Levitron and Larmor Precession (Zac Cook)

Principles Investigated:

• Students will be able to use Newton’s Second Law.
• Students will be able to identify magnetic force.
• Students will be able to use torque to resolve a discrepant event.
• Students will be able to understand precession.

Standards:

8.2. Unbalanced forces cause changes in velocity. As a basis for understanding this concept:

a. Students know a force has both direction and magnitude.

b. Students know when an object is subject to two or more forces at once, the result is the cumulative effect of all the forces.

c. Students know when the forces on an object are balanced, the motion of the object does not change.

9-12.1. Newton’s laws predict the motion of most objects. As a basis for understanding this concept:

b. Students know that when forces are balanced, no acceleration occurs; thus an object continues to move at a constant speed or stays at rest (Newton’s first law).

c. Students know how to apply the law ΣF=ma to solve one-dimensional motion problems that involve constant forces (Newton’s second law).

9-12.2. The laws of conservation of energy and momentum provide a way to predict and describe the movement of objects. As a basis for understanding this concept:

f. Students know an unbalanced force on an object produces a change in its momentum.

9-12.5. Electric and magnetic phenomena are related and have many practical applications. As a basis for understanding this concept:

j.* Students know electric and magnetic fields contain energy and act as vector force fields.

Materials:

• Levitron, available at http://www.amazon.com/Fascinations-Levitron-Platinum-Pro-Combo/dp/B0008076B8/ref=sr_1_1?s=toys-and-games&ie=UTF8&qid=1290049714&sr=1-1
• Wide mouth glass
• Circular disk magnets with holes in the center (optional, but recommended)
• Stand to hold disk magnets (optional, but recommended)

Procedure:

1. Show that a disk magnet will flip over and stick to another disk magnet when placed above it and let go (Optional, but very good to setup the discrepant event)
2. Place a group of disk magnets on the stand with another one suspended above them by flipping its pole. (Optional)
3. Setup Levitron beforehand, making sure that the base is balanced and the top has the correct amount of weight.
4. Drop the top onto the base from the approximate height it floats at to show that it should fall.
5. Spin the top on the lifting plate (included with Levitron) and lift the plate/top up to make it levitate.
6. If the top looks stable, take the wide mouth glass and carefully enclose the top.
7. Place the lifting plate on the rim of the glass so that the top is completely enclosed and still spinning.
8. Carefully remove the glass and plate so the top is still floating.
9. Point out how the top begins to wobble right before it falls down.

Student Prior Knowledge:

This depends on the grade you are teaching. For 8^th graders, just make sure they know Newton’s Second Law. For high school students, the best time to introduce this demonstration would be when they begin magnetism, so they should know Newton’s Laws, torque and energy conservation.

Explanation:

Newton’s 2^nd Law Perspective (8^th grade only): The gravitational force is balanced by an upward force created by the magnets in the top and in the base. Since the net force is zero, there is no acceleration and the top floats.

High School Explanation: The top flips over when simply dropped from above, yet it floats when spun then lifted above. This discrepancy is resolve through torque and precession. The magnetic field, which is basically vertical, causes a torque on the magnet in the top. Normally this torque would simply cause the magnet to flip over, but since the top is spinning the torque causes the top to precess similar to how gravity causes a gyroscope to remain vertical when spun. Eventually friction from the air slows the top down causing it to flip over and fall.

Equations:

τ = dL/dt = d(Iω)/dt = Iα = μBsinθ (1)

τ = Ω_PLsinθ (2)

Where τ is torque, L is angular velocity, I is the moment of inertia (of the top), ω is the angular velocity, α is the angular acceleration, μ is the magnetic moment (of the top), B is the magnetic field (from the base) and Ω_P is the speed of precession.

Note that (2) is only valid when the torque is constant (which in our case it is since B is not changing). We can also see from (2) that the speed of precession is inversely proportional to the angular momentum (Ω_P~ τ/L). This explains why the top begins to wobble right before it falls. The air friction decreases L which increases Ω_P.

Questions and Answers:

1. Why does the top sometimes fly off to one side?

A. If the base isn’t level the magnetic field will no longer be completely vertical and the torque won’t cause the correct precession.

2. Why does the top sometimes fall straight down immediately after you lift it?

A. The levitation point is extremely precise. If you rush the procedure you can impart a small amount of vertical momentum to the top which will cause it not to remain levitating.

3. What would happen if the device were to be placed in a vacuum?

A. Theoretically, there would be no friction, and the top should continue floating indefinitely.

Applications to Everyday Life:

Physics - Particle with magnetic spin (such as neutrons) can be trapped this way.

Chemistry - Nuclear magnetic resonance (NMR) spectroscopy. Each atom/isotope has a unique Larmor frequency when placed in the magnetic field. Since they have magnetic moments they can be captured, identified and analyzed.

Biology - NMR spectroscopy is used to study protein and nucleic acid structures and their functions.

Astronomy - Though not a result of magnetism, there are other types of precession including axial precession which accounts for the change in the North Star’s position, perihelion precession which explains how a planet’s orbit gradually rotates over time and is the astronomical explanation used to explain ice ages.

Photographs:

Videos:

See attachments below.

References:

M. V. Berry, The Levitron^TM: an adiabatic trap for spins, Proc. Roy Soc. Lond., A (1996) 452, 1207-1220.

http://www.phy.bris.ac.uk/people/berry_mv/publications.html

“Gyroscope.” Wikipedia, The Free Encyclopedia. Wikimedia Foundation, Inc. 22 July 2004. Web. 06 Oct. 2010.

http://en.wikipedia.org/wiki/Gyroscope

“Precession.” Wikipedia, The Free Encyclopedia. Wikimedia Foundation, Inc. 22 July 2004. Web. 28 Sep. 2010.

http://en.wikipedia.org/wiki/Precession

“NMR spectroscopy.” Wikipedia, The Free Encyclopedia. Wikimedia Foundation, Inc. 22 July 2004. Web. 07 Oct. 2010.

http://en.wikipedia.org/wiki/NMR_spectroscopy