## Objective

As a result of this lesson you will be able to

- Define what it means for a force to do work on an object and apply the work equation to calculate the amount of work done in a situation.
- Construct the equation of kinetic energy from the concept of work and kinematics.
- Describe how work done on an object changes the body’s kinetic energy, and apply this principle to solve mechanics problems.
- Apply the concept of power to solve problems involving work.
- Value the amount of work done by applying the concept of power.
- Describe the relationship between work, energy, and power.
- Apply gravitational potential energy to problems involving vertical motion.
- Differentiate between conservative and nonconservative forces, and how to solve problems in which both kinds of forces act on a moving body

## Prerequisites

This topic expects you to have an understanding of the following in order to successfully learn all the material.

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# Question

If you aim a gun toward a body of water and shoot, the gas expands and the shell is pushed out. Newton’s third law says that the shell pushes back with the same force. Does the shell perform work on the gas? Once the shell enters the water is the work done by gravity or the water positive or negative on the shell?

# Introduction

There are many forms of energy and we will cover a few in this section. One important concept that should be used when dealing with energy is the Law of Conservation of Energy. You can use the conservation law to keep track of energy at any number of moments within a system because the sum of the energy in the system will always remain the same but it will be in different forms.

# Connection

Newton’s laws of motion explain how a force causes an object to move when the net force is unbalanced. Now we can uses forces to also determine how much energy was given to an object as a result of a force being applied to it. Forces indeed cause change and we use the idea of energy to measure the amount of change that occurs. We then use the idea of work to describe the transferring of energy from one system to another. Finally, the term power describes the rate at which work is applied. What we are now attempting to do is look at the energy forces cause and not just the motion that results applying a force.

# Role

In mechanics we are dealing with objects that are undergoing some sort of change. We can use Newton’s laws of motion and any other problem solving techniques to determine specific information about how the object is changing and predict where it will be. There are some situations where we cannot apply Newton’s laws because the force being applied will vary over time. When the force varies we will then use the concepts of work and energy to tackle the problem.

# Work

Work is the result of a force over a distance. A force needs to be applied in the direction of motion in order to perform work on an object. When you perform work on the system you transfer energy to it in the form of motion.

## Equation/Definition

If a force F acts over a distance d, and F is parallel to d, then the work done by F is the product of force and distance:

(insert latex equation here)

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## Example

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# Kinetic Enerty

Kinetic energy is the energy an object has from its motion. You can combine the techniques from kinematic equations, forces, and work to determine the energy from motion. If an object has initially no velocity and it has an acceleration of a=F/m over a distance ∆s we can follow the following steps to create an equation for kinetic energy. Starting with equation 5 of the kinetic equationsv^2= v_0^2+2a∆s

and then substituting the known acceleration from Newton’s second law with a zero initial velocity

v^2= 0+2 F/m ∆s

We then solve the equation for F∆s and get

1/2 〖mv〗^2= F∆s=W

This result for work is called kinetic energy.

## Equation/Definition

(insert latex equation here and above with definition)

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## Example

An example scaffolding sub question

# Work-Energy Theorem

Since energy and work are both expressed in the same units and the total work done on an object will be the same as the change in its kinetic energy. The Work-Energy theorem is used to relate the work done on an object to its energy of motion. So far we have experienced problems where a force is applied to objects.

## Equation/Definition

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# Potential Energy

Instead of having energy in the form of motion an object can also have energy by virtue of its position or system configuration. Potential energy is related to kinetic energy but potential energy is independent of motion. The potential energy we are going to deal with is a result of gravitational potential energy. Potential energy depends on the mass of the object, its distance from the referenced ground, and gravity. One important note about potential energy is that it is conservative and that means it does not depend on the path it takes as it increases or decreases its distance from the referenced ground.

## Equation/Definition

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# Conservation of Mechanical Energy

So far we have dealt with energy in terms of its position and motion. When we combine potential and kinetic energy we identify it as mechanical energy. It’s important to note that mechanical energy is dependent on its configuration since it is made up of potential energy. When we combine idea of conservation of energy we can then state that an objects initial mechanical energy is the same as an objects final mechanical energy. Our job will be to determine where the energy has gone and it is convenient to choose points where either potential or kinetic energy is zero.

## Equation/Definition

E=K+U=K_i+U_i=K_f+U_f

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# Power

Power is the rate at which work gets done.

## Equation/Definition

Power (P)=work/time=W/t

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## Example

An example scaffolding sub question

## Review Questions

- You have the crazy idea to finally study for your exam and you lift your 2kg book vertically upward with a constant velocity. How much work are using to lift the book 2m?
- You pull a 20kg box containing your favorite items with a force of 40N. The rope makes an angle of 20 degrees along the horizontal. How much work does it take you to move it 10m?
- You pull a 20kg box containing your favorite items with a force of 40N. The rope makes an angle of 20 degrees along the horizontal. How much work does it take you to move it 10m?
- You decide to move your 20kg box of items across a rough surface. The surface has a kinetic friction value of .4 and you apply a 40N force at an angle of 20 degrees to the horizontal. How much work is done by the normal force and how much work is done by friction?
- When you displace a spring from its normal state, it has a restoring force of (k is a positive constant). How much work would it take to move a spring from? Remember this force varies as the displacement varies so it is not a constant force.
- A ball is traveling in the air with a velocity of . What is the kinetic energy of the ball as it travels through the air?
- If you toss a ball straight upward with an initial velocity of 40 how high will it go with no air resistance?
- A 35kg block is placed on a ramp that has an incline of 35 degrees. It then slides down and the vertical component of its fall is 8m. The ramp has a kinetic friction value of .3. What is the total work?
- What is the potential energy of a 5kg rock sitting near the edge of a 100m cliff (relative to the ground)?
- Another 5kg rock sitting near the edge of a 100m cliff (relative to the ground) falls off the cliff. What is the speed of the rock as it hits the ground?
- A 40kg rock falls off a cliff 50m high and is subject to an average force of air resistance equal to 100N. What is the speed of the rock as it hits the ground?
- How much power is used to move a 50kg box a distance of 9m if it takes you 20s to move the box with a force of 200N.
- What does the power output have to be to lift a 1000kg elevator with a constant speed of 9 .
- Derive the definition of kinetic energy by combining the concepts of Newton’s second law, work, and kinematics. Assume an object has been displaced from and the speed changes from and the object is initially at rest.