UTPA STEM/CBI Courses/Physics (Calculus Based)/Angular Momentum and its Conservation
Course Title: Calculus Based Physics I
Lecture Topic: Angular Momentum and its Conservation
Instructor: Liang Zeng
Institution: University of Texas-Pan American
- Primary Objectives- By the next class period students will be able to:
- Know the definition for angular momentum L ⃗=r ⃗ ×p ⃗=r ⃗ ×mv ⃗ (direction, right-hand rule) for a particle
- Know how to calculate angular momentum for multi-particle system L ⃗=∑▒ r ⃗_i ×m_i v ⃗_i
- Know why we need and how to calculate moment of inertia for rigid bodies (continuous mass distribution): L ⃗=Iω ⃗ where I= ∑▒〖m_i r_i^2 〗 becomes the integration of r2 with regard to dm; how to determine the direction of angular momentum (right-hand rule)
- Know the formula for moment of inertia and its values for rigid objects of the most common shapes: rod, cylinder, ball, thin-shell type, plate
- Know that torque equals change of angular momentum, definition of torque τ ⃗=r ⃗ ×F ⃗ , relate this to physical problems; determine the direction of τ ⃗ (right-hand rule)
- Know how to geometrically determine center of mass in order to calculate the magnitude of position vector r ⃗ in the torque formula
- Know how to calculate center of mass for more complicated shapes based on calculus (line density, area density, etc.)
- Know the relation τ ⃗=(dL ⃗)/dt=d(Iω ⃗ )/dt=I × α ⃗ ( assume I is constant); α ⃗ is related to the change of angular speed
- Know the relationship between α ⃗ and tangential acceleration: a ⃗= α ⃗r
- Know when the law of Conservation of Angular Momentum applies (when net torque = 0)
- Discern in the some physical systems whether net torque equals or is not equal to 0 (revolution of planets around the Sun in solar system, neutron star spinning faster when it collapses, etc.)
- Know when net torque does not equal 0, total angular momentum does not conserve, consequently, change of angular momentum: phenomena such as a spinning wheel falling down, precession, a spinning wheel stabilize itself better than a non-spinning wheel, a fast-moving bicycle is not as easy to fall as a slow-moving bicycle
- When net torque is not equal to 0, a spinning object will undergo precession (earth, gyroscope, top)
- Sub Objectives- The objectives will require that students be able to:
- Difficulties- Students may have difficulty:
- Calculating center of mass and moment of inertia using calculus
- Determining the directions of angular momentum and torque using right-hand rule
- Determining the direction of torque in seesaw (non-intuitive)
- Why we use the right hand rule to determine the direction of torque - counter-clockwise, clockwise depends on observer’s position, more ambiguous
- Understanding why net torque can change angular momentum
- Understanding why we need to introduce angular momentum
- In rotation (m,r ⃗,ω ⃗) using an example opening a door: your hand pushing at the knob versus pushing in the middle
- Earth spinning: somebody standing on the equator versus at another altitude, r and v are both changing, but omega is the same (one circle per 24 hours)
- Compare linear momentum (kg m/s) with angular momentum (rotational/circular momentum) ( kg m2/s) – units are different
- Terms: circular (of a circle), rotational (of rotation), and angular (of angle) momentum mean pretty much the same, different emphasis
- When a person walks toward center of the merry-go-around or visa versa, r changes, v changes, moment of inertia changes between the initial and final positions
- Determining when net torque equals zero: Earth revolution example
- Net torque of earth’s revolution around the Sun: τ ⃗=r ⃗ ×F ⃗=0 (r is parallel to F, so τ ⃗ = 0, so the law of conservation of angular momentum applies). More understanding of Kepler’s second law: Earth covers same area between the Sun and the earth revolution orbit per unit time. Area is a function of angular momentum and time span
- Conservation of angular momentum but not angular acceleration: LEarth is constant, L = mvr = mωrr = mr2ω but r is changing with t, so I = mr(t)2 is changing, then ω(t) is changing. There is an angular acceleration as a function of time, and earth revolves faster nearer the sun than farther away
- Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
- Gyroscope applications: orientation in Gravity Probe 2; Tedco Toy Gyroscope – from gyroscope.com*
- Fly wheel on a boat (insert boat models and specifications), NASA fly wheel picture*
- A faster spinning wheel has a larger angular momentum and requires a larger torque to affect it, compared to a slower spinning wheel (with a smaller angular momentum), so a faster spinning wheel is more stable. Example: riding a bike quickly versus riding a bike slowly.
- Analyze speed of earth when it moves near the Sun and farther away from the Sun (speed and distance data: nearest, farthest positions); conservation of angular momentum (m1v1r1=m2v2r2) ω is not the same m1 = m2 r1 > r2 v1 < v2
- Ice skaters and divers – body positioning and arm/hand positioning affects angular speed (needs more research and specification on positioning – try to get a video for slow-motion view of ice skaters and divers)*
- Spinning of yo-yo – angular momentum – play the Toys in Space video clip, use text reference (Wilson)*
- Astrojax – string with two balls secured at each end of the string and one free to move along the length of the string. Play the Toys in Space video clip*
Model of Knowledge
- Concept Map
- Content Priorities
- Enduring Understanding
- Definition for angular momentum for a particle;direction of angular momentum (right-hand rule)
- Calculate angular momentum for multi-particle system
- Calculate moment of inertia for rigid bodies (continuous mass distribution)
- Formula for moment of inertia and it values for rigid objects of most common shapes : rod, cylinder, ball, thin-shell type, plate
- Torque equals change of angular momentum, definition of torque and determine direction (right-hand rule)
- Calculate net torque
- Geometrically determine center of mass for very simple shapes in order to calculate the magnitude of position vector in torque formula
- Calculate center of mass for more complicated shapes based on calculus (line density, area density, etc.)
- Angular acceleration α→ = a→/r
- When the law of Conservation of Angular Momentum applies (when net torque = 0)
- Whether net torque equals or not equal to 0 in some physical systems (all planets revolution around the Sun in solar system, neutron star spinning faster when it collapses, etc.)
- Stabilization: A faster spinning wheel has a larger angular momentum and requires a larger torque to affect it, compared to a slower spinning wheel (with a smaller angular momentum), so a faster spinning wheel is more stable. Example: riding a bike quickly versus riding a bike slowly.
- Important to Do and Know
- Calculate moment of inertia for most common shaped rigid objects: rods, plates, sphere, thin-wall sphere, cylinder, thin-wall cylinder
- How gyroscopes work
- Precession: when torque does not equal 0, total angular momentum does not conserve, consequently, change of angular momentum: a spinning object with nonzero net torque will undergo precession (earth, gyroscope, top). Direction of angular momentum follows direction of torque.
- Worth Being Familiar with
- Center of mass
- When applying a lifting force on one handle of the axis of the spinning wheel, determine net torque, describe your feeling about the immediate behavior of the spinning wheel and why
- Earth's revolution around the sun: conservation of angular momentum but not angular acceleration: LEarth is constant, L = mvr = mωrr = mr2ω but r is changing with t, so I = mr(t)2 is changing, then ω(t) is changing. There is an angular acceleration as a function of time
- Enduring Understanding
Assessment of Learning
- Formative Assessment
- In Class (groups)
- Homework (individual)
- In Class (groups)
- Summative Assessment
By the next class period, students will be able to:
The objectives will require that students be able to:
- 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 amount of tension and the contributions to the overall torque before the rope is burned (4 points). 2. Correct solution for the contributions to the overall torque after the rope is burned (3 points). 3. Correct solution for the direction of precession of the wheel (3 points).
Format: 1. Put down your name. 2. State challenge question and its number. 3. Show all your work.
Challenge Question #6: Watch the Youtube video: “Do Try This At Home: Episode 7 – The Wheel”. Two looped ropes are hanging from a beam and a wheel is placed between the two ropes. The wheel is given a spin and one of the ropes is burned so that it no longer supports the wheel. The wheel does not fall, but continues to spin and begins precessing.
a) Analyze using variables the amount of tension in each rope and the various contributions to the overall torque before the rope is burned.
b) Analyze using variables the various contributions to the overall torque in the system after the rope is burned.
c) Analzye why the wheel precesses in the direction that it does in the video after the rope is burned.
RESEARCH & REVISE
TEST YOUR METTLE
Test Your Mettle Quiz
1. (Moment of inertia in a multi particle system: Knight 2nd edition page 378 #14). The four masses shown in the figure are connected by massless, rigid rods (the lines extending horizontally to the right of D and vertically above B are to indicate an xy coordinate system).
a)Find the coordinates of the center of mass b)Find the moment of inertia about an axis that passes through mass A and is perpendicular to the page.
2. (Moment of inertia for a rigid body: Knight 2nd edition page 381 #56). Determine the moment of inertia about t5he axis of the object shown in the following figure.
3. (Angular momentum for a single particle: Knight 2nd edition page 380 #46). What are the magnitude and direction of the angular momentum relative to the origin of the 200 g particle in the following figure.
4. (Angular momentum for a multi particle system: Serway 7th edition page 330 #11). A light, rigid rod 100 m in length joins two particles, with masses 4.00 kg and 3.00 kg, at its ends. The combination rotates in the xy plane about a pivot through the center of the rod (see figure below). Determine the angular momentum of the system about the origin when the speed of each particle is 5.00 m/s.
5. (Net torque: Serway 7th edition page 302-303 #33). Find the net torque on the wheel in the following figure about the axle through O, taking a = 10.0 cm and b = 25.0 cm.
6. (Practice on mvr variables: Serway 7th edition page 332 #32a). A student sits on a freely rotating stool holding two dumbbells, each of mass 3.00 kg (see figure below). 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 is 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 new angular speed of the student.
7. (Conservation of angular momentum: Knight 2nd edition page 383 #81). A satellite follows the elliptical orbit shown. The only force on the satellite is the gravitational attraction of the planet. The satellite’s speed at point a is 8000 m/s.
a)Is there any torque on the satellite? Explain. b)What is the satellite’s speed at point b? c)What is the satellite’s speed at point c?
8. (Torque and change of angular momentum: Serway 7th edition page 304 #49). This problem describes one experimental method for determining the moment of inertia of an irregularly shaped object such as the payload for a satellite. The figure below shows a counterweight of mass m suspended by a cord wound around a spool of radius r, forming part of a turntable supporting the object. The turntable can rotate without friction. When the counterweight is released from rest, it descends through a distance h, acquiring a speed v. Show that the moment of inertia I of the rotating apparatus (including the turntable) is mr2(2gh/v2 – 1).