Work and Energy Concept Review Answer Key Chapter 13

Section Learning Objectives

By the end of this section, yous will be able to do the following:

• Depict and apply the work–energy theorem
• Describe and summate piece of work and ability

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Teacher Support

The learning objectives in this department will assistance your students master the following standards:

• (6) Science concepts. The student knows that changes occur within a concrete organization and applies the laws of conservation of energy and momentum. The student is expected to:
  • (A) describe and apply the work–free energy theorem;
  • (C) describe and calculate work and power.

In addition, the High Schoolhouse Physics Laboratory Transmission addresses the post-obit standards:

• (6) Science concepts. The student knows that changes occur within a physical arrangement and applies the laws of conservation of energy and momentum. The student is expected to:
  • (C) calculate the mechanical free energy of, power generated within, impulse practical to, and momentum of a concrete system.

Utilize the lab titled Work and Energy as a supplement to address content in this section.

Section Key Terms

free energy	gravitational potential energy	joule	kinetic energy	mechanical energy
potential energy	power	watt	work	work–energy theorem

Teacher Support

Teacher Support

In this department, students larn how work determines changes in kinetic free energy and that power is the rate at which work is done.

[BL] [OL] Review understanding of mass, velocity, and acceleration due to gravity. Define the full general definitions of the words potential and kinetic.

[AL] [AL] Remind students of the equation $\mathrm{Westward}=P{E}_e=fmg$ . Point out that acceleration due to gravity is a constant, therefore PE_e that results from work washed by gravity will also be abiding. Compare this to acceleration due to other forces, such as applying muscles to lift a rock, which may not be abiding.

The Work–Energy Theorem

In physics, the term piece of work has a very specific definition. Work is application of forcefulness, $f$ , to motility an object over a altitude, d, in the management that the force is applied. Work, W, is described by the equation

$$W=fd\text{.}$$

Some things that we typically consider to be work are not work in the scientific sense of the term. Allow's consider a few examples. Think about why each of the post-obit statements is truthful.

• Homework is not work.
• Lifting a rock upwards off the basis is work.
• Carrying a rock in a straight path across the lawn at a constant speed is not piece of work.

The beginning two examples are adequately simple. Homework is not work considering objects are not beingness moved over a distance. Lifting a rock upwards off the footing is work because the rock is moving in the direction that forcefulness is practical. The final example is less obvious. Call up from the laws of motion that force is not required to movement an object at constant velocity. Therefore, while some force may be applied to keep the rock upwards off the ground, no net force is applied to keep the rock moving forrard at constant velocity.

Teacher Back up

Teacher Support

[BL] [OL] Explain that, when this theorem is applied to an object that is initially at rest then accelerates, the $\frac{1}{2}\mathrm{thou}{v}_{\mathrm{one}}^2$ term equals zero.

[OL] [AL] Work is measured in joules and $W=fd$ . Force is measured in newtons and distance in meters, and then joules are equivalent to newton-meters $\left(\text{North}\cdot \text{m}\right)$

Piece of work and energy are closely related. When you do work to move an object, you lot change the object'southward energy. You (or an object) also expend free energy to do piece of work. In fact, energy tin can be defined as the ability to do work. Energy can take a variety of different forms, and 1 form of free energy can transform to another. In this affiliate nosotros will be concerned with mechanical energy, which comes in two forms: kinetic free energy and potential free energy.

• Kinetic energy is too called energy of motion. A moving object has kinetic energy.
• Potential free energy, sometimes chosen stored energy, comes in several forms. Gravitational potential energy is the stored energy an object has as a result of its position above Earth's surface (or another object in space). A roller coaster car at the meridian of a hill has gravitational potential energy.

Let's examine how doing piece of work on an object changes the object'south energy. If we apply force to lift a stone off the ground, nosotros increase the rock'southward potential energy, PE. If we drop the stone, the force of gravity increases the rock'south kinetic energy as the rock moves downward until information technology hits the basis.

The force we exert to lift the rock is equal to its weight, westward, which is equal to its mass, grand, multiplied by dispatch due to gravity, one thousand.

$$f=w=\mathrm{thousand}g$$

The work we practise on the stone equals the force we exert multiplied by the distance, d, that we lift the rock. The work we practise on the stone too equals the rock's gain in gravitational potential energy, PE_east .

$$W=P{E}_{\mathrm{due\; east}}=mkd$$

Kinetic free energy depends on the mass of an object and its velocity, v.

$$ME=\frac{1}{2}m{\mathrm{five}}^2$$

When nosotros drib the rock the force of gravity causes the stone to fall, giving the rock kinetic energy. When piece of work done on an object increases only its kinetic energy, then the cyberspace work equals the change in the value of the quantity $\frac{1}{\mathrm{two}}m{v}^2$ . This is a statement of the work–energy theorem, which is expressed mathematically as

$$W=\text{Δ}KE\text{ =}\frac{1}{2}\mathrm{one\; thousand}{\mathrm{five}}_2^{\mathrm{two}}-\frac{1}{2}m{v}_1^2\text{.}$$

The subscripts ₂ and ₁ indicate the terminal and initial velocity, respectively. This theorem was proposed and successfully tested by James Joule, shown in Figure 9.two.

Does the name Joule sound familiar? The joule (J) is the metric unit of measurement for both work and energy. The measurement of work and energy with the same unit reinforces the idea that piece of work and free energy are related and can be converted into one another. 1.0 J = one.0 N∙one thousand, the units of force multiplied past altitude. one.0 Due north = one.0 kg∙m/sⁱⁱ, so 1.0 J = 1.0 kg∙m²/s². Analyzing the units of the term (1/2)m five ⁱⁱ will produce the aforementioned units for joules.

Figure 9.2 The joule is named later on physicist James Joule (1818–1889). (C. H. Jeens, Wikimedia Commons)

Watch Physics

Work and Energy

This video explains the piece of work energy theorem and discusses how work done on an object increases the object'south KE.

Grasp Cheque

True or false—The energy increment of an object acted on only by a gravitational force is equal to the product of the object's weight and the distance the object falls.

1. True
2. False

Teacher Support

Teacher Back up

Echo the information on kinetic and potential energy discussed earlier in the section. Take the students distinguish between and understand the two ways of increasing the energy of an object (1) applying a horizontal forcefulness to increase KE and (ii) applying a vertical force to increase PE.

Calculations Involving Piece of work and Power

In applications that involve work, nosotros are often interested in how fast the work is done. For case, in roller coaster design, the amount of time information technology takes to elevator a roller coaster car to the peak of the first loma is an important consideration. Taking a half hour on the rise will surely irritate riders and subtract ticket sales. Let's take a look at how to calculate the fourth dimension information technology takes to do work.

Recall that a charge per unit can be used to depict a quantity, such as work, over a menstruation of time. Ability is the rate at which work is washed. In this case, rate means per unit of time. Ability is calculated by dividing the work done by the fourth dimension it took to practice the work.

$$P=\frac{W}{t}$$

Let'southward consider an example that tin can help illustrate the differences among piece of work, force, and power. Suppose the woman in Figure ix.three lifting the Television with a caster gets the TV to the quaternary floor in 2 minutes, and the homo carrying the TV upward the stairs takes five minutes to arrive at the aforementioned identify. They accept done the same amount of piece of work $\left(fd\right)$ on the TV, because they have moved the aforementioned mass over the same vertical altitude, which requires the same amount of upward forcefulness. However, the woman using the pulley has generated more ability. This is because she did the work in a shorter amount of time, so the denominator of the ability formula, t, is smaller. (For simplicity's sake, we will go out bated for at present the fact that the man climbing the stairs has also done piece of work on himself.)

Figure nine.three No matter how you movement a Television set to the 4th flooring, the amount of work performed and the potential energy gain are the same.

Power can exist expressed in units of watts (Due west). This unit can exist used to measure out power related to any form of energy or work. You accept nearly likely heard the term used in relation to electric devices, particularly light bulbs. Multiplying power past time gives the amount of free energy. Electricity is sold in kilowatt-hours because that equals the amount of electrical energy consumed.

The watt unit was named after James Watt (1736–1819) (see Figure 9.4). He was a Scottish engineer and inventor who discovered how to coax more power out of steam engines.

Figure nine.four Is James Watt thinking about watts? (Carl Frederik von Breda, Wikimedia Commons)

Teacher Back up

Instructor Support

[BL] [OL] Review the concept that work changes the energy of an object or system. Review the units of work, energy, force, and distance. Use the equations for mechanical energy and work to show what is work and what is not. Brand it clear why property something off the ground or conveying something over a level surface is not work in the scientific sense.

[OL] Ask the students to employ the mechanical energy equations to explain why each of these is or is not work. Ask them to provide more examples until they empathize the difference between the scientific term work and a task that is simply difficult only not literally work (in the scientific sense).

[BL] [OL] Stress that ability is a rate and that rate ways "per unit of time." In the metric system this unit is normally seconds. Stop the section by immigration up whatever misconceptions about the distinctions betwixt forcefulness, work, and power.

[AL] Explicate relationships between the units for forcefulness, work, and ability. If $W=fd$ and work tin can be expressed in J, so $P=\frac{\mathrm{Due\; west}}{t}=\frac{fd}{t}$ and so power tin can be expressed in units of $\frac{\text{Due north}\cdot \text{m}}{\text{southward}}$

Too explicate that nosotros buy electricity in kilowatt-hours because, when power is multiplied by time, the time units cancel, which leaves work or free energy.

Links To Physics

Watt's Steam Engine

James Watt did not invent the steam engine, but by the fourth dimension he was finished tinkering with it, it was more useful. The outset steam engines were not merely inefficient, they merely produced a back and forth, or reciprocal, motion. This was natural because pistons move in and out as the pressure level in the chamber changes. This limitation was okay for simple tasks similar pumping water or mashing potatoes, only did non work and so well for moving a railroad train. Watt was able build a steam engine that converted reciprocal motion to round motion. With that one innovation, the industrial revolution was off and running. The earth would never be the same. Ane of Watt's steam engines is shown in Figure 9.5. The video that follows the figure explains the importance of the steam engine in the industrial revolution.

Effigy 9.5 A late version of the Watt steam engine. (Nehemiah Hawkins, Wikimedia Commons)

Instructor Back up

Teacher Back up

Initiate a discussion on the historical significance of of a sudden increasing the corporeality of power available to industries and transportation. Take students consider the fact that the speed of transportation increased roughly tenfold. Changes in how appurtenances were manufactured were just equally not bad. Inquire students how they think the resulting changes in lifestyle compare to more than contempo changes brought nearly by innovations such as air travel and the Internet.

Lookout Physics

Watt's Role in the Industrial Revolution

This video demonstrates how the watts that resulted from Watt'southward inventions helped brand the industrial revolution possible and allowed England to enter a new historical era.

Click to view content

Grasp Check

Which form of mechanical free energy does the steam engine generate?

1. Potential energy
2. Kinetic energy
3. Nuclear energy
4. Solar energy

Before proceeding, be sure you understand the distinctions among forcefulness, work, energy, and ability. Force exerted on an object over a altitude does work. Work can increment energy, and energy tin practise work. Ability is the rate at which work is washed.

Worked Case

Applying the Work–Energy Theorem

An ice skater with a mass of 50 kg is gliding across the ice at a speed of viii chiliad/s when her friend comes up from behind and gives her a push button, causing her speed to increase to 12 m/s. How much work did the friend practice on the skater?

Strategy

The piece of work–energy theorem tin be applied to the problem. Write the equation for the theorem and simplify it if possible.

$$W=\text{Δ}\text{KE =}\frac{1}{2}m{5}_2^2-\frac{1}{\mathrm{ii}}\mathrm{grand}{\mathrm{five}}_1^2$$

$$\text{Simplify to}W=\frac{\mathrm{ane}}{2}m(v_2^2-{\mathrm{five}}_{\mathrm{ane}}^2)$$

Discussion

Work done on an object or system increases its energy. In this case, the increase is to the skater's kinetic energy. It follows that the increment in energy must be the difference in KE before and later the push.

Tips For Success

This problem illustrates a general technique for budgeted issues that require you to apply formulas: Identify the unknown and the known variables, limited the unknown variables in terms of the known variables, and and then enter all the known values.

Instructor Support

Teacher Support

Identify the three variables and cull the relevant equation. Distinguish betwixt initial and concluding velocity and pay attention to the minus sign.

Identify the variables. m = 50 kg,

$$5_2=12\frac{\text{m}}{\text{s}}\text{, and}{v}_1=8\frac{\text{m}}{\text{due south}}$$

Substitute.

$$\mathrm{Due\; west}=\frac{1}{\mathrm{ii}}\mathrm{fifty}({12}^{\mathrm{two}}-8^2)=\mathrm{two},000\text{ J}$$

Practice Bug

i .

(credit: modification of work by Pass My Exams, CC BY-SA 4.0)

Figure 9.6

A weightlifter lifts a 200 N barbell from the floor to a height of 2 m. How much work is done?

1. 0\,\text{J}

2. 100\,\text{J}

3. 200\,\text{J}

4. 400\,\text{J}

ii .

Identify which of the following actions generates more power. Show your work.

• conveying a 100\,\text{N} TV to the second floor in 50\,\text{due south} or
• conveying a 24\,\text{N} watermelon to the second flooring in 10\,\text{s}?

1. Conveying a 100\,\text{North} TV generates more than power than carrying a 24\,\text{North} watermelon to the aforementioned superlative because power is defined every bit work done times the time interval.

2. Conveying a 100\,\text{N} Television set generates more power than carrying a 24\,\text{N} watermelon to the same tiptop because power is defined every bit the ratio of work done to the fourth dimension interval.

3. Conveying a 24\,\text{N} watermelon generates more power than carrying a 100\,\text{N} Television receiver to the same height because power is defined every bit work done times the time interval.

4. Carrying a 24\,\text{N} watermelon generates more than power than carrying a 100\,\text{N} Boob tube to the same height because power is defined as the ratio of work washed and the time interval.

Check Your Agreement

3 .

Identify two properties that are expressed in units of joules.

1. work and force

2. free energy and weight

3. piece of work and free energy

4. weight and force

4 .

When a coconut falls from a tree, work West is done on information technology as information technology falls to the beach. This piece of work is described by the equation

$$\mathrm{Westward}=Fd\text{ =}\frac{1}{2}m{v}_2^{\mathrm{two}}-\frac{1}{2}m{v}_{\mathrm{one}}^2.$$

ix.3

Place the quantities F, d, grand, v ₁, and v _two in this upshot.

1. F is the force of gravity, which is equal to the weight of the coconut, d is the altitude the nut falls, thou is the mass of the world, v ₁ is the initial velocity, and v ₂ is the velocity with which it hits the beach.
2. F is the force of gravity, which is equal to the weight of the coconut, d is the altitude the nut falls, grand is the mass of the coconut, v ₁ is the initial velocity, and v _ii is the velocity with which it hits the beach.
3. F is the forcefulness of gravity, which is equal to the weight of the coconut, d is the distance the nut falls, m is the mass of the globe, five ₁ is the velocity with which information technology hits the beach, and v ₂ is the initial velocity.
4. F is the force of gravity, which is equal to the weight of the coconut, d is the distance the nut falls, one thousand is the mass of the kokosnoot, v ₁ is the velocity with which information technology hits the beach, and v ₂ is the initial velocity.

Teacher Support

Teacher Support

Utilize Check Your Understanding questions to appraise students' achievement of the section's learning objectives. If students are struggling with a specific objective, the Check Your Understanding volition help identify which one and direct students to the relevant content.

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Source: https://openstax.org/books/physics/pages/9-1-work-power-and-the-work-energy-theorem

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