Work, Energy, and Power

Work and Kinetic Energy

Work is the transfer of energy that occurs when a force is applied over a distance. The kinetic energy of an object is the energy it possesses due to its motion.

• Work (W): where is force, is displacement, and is the angle between force and displacement.

• Kinetic Energy (KE): where is mass and is velocity.

• Work-Energy Theorem: The net work done on an object equals its change in kinetic energy: .

• Example: To stop a moving object, the work done by friction equals its initial kinetic energy.

Potential Energy and Conservation of Energy

Potential energy is stored energy due to position. The law of conservation of energy states that energy cannot be created or destroyed, only transformed.

• Gravitational Potential Energy (PE): where is height above a reference point.

• Elastic Potential Energy (Spring): where is spring constant and is compression/stretch.

• Conservation of Mechanical Energy: (if no non-conservative forces like friction are present).

• Example: A dart gun uses spring potential energy to launch a dart upward, converting it to kinetic and then gravitational potential energy.

Power

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

• Power (P): where is time.

Forces and Motion

Newton's Laws and Friction

Newton's laws describe the relationship between forces and motion. Friction is a force that opposes motion between surfaces.

• Newton's Second Law:

• Kinetic Friction: where is the coefficient of kinetic friction and is the normal force.

• Example: Calculating work done by friction as a block moves over a rough surface.

Fluid Statics and Dynamics

Buoyancy and Archimedes' Principle

Buoyancy is the upward force exerted by a fluid on a submerged object. Archimedes' principle states that the buoyant force equals the weight of the displaced fluid.

• Buoyant Force (F_b):

• Floating Objects: An object floats if its average density is less than the fluid's density.

• Example: Comparing buoyant forces on blocks of different materials submerged in water.

Pressure in Fluids

Pressure in a fluid increases with depth and is given by the hydrostatic pressure equation.

• Hydrostatic Pressure: where is atmospheric pressure, is fluid density, is gravity, and is depth.

• Example: Calculating the thickness of an oil layer needed to achieve a certain pressure at the bottom of a tank.

Circular Motion and Springs

Uniform Circular Motion

Objects moving in a circle at constant speed experience a centripetal force directed toward the center.

• Centripetal Force:

• Example: Blocks connected by springs in circular motion, with spring force providing the centripetal force.

Springs and Hooke's Law

Springs obey Hooke's law, which relates the force exerted by a spring to its displacement.

• Hooke's Law:

• Spring Constant (k): A measure of the stiffness of a spring.

• Example: Determining the spring constant from the compression caused by a falling mass.

Applications and Problem Solving

Energy Changes in Vehicles

Changing the speed of a vehicle involves a change in kinetic energy, which can be calculated using the mass and velocity.

• Example: Calculating the energy needed to accelerate a 1600 kg vehicle from 15.0 m/s to 40.0 m/s.

Density and Floating Objects

The density of an object determines whether it will float or sink in a fluid. The fraction of volume above the surface can be found using the ratio of densities.

• Density:

• Fraction Above Surface:

• Example: A board floating in water with a known density; calculate the fraction above water.

Tables

Comparison of Buoyant Forces

Summary Table: Energy Types