table: Forms of Energy

Additional info: Some example problems and handwritten solutions were included to illustrate the application of work and energy principles in real-world scenarios. The notes cover all major aspects of Chapter 10: Energy and Work, including definitions, formulas, and problem-solving strategies.