# UTPA STEM/CBI Courses/Physics (Calculus Based)/Circular Motion

Course Title: Calculus Based Physics I

Lecture Topic: Circular Motion

Instructor: Liang Zeng

Institution: University of Texas-Pan American

## Backwards Design

Course Objectives

• Primary Objectives- By the next class period students will be able to:
• Know why an object goes in circular motion (centripetal force, constantlypointing towards the center of the circle)
• Know how to analyze and calculate the resultant force, centripetal force and acceleration
• Know how to calculate instantaneous velocity (along tangential line)
• Know how to convert degrees to radians and revolutions to radians
• Know how to calculate angular velocity and determine its direction
• Know how to calculate angular acceleration and determine its direction from the resultant force
• Know how to do cross product using the right hand rule (pay attention to unit vectors)
• Know how to use the right-hand rule to determine the direction of velocity
• Know how to do rotational kinematics with constant angular acceleration
• Compare angular kinematics with translational kinematics
• Sub Objectives- The objectives will require that students be able to:
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• Difficulties- Students may have difficulty:
• Finding centripetal force on a banked surface ( students need to be taught how to determine the direction of normal force, direction of centripetal force pointing towards a virtual center on a horizontal plane)
• The direction of angular velocity (curl four fingers and thumb points toward angular velocity)
• Right-hand rule (cross-product) applied to rigid bodies such as a spinning wheel
• Determining the direction of instantaneous velocity (along the tangential line)
• Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
• Drive a car and turn a sharp curve (easier to turn a large curve than a sharp curve)*
• Banked curve versus unbanked curve simulation
• Centrifuge (biology & chemistry labs, separate particles of different density, set test tube at an angle)*
• Hula hoop
• CD player
• Pilots fly loop-the-loop
• Roller coaster
• Merry-go-round
• Spinning wheels: bicycle and car wheels, hands of a watch, big gears in big motors, fly wheel in a boat
• Earth's spin around its own axis, spinning top
• Satellite goes almost circular motions
• Swing something attached to a cord

Model of Knowledge

• Concept Map
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• Content Priorities
• Enduring Understanding
• Centripetal force, centripetal acceleration (how to calculate them)
• Centrifugal phenomena
• Conversion of units from degrees to radians, angular displacement
• Determining magnitude of angular velocity
• Relation between angular velocity and instantaneous velocity and their directions (right hand rule, cross product)
• Rotational kinemtaic equations
• Comparison between rotational and translational kinematic equations
• Important to Do and Know
• Identify the normal force at different locations on a circular loop
• Analyze the normal force for more complicated multi-curve problems, such as roller coasters
• Worth Being Familiar with
• Elliptical motion: Earth revolves around the Sun, Sun is in one of the foci
• Radial and tangential acceleration, example birds scoop up fish and fly away

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 how a passenger feels at any time on the ride and how the force a passenger feels changes over different positions along the track (4 points). 2. Correct solution for the relationship between the angular speed of the apparatus and the position of the ball-shaped room (4 points). 3. Reasonable analysis for the maximum allowable angular speed of the apparatus (2 points). Format: 1. Put down your name. 2. State challenge question and its number. 3. Show all your work.

Challenge Question #5: As an engineer for Disney World you are asked to design a new amusement ride. This device consists of a ball-shaped room where the passengers strap themselves to seats. The ball is placed in a hemispherical bowl-shaped apparatus. When the apparatus is rotating the ball is allowed to move up along the wall of the apparatus by means of a track that extends from the bottom of the apparatus to the top. Suppose the mass of the ball including the passengers is m and suppose the radius of the ball is much smaller than the radius of the apparatus. Analyze how a passenger feels at any time on the ride and how the force a passenger feels changes over different positions along the track? What is the relationship between the angular speed of the apparatus and the position of the ball-shaped room? To ensure that none of the passengers blacks out or loses consciousness during the ride, what is the maximum allowable angular speed of the apparatus? Show all you mathematical work to support your analysis and descriptions.

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.(Centripetal force, banked surface: Knight 2nd edition page 231 #8). A highway curve of radius 500 m is designed for traffic moving at 90 km/hr. What is the correct banking angle of the road?

2.(Centripetal force, unbanked surface: Knight 2nd edition page 231 #5). A 1500 kg car takes a 50-m-radius unbanked curve at 15 m/s. What is the size of the friction force on the car?

3.(Centripetal force, looped surface: Knight 2nd edition page 233 #41). In an amusement park ride called the Roundup, passengers stand inside a 16-m-diameter rotating ring. After the ring has acquired sufficient speed, it tilts into a vertical plane, as shown in the figure.

a)Suppose the ring rotates once every 4.5 s. If a rider’s mass is 55 kg, with how much force does the ring push on her at the top of the ride? At the bottom?

b)What is the longest rotational period of the wheel that will prevent the riders from falling off at the top?

4.(Centripetal force, tension in a cord, horizontal plane: Serway 7th edition page 159 #50). An air puck of mass m1 is tied to a string and allowed to revolve in a circle of radius R on a frictionless horizontal table. The other end of the string passes through a small hole in the center of the table, and a load of mass m2 is tied to the string, as shown in the figure. The suspended load remains in equilibrium while the puck on the tabletop revolves.

a)What is the tension in the string?

b)What is the radial force acting on the puck?

c)What is the speed of the puck?

d)Qualitatively describe what will happen in the motion of the puck if the value of m2 is somewhat increased by placing an additional load on it.

e)Qualitatively describe what will happen in the motion of the puck if the value of m2 is instead decreased by removing a part of the hanging load.

5.(Centripetal force, tension in a cord, at angle to horizontal plane: Serway 7th edition page 160 #53). An amusement park ride consists of a rotating circular platform 8.00 m in diameter from which 10.0-kg seats are suspended at the end of 2.50-m massless chains, as shown in the figure. When the system rotates, the chains make an angle θ = 28.0° with the vertical.

a)What is the speed of each seat?

b)Draw a free-body diagram of a 40.0-kg child riding in a seat and find the tension in the chain.

6.(Centripetal acceleration, conceptual: Knight 2nd edition page 230 #3). The following figure is a bird’s eye view of particles moving in horizontal circles on a tabletop (scan and insert figure). All are moving at the same speed.

a)Rank in order, from largest to smallest, the tensions Ta to Td. Give your answer in the form a > b = c > d and explain your ranking.

b)Calculate the angular speeds ωa, ωb, ωc, and ωd.

7(Rotational kinematics with constant acceleration: Serway 7th edition page 300 #6). A rotating wheel requires 3.00 s to rotate through 37.0 revolutions. Its angular speed at the end of the 3.00-s interval is 98.0 rad/s. What is the constant angular acceleration of the wheel?

8.(Multi-curve problem: Serway 7th edition page 156 #14). A roller-coaster car (see figure below) has a mass of 500 kg when fully loaded with passengers.

a)If the vehicle has a speed of 20.0 m/s at point A , what is the force exerted by the track on the car at this point?

b)What is the maximum speed the vehicle can have at point B and still remain on the track?