Warm-Up: February 27, 2015 Write down everything you remember about uniform circular motion.

AP Physics C

Rotational Motion and Angular Momentum
OpenStax Chapter 10 AP Physics 1

OpenStax Section 10.1 AP Physics 1
Angular Acceleration OpenStax Section 10.1 AP Physics 1

Uniform Circular Motion
Motion in a circle at constant speed, constant angular velocity. According to convention, counterclockwise is positive, clockwise is negative.

Angular Acceleration Angular acceleration, α, is the rate of change of angular velocity. Measured in units of radians per second squared.

You-Try 10.1 Suppose a teenager puts her bicycle on its back and starts the rear wheel spinning from rest to a final angular velocity of 250. rpm in s. Calculate the angular acceleration in rad/s2. If she now slams on the brakes, causing an angular acceleration of rad/s2, how long does it take the wheel to stop?

Suppose a teenager puts her bicycle on its back and starts the rear wheel spinning from rest to a final angular velocity of 250. rpm in 5.00 s. Calculate the angular acceleration in rad/s2. If she now slams on the brakes, causing an angular acceleration of rad/s2, how long does it take the wheel to stop?

Tangential Acceleration
In uniform circular motion, the only acceleration is centripetal (towards the center). This is perpendicular to the velocity. In non-uniform circular motion, there can also be a component of acceleration in the same direction as velocity. Tangential acceleration is the linear acceleration that is tangent to the circle.

Acceleration

You-Try 10.2 A powerful motorcycle can accelerate from rest to 30.0 m/s in 4.20 s. What is the angular acceleration of its m radius wheels?

Linear-Angular Analogues

You-Try #2 An ultracentrifuge accelerates from rest to 1.00x105 rpm in 2.00 min. What is its angular acceleration in rad/s2? What is the tangential acceleration of a point 9.50 cm from the axis of rotation? What is the radial acceleration in m/s2 and multiples of g of the point at full rpm?

Assignment Current: Read OpenStax section 10.1
OpenStax page 352 #1-4 OpenStax page 356 #1-3 odd Final: Read OpenStax Chapter 10 OpenStax page 352 #1-30 OpenStax page 356 #1-47 odd

Homework Questions?

Warm-Up 4: March 2, 2015 You are told that a basketball player spins the ball with an angular acceleration of 100. rad/s2. What is the ball’s final angular velocity if the ball starts from rest and the acceleration lasts s? What is unreasonable about the result? Which premises are unreasonable or inconsistent?

You are told that a basketball player spins the ball with an angular acceleration of 100. rad/s2.
What is the ball’s final angular velocity if the ball starts from rest and the acceleration lasts 2.00 s? What is unreasonable about the result? Which premises are unreasonable or inconsistent?

Kinematics of Rotational Motion
OpenStax section 10.2 AP Physics 1

Rotational & Translational Motion

Rotational Kinematics Problem Solving
Examine the situation to determine that rotational kinematics is involved. Make a list of what is given or can be inferred. Identify exactly what needs to be determined in the problem. Solve the appropriate equation or equations. Substitute the known values into the equation and obtain a numerical solution, with units. Check your answer to see if it is reasonable.

You-Try 10.3 A deep-sea fisherman hooks a big fish that swims away from the boat pulling the fishing line from his fishing reel. The whole system is initially at rest and the fishing line unwinds from the reel at a radius of 4.50 cm from its axis of rotation. The reel is given an angular acceleration of 110. rad/s2 for 2.00 s. What is the angular velocity of the reel? At what speed is fishing line leaving the reel after 2.00 s elapses? How many revolutions does the reel make? How many meters of fishing line come off the reel in this time?

A deep-sea fisherman hooks a big fish that swims away from the boat pulling the fishing line from his fishing reel. The whole system is initially at rest and the fishing line unwinds from the reel at a radius of cm from its axis of rotation. The reel is given an angular acceleration of 110. rad/s2 for 2.00 s. What is the angular velocity of the reel? At what speed is fishing line leaving the reel after 2.00 s elapses? How many revolutions does the reel make? How many meters of fishing line come off the reel in this time?

You-Try 10.4 Now let us consider what happens if the fisherman applies a brake to the spinning reel, achieving an angular acceleration of rad/s2. How long does it take the reel to come to a stop?

You-Try 10.5 Large freight trains accelerate very slowly. Suppose one such train accelerates from rest, giving its m radius wheels an angular acceleration of rad/s2. After the wheels have made 200 revolutions (assume no slippage): How far has the train moved down the track? What is the final angular velocity of the wheels? What is the final linear velocity of the train?

Large freight trains accelerate very slowly
Large freight trains accelerate very slowly. Suppose one such train accelerates from rest, giving its m radius wheels an angular acceleration of rad/s2. After the wheels have made 200. revolutions (assume no slippage): How far has the train moved down the track? What is the final angular velocity of the wheels? What is the final linear velocity of the train?

You-Try 10.6 A person decides to use a microwave oven to reheat some lunch. In the process, a fly accidentally flies into the microwave and lands on the outer edge of the rotating plate and remains there. If the plate has a radius of 15.0 cm and rotates at 6.00 rpm, calculate the total distance traveled by the fly during a min cooking period. (Ignore the start- up and slow-down times).

A person decides to use a microwave oven to reheat some lunch
A person decides to use a microwave oven to reheat some lunch. In the process, a fly accidentally flies into the microwave and lands on the outer edge of the rotating plate and remains there. If the plate has a radius of 15.0 cm and rotates at 6.00 rpm, calculate the total distance traveled by the fly during a 2.00 min cooking period. (Ignore the start-up and slow-down times).

Assignment Current: Read OpenStax sections 10.1-10.2

Homework Questions?

Warm-Up 6: March 3, 2015 Suppose a piece of dust finds itself on a rotating CD. If the spin rate of the CD is 500. rpm, and the piece of dust is 4.30 cm from the center, what is the total distance traveled by the dust in minutes?

AP Physics C

Dynamics of Rotational Motion: Rotational Inertia

Think/Write-Pair-Share
You push a door open. The door is attached to the wall with hinges. What could affect how quickly it opens? List as many things as possible.

Torque on a point mass 𝜏= 𝑟 ⊥ 𝐹 𝜏=𝑟𝑚𝑎 𝜏=𝑟𝑚 𝑟𝛼 𝜏=𝑚 𝑟 2 𝛼 𝜏=𝐼𝛼
This is analogous to Newton’s second law, 𝐹=𝑚𝑎

Rotational Inertia Rotational inertia, also called moment of inertia, is the rotational analogue of inertial mass. Resistance to angular acceleration. Represented by 𝐼 Has units of kg∙m2 For a point mass, 𝐼=𝑚 𝑟 2 In general, 𝐼= 𝑚 𝑟 2 𝜏 𝑛𝑒𝑡 =𝐼𝛼

Some Rotational Inertias

You-Try 10.7 Consider the father pushing a playground merry-go-round shown. He exerts a force of N at the edge of the 50.0 kg merry-go- round, which has a 1.50 m radius. Consider the merry-go-round to be a uniform disk with negligible retarding friction. Calculate the angular acceleration when no one is on the merry-go-round. Calculate the angular acceleration when an kg child sits 1.25 m away from the center.

Consider the father pushing a playground merry-go-round shown
Consider the father pushing a playground merry-go-round shown. He exerts a force of 250. N at the edge of the kg merry-go-round, which has a 1.50 m radius. Consider the merry-go-round to be a uniform disk with negligible retarding friction. Calculate the angular acceleration when no one is on the merry-go-round. Calculate the angular acceleration when an 18.0 kg child sits 1.25 m away from the center.

You-Try 10 Consider the same merry-go-round with child from You-Try 10.7. How long does it take the father to give the merry-go-round an angular velocity of rad/s. How many revolutions must he go through to generate this velocity? If he exerts a slowing force of 300. N at a radius of 1.35 m, how long would it take him to stop them?

Consider the same merry-go-round with child.
How long does it take the father to give the merry- go-round an angular velocity of 1.50 rad/s. How many revolutions must he go through to generate this velocity? If he exerts a slowing force of 300. N at a radius of m, how long would it take him to stop them?

Homework Questions?

Warm-Up 16: March 4, 2015 Zorch, an enemy of Superman, decides to slow Earth’s rotation to once per 28.0 hours by exerting an opposing force at and parallel to the equator. Superman is not immediately concerned, because he knows Zorch can only exert a 4.00× N force. How long must Zorch push with this force to accomplish his goal? 𝐼 sphere = 2 5 𝑀 𝑅 2

Zorch, an enemy of Superman, decides to slow Earth’s rotation to once per 28.0 hours by exerting an opposing force at and parallel to the equator. Superman is not immediately concerned, because he knows Zorch can only exert a 4.00× N force. How long must Zorch push with this force to accomplish his goal? 𝐼 sphere = 2 5 𝑀 𝑅 2

AP Physics C

Homework Questions?

Warm-Up 10: March 5, 2015 Consider the same merry-go-round with child from You-Try 10.7. How long does it take the father to give the merry-go-round an angular velocity of rad/s. How many revolutions must he go through to generate this velocity? If he exerts a slowing force of 300. N at a radius of 1.35 m, how long would it take him to stop them?

Consider the same merry-go-round with child.
How long does it take the father to give the merry- go-round an angular velocity of 1.50 rad/s. How many revolutions must he go through to generate this velocity? If he exerts a slowing force of 300. N at a radius of m, how long would it take him to stop them?

Homework Questions?

Rotational Kinetic Energy: Work and Energy Revisited

Work Assuming a force that is always tangent to the circle of motion,
𝑊 𝑛𝑒𝑡 = 𝐹 𝑛𝑒𝑡 ∆𝑠 𝑊 𝑛𝑒𝑡 =𝑟 𝐹 𝑛𝑒𝑡 ∆𝑠 𝑟 𝑊 𝑛𝑒𝑡 = 𝜏 𝑛𝑒𝑡 ∆𝜃 𝑊 𝑛𝑒𝑡 =𝐼𝛼∆𝜃 Since 𝜔 2 = 𝜔 𝛼𝜃, 𝛼𝜃= 𝜔 2 − 𝜔 0 2 , 𝑊 𝑛𝑒𝑡 = 1 2 𝐼 𝜔 2 − 𝜔 0 2 𝑊 𝑛𝑒𝑡 = 1 2 𝐼 𝜔 𝐼 𝜔 0 2

Rotational Kinetic Energy
𝐾 𝑟𝑜𝑡 = 1 2 𝐼 𝜔 2 Work-Energy Theorem for rotational motion: 𝑊 𝑛𝑒𝑡 = 1 2 𝐼 𝜔 𝐼 𝜔 0 2

You-Try 10.8 A person spins a large grindstone by placing her hand on its edge and exerting a force through part of a revolution. How much work is done if she exerts a force of 200. N through a rotation of 1.00 rad? The force is kept perpendicular to the grindstone’s m radius. What is the final angular velocity if the grindstone has a mass of 85.0 kg? What is the final rotational kinetic energy?

A person spins a large grindstone by placing her hand on its edge and exerting a force through part of a revolution. How much work is done if she exerts a force of N through a rotation of 1.00 rad? The force is kept perpendicular to the grindstone’s m radius. What is the final angular velocity if the grindstone has a mass of 85.0 kg? What is the final rotational kinetic energy?

You-Try 10.9 A typical small rescue helicopter has four blades; each is 4.00 m long and has a mass of 50.0 kg. The blades can be approximated as thin rods that rotate about one end 𝐼= 𝑀 𝑙 The helicopter has a total loaded mass of kg. Calculate the rotational kinetic energy of the blades when they rotate at 300 rpm. Calculate the translational kinetic energy of the helicopter when it flies at 20.0 m/s. To what height could the helicopter be raised if all of the rotational kinetic energy could be used to lift it?

A typical small rescue helicopter has four blades; each is 4
A typical small rescue helicopter has four blades; each is 4.00 m long and has a mass of 50.0 kg. The blades can be approximated as thin rods that rotate about one end 𝐼= 𝑀 𝑙 The helicopter has a total loaded mass of kg. Calculate the rotational kinetic energy of the blades when they rotate at 300 rpm. Calculate the translational kinetic energy of the helicopter when it flies at 20.0 m/s. To what height could the helicopter be raised if all of the rotational kinetic energy could be used to lift it?

Warm-Up 24: March 6, 2015 Calculate the rotational kinetic energy in the motorcycle wheel if its angular velocity is rad/s, its mass is 12.0 kg, 𝑅 1 =0.280 m, 𝑅 2 =0.330 m.

Calculate the rotational kinetic energy in the motorcycle wheel if its angular velocity is 120. rad/s, its mass is kg, 𝑅 1 =0.280 m, 𝑅 2 =0.330 m.

Chemistry Olympiad March 21 or 22 at 9 AM (you may take the test either day) USD Shiley Center for Science and Technology – rooms 129, 130, 133 Approximately 2 hours Registration is due today!

Conservation of Energy
𝑈 0 + 𝐾 0 =𝑈+𝐾 For an object starting at rest 𝐾 0 =0 , and rolling down to the bottom of an incline 𝑈=0 , 𝑈 0 =𝐾 𝑚𝑔ℎ= 1 2 𝑚 𝑣 𝐼 𝜔 2

Think-Pair-Share Three cans of the same size and mass are at the top of an incline. Can 1 has a very low friction coating. Can 2 contains thin soup. Can 3 contains thick soup. If all cans are released at the same time, in what order do they reach the bottom?

Think-Pair-Share Answer

You-Try 10.10 Calculate the final speed of a solid cylinder that rolls down a 2.00 m high incline. The cylinder starts from rest, has a mass of kg, and has a radius of 4.00 cm.

Homework Questions?

Warm-Up 28: March 9, 2015 A ball with an initial velocity of 8.00 m/s rolls up a hill without slipping. Treating the ball as a spherical shell, calculate the vertical height it reaches. Repeat the calculation for the same ball if it slides up the hill without rolling.

A ball with an initial velocity of 8
A ball with an initial velocity of 8.00 m/s rolls up a hill without slipping. Treating the ball as a spherical shell, calculate the vertical height it reaches. Repeat the calculation for the same ball if it slides up the hill without rolling.

Lab soon Lab requires two rolls of toilet paper per lab group.

Homework Questions?

PhET Simulations Ladybug Revolution My Solar System

Warm-Up 22: March 10, 2015 What is the final velocity of a hoop that rolls without slipping down a 5.00 m high hill, starting from rest?

Angular Momentum and Its Conservation

Think/Write-Pair-Share
Write down an expression that you think would equal angular momentum.

Angular Momentum The rotational analogue of linear momentum
Represented by 𝐿 𝐿=𝐼𝜔 Has units of kg⋅m2/s Conserved when net external torque is zero.

You-Try 10.11 Calculate the angular momentum of the Earth due to its rotation. 𝑀=5.979× kg, 𝑅=6.376× m

Newton’s 2nd Law 𝜏 𝑛𝑒𝑡 = ∆𝐿 ∆𝑡 Analogous to 𝐹 𝑛𝑒𝑡 = ∆𝑝 ∆𝑡 = ∆ 𝑚𝑣 ∆𝑡

You-Try 10.12 Suppose a person exerts a 2.50 N force perpendicular to the Lazy Susan’s m radius for s. What is the final angular momentum of the Lazy Susan if it starts from rest, assuming friction is negligible? What is the final angular velocity of the Lazy Susan, given that its mass is 4.00 kg and assuming its moment of inertia is that of a disk?

Suppose a person exerts a 2
Suppose a person exerts a 2.50 N force perpendicular to the Lazy Susan’s m radius for s. What is the final angular momentum of the Lazy Susan if it starts from rest, assuming friction is negligible? What is the final angular velocity of the Lazy Susan, given that its mass is 4.00 kg and assuming its moment of inertia is that of a disk?

Conservation of Angular Momentum
𝜏 𝑛𝑒𝑡 = ∆𝐿 ∆𝑡 If there is no net torque, then ∆𝐿=0. There is no change in angular momentum. Angular momentum is conserved. 𝐿 0 =𝐿 𝐼 0 𝜔 0 =𝐼𝜔

Think-Pair-Share Think of at least one real-world example where a change in rotational inertia, 𝐼, causes a change in rotational velocity, 𝜔.

Examples Ice skaters spinning Elliptical orbits Storms
Formation of solar system

You-Try 10.14 Suppose an ice skater is spinning at rev/s with her arms extended 𝐼=2.34 kg∙ m 2 . What is her angular velocity after she pulls in her arms? 𝐼=0.363 kg∙ m 2 What is her rotational kinetic energy before and after she pulls in her arms?

Suppose an ice skater is spinning at 0
Suppose an ice skater is spinning at rev/s with her arms extended 𝐼=2.34 kg∙ m 2 . What is her angular velocity after she pulls in her arms? 𝐼=0.363 kg∙ m 2 What is her rotational kinetic energy before and after she pulls in her arms?

Homework Questions?

Warm-Up: March 11, 2015 You-Try 10.14

Suppose an ice skater is spinning at 0

Homework Questions?

Collisions of Extended Bodies in Two Dimensions
OpenStax Section 10.6 AP Physics 1

Collisions Momentum can be transferred between objects.
Both linear and angular momentum are conserved if there are no external forces. Angular momentum is conserved if there are no external torques.

You-Try 10.15 Suppose the disk shown has a mass of 50.0 g and an initial velocity of 30.0 m/s when it strikes the stick that is m long and 2.00 kg. The disk adheres to the stick, and they rotate together, pivoting around the nail. What is the angular velocity of the two after the collision? What is the kinetic energy before the collision? What is the kinetic energy after the collision? What is the total linear momentum before the collision? What is the total linear momentum after the collision?

Suppose the disk shown has a mass of 50
Suppose the disk shown has a mass of 50.0 g and an initial velocity of m/s when it strikes the stick that is 1.20 m long and 2.00 kg. The disk adheres to the stick, and they rotate together, pivoting around the nail. What is the angular velocity of the two after the collision? What is the kinetic energy before the collision? What is the kinetic energy after the collision? What is the total linear momentum before the collision? What is the total linear momentum after the collision?

Percussion point The place on a pivoting object where a collision causes no force on the pivot point. In sports, also called the sweet spot.

Homework Questions?

Warm-Up: March 12, 2013 A 25.0 N⋅m torque is applied to a cylinder, causing an angular acceleration of 1.25 rad/s2. What is the moment of inertia of the cylinder?

Lab You will be assigned a group.
Part 1: Determine the moment of inertia of a roll of toilet paper. Create a graph where the slope of the best fit line is equal to 𝐼. Allowed materials: Stopwatch, masses of known mass, rods (to hold the toilet paper), tape, ruler, meterstick Part 2: Details tomorrow

Uriostegui Parra, Karla
Lab Groups Group 1 Cdebaca, Paul Nguyen, Peter Kibret, Elroi Group 2 Dolphin, Jeremy Nguyen, Christina Randazzo, Joseph Group 3 Lenhoff, Shane To, Frank Le, Tiffany Uriostegui Parra, Karla Group 4 Nava Saucedo, Hector Galac, Roger Regge Lopez, Isabel Hy, Kevin Group 5 Kirk, John Peek, Angela Wu, Tong Moreno, Mark Group 6 Lenhoff, Erin Sutton, Foster Garcia-Ayala, Diana Gorman, Courtney Group 7 Nguyen, Phat Jezycki, Jocelyn Basinger, Shelby Lee, Justin

March 13, 2015 You have all of class to work on the lab from yesterday. Stay in your assigned seats until after Mr. Szwast takes attendance.

Warm-Up: March 16, 2015 You-Try 10.15
Suppose the disk shown has a mass of 50.0 g and an initial velocity of 30.0 m/s when it strikes the stick that is m long and 2.00 kg. The disk adheres to the stick, and they rotate together, pivoting around the nail. What is the angular velocity of the two after the collision? What is the kinetic energy before the collision? What is the kinetic energy after the collision? What is the total linear momentum before the collision? What is the total linear momentum after the collision?

Suppose the disk shown has a mass of 50
Suppose the disk shown has a mass of 50.0 g and an initial velocity of m/s when it strikes the stick that is 1.20 m long and 2.00 kg. The disk adheres to the stick, and they rotate together, pivoting around the nail. What is the angular velocity of the two after the collision? What is the kinetic energy before the collision? What is the kinetic energy after the collision? What is the total linear momentum before the collision? What is the total linear momentum after the collision?

Percussion point The place on a pivoting object where a collision causes no force on the pivot point. In sports, also called the sweet spot.

𝐼= 𝐼 𝐶𝑀 +𝑚 𝑑 2 Parallel Axis Theorem
If a body is rotating about an axis parallel to an axis through its center of mass: 𝐼= 𝐼 𝐶𝑀 +𝑚 𝑑 2 Where 𝑑 is the distance between the axis of rotation and a parallel axis through the center of mass.

Lab: Part 2

Lab: Part 2 Determine two heights to drop the two rolls of toilet paper (holding on to the end of one roll) at the same time, so that they hit the ground at the same time. At least one height must be greater than or equal to 1.oo meters You will get a second roll of toilet paper after you complete your calculations. The calculations must be recorded in your lab notebook.

Warm-Up 40: March 19, 2015 Three children (masses 22.0, 28.0 and 33.0 kg) are riding on the edge of a merry-go-round that is kg, has a m radius, and is spinning at 20.0 rpm. If the child of mass 28.0 kg moves to the center of the merry-go-round, what is the new angular velocity in rpm?

Homework Questions

Gyroscopic Effects: Vector Aspects of Angular Momentum
OpenStax Section 10.7 AP Physics 1

Think-Pair-Share A force is being applied to the outside of a disk, causing it to spin in a counterclockwise direction. What is the direction of the torque?

𝝉 , 𝝎 , and 𝑳 𝝎 and 𝐿 are in the same direction.
The direction is determined by the right-hand rule.

Right Hand Rule (one of them)

Gyroscopic Effect A person is holding a spinning wheel. She lifts it with her right hand and pushes down with her left hand in an attempt to rotate the wheel. This action creates a torque directly toward her. This torque causes a change in angular momentum ΔL in exactly the same direction. The wheel moves toward the person, perpendicular to the forces she exerts on it.

Gyroscope As seen in figure (a), the forces on a spinning gyroscope are its weight and the supporting force from the stand. These forces create a horizontal torque on the gyroscope, which create a change in angular momentum ΔL that is also horizontal. In figure (b), ΔL and L add to produce a new angular momentum with the same magnitude, but different direction, so that the gyroscope precesses in the direction shown instead of falling over.

Think-Pair-Share Rotational kinetic energy is associated with angular momentum. Does that mean that rotational kinetic energy is a vector?

Assignment Read OpenStax Chapter 10 OpenStax page 352 #1-30
OpenStax page 356 #1-47 odd

Lab You have the rest of class to work on the lab.

March 20, 2015 You have all of class to work on the lab. Stay in your assigned seat until after Mr. Szwast has taken attendance. Today is the last day of class for the lab. Labs due Wednesday, March 25

Warm-Up: February 27, 2015 Write down everything you remember about uniform circular motion.

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Warm-Up: February 27, 2015 Write down everything you remember about uniform circular motion.

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