# Chapter 4: Energy

## Work:

• Work is defined as the transfer of energy when a force is applied to an object and causes it to move.

• The unit of work is the joule (J), which is defined as the work done when a force of one newton is applied to an object and moves it through a distance of one meter.

• Work can be positive, negative, or zero, depending on the direction of the force and the displacement of the object.

• Work can be calculated using the formula: W = F x d x cos(theta), where F is the force applied, d is the distance moved, and theta is the angle between the force and the displacement vectors.

• Work can also be calculated as the change in kinetic energy of an object, using the work-energy theorem.

• Work done by a conservative force, such as gravity, depends only on the initial and final positions of the object and is independent of the path taken.

• Work done by a non-conservative force, such as friction, depends on the path taken and can result in a loss of mechanical energy.

• The principle of work-energy conservation states that the total work done on an object is equal to the change in its kinetic and potential energy.

Sample Problem 2

Calculate the total work (or net work) done on object M by the four forces indicated.

Solution

Total, or net, work is the sum of the individual works done by each of the individual forces.

## Power:

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

• The unit of power is watt (W), which is defined as one joule of work done per second.

• Power can be calculated using the formula: P = W/t, where W is the work done and t is the time taken.

• Power is a scalar quantity and can be positive, negative, or zero, depending on the direction of the work done and the time taken.

• Power is important in many practical applications, such as engines, motors, and generators.

TIP:

When a constant force Fis applied to an object moving at constant velocity v, the power P is the product of the two values: P= Fiv.

## Kinetic Energy and the Work-Energy Theorem:

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

• The formula for kinetic energy is KE = 0.5 x m x v^2, where m is the mass of the object and v is its velocity.

• The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy.

• The work-energy theorem can be written as Wnet = KEf - KEi, where Wnet is the net work done, KEf is the final kinetic energy, and KEi is the initial kinetic energy.

## Potential Energy and Conservative Forces:

• Potential energy is the energy possessed by an object due to its position or configuration in a conservative force field.

• Conservative forces are forces that do not dissipate energy and do not depend on the path taken by the object. Examples include gravitational and elastic forces.

• The potential energy associated with a conservative force can be calculated using the formula: PE = -Wc, where Wc is the work done by the conservative force and PE is the potential energy.

• The potential energy can also be calculated as the negative of the work done by an external force to move the object from a reference position to its current position.

• The total mechanical energy of an object in a conservative force field is the sum of its kinetic and potential energy and remains constant as the object moves within the force field.

• The law of conservation of energy states that the total energy of a closed system, including all forms of energy (kinetic, potential, thermal, etc.), remains constant over time.

Sample Problem

An arrow is shot from the roof of a building 30 meters high at 5 m/s and at an angle of 45 degrees. How fast will the arrow be going when it hits the ground?

Solution

From the information given, we know that:

Sample Problem 8

A 2-kg mass sliding along a frictionless floor at 3 m/s hits a spring (with k=200 N/m), compresses it, and bounces back. What is the maximum compression of the spring?

Solution

In this problem, we are dealing with the potential energy of a spring. So

Note that the final state refers to the spring at maximum compression. From the information given, we know:

REMEMBER:

Velocity is 0 m/s when the spring is at maximum compression (V = 0 during any turnaround).

## Conservation of Energy:

• The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another.

• Energy is a scalar quantity and can exist in many different forms, including kinetic, potential, thermal, electromagnetic, and chemical energy.

• The total energy of a system is the sum of all its forms of energy, and this total energy is conserved over time.

• Energy can be transferred between objects or systems through work or heat transfer.

• In an isolated system, where there is no external work or heat transfer, the total energy remains constant, and the system is said to be in a state of internal energy equilibrium.

• The law of conservation of energy is a fundamental principle in physics and is applicable to all physical systems, from the microscopic to the macroscopic scale.