3-Dimensional Rotation: Gyroscopes

8.01

W13D2

Torque and Time Derivative of Angular Momentum

Torque about S is equal to the time derivative of the angular momentum about S

If the magnitude of the angular momentum is constant then the torque can cause the direction of the angular momentum to change

ext SSS

d

dt

L

Time Derivative of a Vector

Consider a vector

where

A vector can change both magnitude and direction.

Example: Suppose does not change magnitude but only changes direction then

ˆ ˆsinr

d d dA

dt dt dt

A

A

ˆ ˆz rA A A k r

sinrA A

coszA A

ˆ ˆz rA A A k r

Time Derivative of Vectors of Constant Length: Circular

MotionCircular Motion: position vector points radially outward, with constant magnitude but changes in direction. The velocity vector points in a tangential direction to the circle with a constant magnitude. The acceleration vector points radially inward.

ˆd dr

dt dt

rv

v r

ddt

ˆrr r

2

ˆ ˆd d d

v rdt dt dt

va r r

Introduction To Gyroscopic Motion

Deflection of a Free Particle by a Small Impulse

If the impulse << the primary effect is to rotate aboutthe x axis by a small angle .

p

1p

I

Deflection of a Free Particle by a Small Impulse

ave t I p F

ave avet t L r F

ave t L r F

L r I

The application of causes a change in the angular momentum through the torque equation.

I

L

Deflection of a Free Particle by a Small Impulse

I

L

As a result, rotates about the x axis by a small angle . Note that although is in the z direction, is in the negative y direction.

L

Effect of a Small Impulse on a Tethered Ball

The ball is attached to a string rotating about a fixedpoint. Neglect gravity.

Effect of a Small Impulse on a Tethered Ball

The ball is given an impulse perpendicular to and to .p

r

Effect of a Small Impulse on a Tethered Ball

As a result, rotates about the x axis bya small angle . Note that although is in thez direction, is in the negative y direction.L

L

I

Effect of a Small Impulse on a Tethered Ball

IThe plane in which the ball moves also rotates about

the x axis by the same angle. Note that although isin the z direction, the plane rotates about the x axis.

Concept Question: Effect of a Large Impulse on a

Tethered Ball

I

What impulse must be given to the ball in orderto rotate its orbit by 90 degrees as shown withoutchanging its speed?

Effect of a Large Impulse on a Tethered Ball

I

What impulse must be given to the ball in orderto rotate its orbit by 90 degrees as shown withoutchanging its speed?

Solution: Effect of a Large Impulse on a Tethered Ball

I

must halt the y motion and provide a momentumof equal magnitude along the z direction.

Solution: Effect of a Large Impulse on a Tethered Ball

cancels the z component of and adds a componentof the same magnitude in the negative y direction.L

L

Effect of a Small Impulse Couple on a Baton

Now we have two equal masses at the ends of amassless rod which spins about its center. We applyan impulse couple to insure no motion of the CM.

Effect of a Small Impulse Couple on a Baton

Again note that the impulse couple is applied in the zdirection. The resulting torque lies along the negative ydirection and the plane of rotation tilts about the x axis.

Effect of a Small Impulse Couple on Massless Shaft

of a Baton

Instead of applying the impulse couple to the masses one could apply it to the shaft to achieve the same result.

Concept Question: Effect of a Small Impulse Couple on Massless Shaft of a Baton

To make the top of the shaft move in the -y directionin which direction should one apply the top half of animpulse couple?

Solution: Effect of a Small Impulse Couple on

Massless Shaft of a Baton

The impulse couple Ib applied to the shaft has thesame effect as the Ia couple applied directly to themasses. Both produce a torque in the - y direction.

Effect of a Small Impulse Couple on Massless Shaft

of a Baton

Trying to twist the shaft around the y axis causesthe shaft and the plane in which the baton movesto rotate about the x axis.

Effect of a Small Impulse Couple on a Disk

The plane of a rotating disk and its shaft behave justlike the plane of the rotating baton and its shaft whenone attempts to twist the shaft about the y axis.

Effect of a Small Impulse Couple on a Non-Rotating

Disc

This unexpected result is due to the large pre-existing .If the disk is not rotating to begin with, is also thefinal . The shaft moves in the direction it is pushed.

L

L L

Effect of a Small Impulse Couple on a Disk

It does not matter where along the shaft the impulsecouple is applied, as long as it creates the same torque.

Effect of a Force Couple on a Rotating Disk

A series of small impulse couples, or equivalently acontinuous force couple, causes the tip of the shaftto execute circular motion about the x axis.

Effect of a Force Couple on a Rotating Disk

d dt L L

d

dt

L

L

I

I

The precession rate of the shaft is the ratio of themagnitude of the torque to the angular momentum.

Precessing Gyroscope

Toy Gyroscope: Forces and Torque

Gravitational force acts at the center of the mass and points downward

Contact force between the end of the axle and the pylon

Torque about the contact point due to gravitational force

The direction of the torque about pivot points into the page in the figure

,cm gravityˆˆˆ ( )S S b mg b mg r F r k

Torque: Magnitude of Angular Momentum Changes

If the flywheel of the gyroscope is not spinning, gyroscope starts to fall downward and the torque about the pivot point S

induces the gyroscope to start rotating about an axis pointing into page.

Torque induces the magnitude of the angular momentum to change.

SS

d

dt

L

bmg I

S

d 2dt2

Direction of Angular Momentum Changes

If the flywheel is spinning, the spin angular momentum about the center of mass of the flywheel points along the axle, radially outward; the torque causes the spin angular momentum to change its direction, with precessional angular frequency

spincm cm ˆSI L r

spin spincm cm

ˆd d

dt dt

L L

ddt

spin spincm cm cm

ˆ ˆS

dI

dt L L

d / dt

Gyroscope: Precession

Torque about the pivot point

induces the angular momentum to change

Precessional angular frequency of the gyroscope

Newton’s Second Law: center of mass remains at rest

spincmb mg L

S

S

d

dt

L

spincmcm S

b mg b mg

I

L

F

vertical m g 0

Frad

mb2

Gyroscopic Approximation

Flywheel is spinning with an angular velocity

Precessional angular velocity

Total angular velocity

Gyroscopic approximation: the angular velocity of precession is much less than the component of the spin angular velocity ,

total spin

spin S r̂

k̂

S

= S

Table Problem: Gyroscope

A gyroscope wheel is at one end of an axle of length l . The axle is pivoted at an angle with respect to the horizontal. The wheel is set into motion so that it executes uniform precession. The wheel has mass m and moment of inertia Icm about its center of mass . Its spin angular velocity is s . Neglect the mass of the shaft. What is the precessional frequency of the gyroscope? Which direction does the gyroscope rotate?