# UTPA STEM/CBI Courses/Physics (Calculus Based)/Work Energy Theorem and the Conservation of Energy

θCourse Title: Calculus Based Physics I

Lecture Topic: Work Energy Theorem and the Conservation of Energy

Instructor: Dr. Linag Zeng

Institution: University of Texas-Pan American

## Backwards Design

Course Objectives

• Primary Objectives- By the next class period students will be able to:
• Know the definition for rotational energy and how to calculate rotational energy for a single particle
• Know the definition for rotational energy and how to calculate rotational energy for a rigid body
• Review kinetic energy
• Know the definition for potential energy and how to calculate potential energy
• Know the definition of work and how to calculate work using the dot product
• Know that the amount of work done by conservative forces (eg. gravity) or restorative forces (eg. spring) does not depend on the path; the amount of work done by non-conservative forces (eg. friction) does depend on the path
• Know the relationship between work and energy – work is not a form of energy (although it has units of Joules), but work can produce or consume energy
• Know the definition for spring potential energy and how to calculate spring potential energy (Hook’s Law)
• Know the definition of the system - anything included in an area/volume within a boundary (examples cell, tree bark, skin on body, ecosystem, sun). Know how to identify the system from reading the question. Know how to determine if the system is isolated or not isolated from the surroundings (isolated means that there is no energy exchange across the boundary).
• Know that most situations in this course are isolated systems. Know how to identify the conservative and non-conservative forces in the system. If all the forces in the system are conservative then conservation of mechanical energy can be applied. If at least one of the forces is non-conservative then the work-energy theorem (net work = change of total energy) must be used.
• Know different forms of energy transfer where energy is not conserved for a system (sound, electrical, electromagnetic radiation, matter transfer, mechanical waves, heat).
• Know definition of power: the change of energy over time, or the change of work over time. Know the unit of power: Watt. Know that your electrical bill comes in units of energy (kW x hour).
• Sub Objectives- The objectives will require that students be able to:
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• Difficulties- Students may have difficulty:
• Using Hook's Law:F = -kx a)Understanding why there is a negative sign: F and x are proportional but in opposite directions
• Identifying properties of a system: isolated versus non-isolated
• Identifying conservative forces versus non-conservative forces
• When to apply conservation of mechanical energy (conservative forces) versus when to apply the work-energy theorem (at least one non-conservative force)
• Identifying the different forms of energy to be considered in the initial state versus the final state
• Understanding that work and energy are both scalar quantities, but signs for work are inherited from the formulas used to calculate work depending on the angle between the force and displacement involved, while the signs for change of energy depend on the difference (subtraction) of initial from final energy (W = ΔE = Ef – Ei).
• Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
• The use of a ramp to push an object onto the back of a truck versus lifting it yourself. The total work is the same, but with the ramp the distance is longer so the force is smaller*
• Waterfalls, skiing, and hydroelectric plants at dams – conversion of potential energy to kinetic energy (and then mechanical and electrical energy) from the top of the waterfall to the bottom of the waterfall*
• Music boxes – wind the spring to create potential energy then the potential energy is converted to other forms of energy to play music or make the figurine twirl etc.
• Car shock absorber springs in the tires – when the car runs over something there is not much bouncing because the spring constant (F = -kx) is very large resulting in a small displacement*
• Car engine: battery has chemical energy converted to electrical energy, causes a spark which ignites the fuel so chemical potential is converted to heat energy, heat expands air which causes pistons to move up and down so heat energy is converted to mechanical energy, pistons are connected to rods and gears which cause the tires to turn*
• Solar energy from the sun is used by plants to grow and people eat plants for energy and then can do work. Continue with discussion from students*

Model of Knowledge

• Concept Map
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• Content Priorities
• Enduring Understanding
• Definition for rotational energy and calculate rotational energy for a single particle
• Definition for rotational energy and calculate rotational energy for a rigid body, including a rotational pulley example
• Review kinetic energy
• Definition for potential energy and calculate potential energy
• Definition of work and calculate work (W = F · d) using the dot product
• Amount of work done by conservative forces (eg. gravity) or restorative forces (eg. spring) does not depend on the path; the amount of work done by non-conservative forces (eg. friction) does depend on the path
• Relationship between work and energy – work is not a form of energy (although it has units of Joules), but work can produce or consume energy
• Definition for spring potential energy and calculate spring potential energy (Hook’s Law)
• Definition of the system - anything included in an area/volume within a boundary. (Give lots of examples – cell, tree bark, skin on body, ecosystem, sun). Identify the system from reading the question. Determine if the system is isolated or not isolated from the surroundings (isolated means that there is no energy exchange across the boundary).
• Most situations in this course are isolated systems. Identify the conservative and non-conservative forces in the system. If all the forces in the system are conservative then conservation of mechanical energy can be applied. If at least one of the forces is non-conservative then the work-energy theorem (work = change of energy) must be used
• Definition of power: the change of energy over time, or the change of work over time (force * velocity). Know the unit of power: Watt. Electric bill comes in units of energy (kW x hour)
• Important to Do and Know
• Different forms of energy transfer where energy is not conserved for a system (sound, electrical, electromagnetic radiation, matter transfer, mechanical waves, heat)
• Worth Being Familiar with
• In other systems there are other forms of transfer of energy – matter transfer, electrical transmission, electromagnetic radiation
• Faraday's universal conservation of energy hypothesis (total energy in the universe is conserved – Physics Today)

Assessment of Learning

• Formative Assessment
• In Class (groups)
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• Homework (individual)
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• Summative Assessment
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## Legacy Cycle

OBJECTIVE

By the next class period, students will be able to:

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The objectives will require that students be able to:

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THE CHALLENGE

• Instructions: students can refer to any resource to answer this question. Each person needs to scan and save the report as a pdf file, and email the report to Mr. Manuel Lara, Teaching Assistant on WebCT which answers the question with supporting data linking back with the relevant physics content. The report is usually due a week from the date when the question is assigned and will be kept in a WebCT folder.

Rubric for grades (on a 0-10 point scale): 1. Correct solution for the speed the driver was going when he got onto the ramp (8 points). 2. Reasonable recommendation to avoid similar accidents at this overpass (2 points). Format: 1. Put down your name. 2. State challenge question and its number. 3. Show all your work.

Challenge Question #7: The westbound lanes of Expresseway 83 in McAllen have an exit leading to a ramp that leads to an overpass along Bicentennial Boulevard that you can use to get to the Miller International Airport. The ramp goes up and ends in a T-junction which merges onto Bicentennial Boulevard (see figures). Several accidents have occurred at this T-junction because drivers coming up the ramp are not able to successfully make the turn at the T-junction, and going forward through the T-junction results in you going through a barrier, dropping off the overpass and landing on the Expresseway 83 below. A driver recently had an accident at the T-junction, where he hit the barrier at the T-junction causing the bumper on the front of his car to be compressed by 1 foot (the bumper collision can be approximated as a spring with a spring constant k = 150 N/m). The width of the T-junction from the end of the ramp to the barrier is 20 feet, and coefficient of friction between the asphalt surface and the car tires when the vehicle is braking is 0.7 (it was a foggy day). The height of the overpass is 18.5 feet above the Expresseway, and the horizontal distance from the beginning of the ramp to the beginning of the T-junction is 115 feet (the rolling friction is .015). There were faint and alternating skid marks across the entire 20 feet span of the T-junction, indicating that the driver slammed on his brakes across the entire span of the T-junction. The driver reported that once he left the Expresseway and got onto the ramp he took his foot off the accelerator and did not hit the brakes until he got to the T-junction. The speed limit for the ramp was 45 mph. How fast was the driver going when he got onto the ramp? Based on your calculation of the velocity, what would you recommend to avoid similar accidents at this overpass? Justify your answer.

GENERATE IDEAS

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MULTIPLE PERSPECTIVES

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RESEARCH & REVISE

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GO PUBLIC

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## Pre-Lesson Quiz

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1. (Rotational energy for a rigid body: Serway 7th edition page 332 #32b). A student sits on a freely rotating stool holding two dumbbells, each of mass 3.00 kg (see figure). When the student’s arms are extended horizontally (figure a), the dumbbells are 1.00 m from the axis of rotation and the student rotates with an angular speed of 0.750 rad/s. The moment of inertia of the student plus stool 3.00 kg•m2 and is assumed to be constant. The student pulls the dumbbells inward horizontally to a position 0.300 m from the rotation axis (figure b). Find the kinetic energy of the rotating system before and after he pulls the dumbbells inward.

2. (Free fall: Shipman, Wilson, and Todd, 12th edition page 98 #17). An object is dropped from a height of 12 m. At what height will its kinetic energy and its potential energy be equal?

3. (Potential and kinetic energy irregular slide: Serway 7th edition page 218 #4). A particle of mass m = 5.00 kg is released from point A and slides on the frictionless track shown in the figure. Determine the following:

a)The particle’s speed at points B and C b)The net work done by the gravitational force as the particle moves from A to C

4. (Work with a force applied at an angle to displacement: Serway 7th edition page 189 #1). A block of mass 2.50 kg is pushed 2.20 m along a frictionless horizontal table by a constant 16.0-N force directed 25° below the horizontal.

a)Determine the work done on the block by the applied force b)Determine the work done on the block by the normal force exerted by the table c)Determine the work done on the block by the gravitational force d)Determine the net work done on the block

5. (Conservative force: Serway 7th edition page 192 #39). A 4.00-kg particle moves from the origin to position C, having coordinates x = 5.00 m and y = 5.00 m (see figure). One force on the particle is the gravitational force acting in the negative y direction. Using equation 7.3 from your textbook, calculate the work done by the gravitational force on the particle as it goes:

a) From O to C along OAC b) From O to C along OBC c) From O to C along OC d) Your results should all be identical. Why?

6. (Non-conservative force: adapted from Serway 7th edition page 192 #39). The same 4.00-kg particle from the previous question moves from the origin to position C, having coordinates x = 5.00 m and y = 5.00 m (see figure). Another force on the particle is a friction force with magnitude 3.00 N and acting in the direction opposite to the particle’s displacement. Calculate the work done by the friction force on the particle as it moves along the following paths:

a)OAC b)OBC c)OC d)Are all of your results identical? If not, why?

7. (Relationship between work and energy: Walker 2nd edition page 194 #22). A 1300-kg car coasts on a horizontal road with a speed of 18 m/s. After crossing an unpaved, sandy stretch of road 30.0 m long its speed decreases to 15 m/s.

a)Was the net work done on the car positive, negative, or zero? Explain. b)Find the magnitude of the average net force on the car in the sandy section.

8. (Spring potential and object kinetic energy: Serway 7th edition page 193 #57). The ball launcher in a pinball machine has a spring that has a force constant of 1.20 N/cm (see figure). The surface on which the ball moves is inclined 10.0° with respect to the horizontal. The spring is initially compressed 5.00 cm. Find the launching speed of a 100-g ball when the plunger is released. Friction and the mass of the plunger are negligible.

9. (Pulley, no mass or rotation of pulley: Serway 7th edition page 219 #7). Two objects are connected by a light spring passing over a light, frictionless pulley as shown in the figure. The object of mass 5.00 kg is released from rest. Using the isolated system model:

a)Determine the speed of the 3.00-kg object just as the 5.00-kg object hits the ground. b)Find the maximum height to which the 3.00-kg object rises.

10. (Pulley, with mass and rotation of pulley: Serway 7th edition page 304 #44). Consider the system shown in the figure with m1 = 20.0 kg, m2 = 12.5 kg, R = 0.200 m, and the mass of the uniform pulley M = 5.00 kg. Object m2 is resting on the floor, and object m1 is is 4.00 m above the floor when it is released from rest. The pulley axis is frictionless. The cord is light, does not stretch, and does not slip on the pulley. Calculate the final velocity of m1 when it is about to hit the floor.

11. (Conservation of mechanical energy and projectile motion: Walker 2nd edition page 230 #57). The water slide shown in the figure (scan and insert figure) ends at a height of 1.50 m above the pool. If the person starts from rest at point A and lands in the water at point B, what is the height h of the water slide? (Assume the water slide is frictionless).

12. (Pendulum-like question: Serway 7th edition page 264 #47a and 47b). A particle is suspended from a post on top of a cart by a light string of length L as shown in the figure (figure a). The cart and particle are initially moving to the right at constant speed vi with the string vertical. The cart suddenly comes to rest when it runs into and sticks to a bumper as shown in the figure (figure b). The suspended particle swings through an angle .

a)Show that the original speed of the cart can be computed from vi = (2gL(1-cosθ))^1/2 b)Find the initial speed implied by L = 1.20 m and = 35.0°.

13. (Power in a moving elevator: Serway 7th edition page 221 #32). A 650-kg elevator starts from rest. It moves upward for 3.00 s with constant acceleration until it reaches its cruising speed of 1.75 m/s.

a)What is the average power of the elevator motor during this time interval? b)How does this power compare with the motor power when the elevator moves at its cruising speed?