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•  Finishing up Chapter 11 and starting Chapter 12

Web page: http://www.colorado.edu/physics/phys1110/

Gyroscopes, precession, and statics

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Clicker question 1 Set frequency to BA Three identical wheels are all spinning with the same angular velocity. The total angular momentum of the 3-wheel system has magnitude L. One of the three wheels is flipped upside-down, while the magnitude of its angular velocity remains constant. The new angular momentum of the 3-wheel system has magnitude…

A.  L (same as before) B.  L/3 C.  2L/3 D.  L/2 E.  Some other value Angular momentum is a vector.

Initial: Linitial = Iω + Iω + Iω = 3Iω Final: Lfinal = Iω + Iω − Iω = Iω

ω ωω

Direction of angular velocity from right hand rule.

Note, there was an external torque needed to flip the wheel. Therefore, conservation of angular momentum does not apply.

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Clicker question 2 Set frequency to BA A disk of mass M has a small red blob of mass m on the rim on top of massless Atom-Ant. The disk is initially rotating with angular velocity . Atom-Ant moves the mass toward the center. What happens to the angular momentum of the disk+ant+blob system? A.  stays the same B.  increases C.  decreases D.  changes direction E.  depends on masses

ω0

ω0

I = 12MR

2 +mr2L = Iω

No external torque so angular momentum is conserved. Since r is decreasing, I is decreasing so is increasing. ω

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Clicker question 3 Set frequency to BA A disk of mass M has a small red blob of mass m on the rim on top of massless Atom-Ant. The disk is initially rotating with angular velocity . Atom-Ant moves the mass toward the center. What happens to the rotational kinetic energy of the disk+ant+blob system? A.  stays the same B.  increases C.  decreases D.  changes direction E.  depends on masses

ω0

ω0

No external torque so angular momentum is conserved. Since r is decreasing, I is decreasing so is increasing.

I = 12MR

2 +mr2L = Iω

ω

Krot =12 Iω

2 = 12 Iω( )ω = 1

2 Lω

Since L is constant and is increasing, the rotational kinetic energy is increasing. Atom-Ant is doing work!

ω

3 gyroscopes (to give direction) and 3 accelerometers (to give velocity) can be combined to form an inertial navigation system.

Gyroscopes

The artificial horizon in planes uses gyroscopes to indicate the pitch and roll of the plane in the absence of external views out the window due to clouds or darkness.

Gyroscopes can also be used as stabilizers: The international space station uses four 600 pound gyroscopes spinning at 6500 RPM to maintain orientation

Conservation of angular momentum allows gyroscopes to work. When you spin up a gyroscope, it gets an angular momentum. As long as no external torques act on the gyroscope, it will continue pointing in the same direction.

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Gyroscopes and precession Remember forces that cause radial acceleration just change the velocity direction but not the magnitude.

Similarly, some torques change the angular momentum vector direction rather than the magnitude.

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Bicycle wheel gyroscope

pivot

T

L!

mg

τ!When the wheel is not spinning the torque

from the weight causes the wheel to rotate downward causing an angular velocity into the page. The torque points into the page as well.

When the wheel is spinning with the angular momentum vector perpendicular to the weight vector the torque causes a change in the angular momentum vector direction causing it to precess which is another form of rotation.

Can work out precession frequency: Ω =mgRIω

R

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Gyroscope precession

Clicker question 4 Set frequency to BA

If there is an object is completely stationary (and remains stationary), which answer is the most correct. A.  Net force on the object is 0 B.  Net torque on the object about the center of mass is 0. C.  Net torque on the object about any point in the universe (that

is in the same inertial frame) is 0. D.  Two of A, B, C are true. E.  All three of A, B, C are true.

If there is a net force on the object it must be accelerating.

A net torque about the center of mass would start it rotating.

A net torque about any point in the universe indicates it is rotating about that point and therefore moving.

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Statics The theory behind statics is very simple. Since nothing is moving, the acceleration and angular acceleration is 0.

Statics has just two equations: !F∑ = 0!τ∑ = 0 (about any point)

These are vector equations so we actually have more equations:

Fx∑ = 0 Fy∑ = 0 Fz∑ = 0 τ x∑ = 0 (about any point)

τ y∑ = 0 (about any point) τ z∑ = 0 (about any point)