Transcript Concept Question: Rotating Rod

Three Dimensional Roation:
Problem Solving
8.01
W14D2
Today’s Reading Assignment
Young and Freedman: 10.7
Announcements
Problem Set 11 Due Thursday Dec 8
9 pm
Sunday Tutoring in 26-152 from 1-5 pm
2
Concept Question: Rotating Rod
Consider a massless rod of length I with two point-like objects of mass m
at each end, rotating about the vertical z-axis with angular speed ω. There
is a sleeve on the axis of rotation. At the moment shown in the figure, two
forces are acting on the sleeve. The direction of the change of the angular
momentum about the center of mass points
1. along the z-axis.
2. along the line formed by the rod.
3. in the plane of the figure, perpendicular
to the line formed by the rod.
4. into the plane of the figure.
5. out of the plane of the figure.
Concept Q. Answer: Rotating Rod
Answer 4. The torque about the
center of mass points into the plane
of the figure. Therefore the direction
of the change of the angular
momentum about the center of mass
points into the plane of the figure.
Problem Solving Strategy
r
1. Calculate torque about appropriate point P,  P
r
2. Calculate angular momentum about P, L P
3. Apply approximation that  =  to decide which
contribution to the angular momentum
about P is
r
changing in time. Calculate dL P / dt
r
r
4. Apply torque law  P  dL P / dt to determine
direction and magnitude of angular precessional
velocity 
5. Apply Newton’s Second Law to center of mass
motion
Table Problem: Suspended
Gyroscope
A gyroscope wheel is at one end of
an axle of length d . The oth er end of
the axle is suspend ed from a string
of length s . The wheel is set into
motion so that it executes uniform
precession in the horizontal plane.
The string makes an angle  with
the vertical. The wheel has mass M
and moment of inertia about its
center of mass I cm . Its sp in angular
speed is  . Neglect the mass of the
shaft and the mass of the string.
Assume    . What is the
direction and magnitude of
precessional angular velocity?
the
Table Prob. Gyroscope on Rotating Platform
A gyroscope c onsists of an axle of
negligible mass and a disk of mass M
and rad ius R mounted on a platform
that ro tates with angular speed  as
shown in the figure below. The
gyroscope is spinning with angular
speed  . Forces Fa and Fb act on the
gyroscopic mounts. The goal of this
problem is to find the magnitudes of the
forces Fa and Fb . You may assume that
the moment of inertia of the gyroscope
about a n axis passing thr ough the
center o f mass normal to the plane of
the d isk is given by I n .
a) Calculate the torque about the center of mass of the gyroscope.
b) Calculate the angular momentum about the center of mass of the gyroscope.
c) Use Ne wtonХs Second Law find a relationship between Fa and Fb , the mass
M of the g yroscope, and the gravitational constant g .
d) Use the to rque equation and Ne wtonХs Second Law to find expressions for Fa
and Fb .
Table Problem: Mill Stone
In a mill, grain is ground by a massive
wheel that ro lls without slipping in a
circle on a flat horizontal mill stone
driven by a vert ical shaft. The rolli ng
wheel has mass M , radius b and is
constra ined to roll in a horizonta l circle
of radius R at angular speed  . The
wheel pushes do wn on the lower mill
stone with a force equal to twice its
weight (normal force). The mass of the
axle of the wheel can be neglected.
Express your ans wers to the following
questions in terms of R , b , M ,  , and
g as need ed. The goal of this problem
is to find  .
a) What is the relation between the angular speed  of the whee l about its axle and
the angu lar speed  about th e vert ical axis?
b) Find the time derivative of the angular momentum about the joint (about the point
P in the figure above) dL P / dt .
c) What is the torque about the joint (ab out the point P in the figure above ?
d) What is the value of  ?
Table Problem: Sopwith Camel
The Sopwith Camel was a single-engine fighter plane flown by
British pilots during WWI (and also by the character Snoopy in
the Peanuts comic strip). It was powered by a radial engine, and
the entire engine rotated with the propeller. The Camel had an
unfortunate property: if the pilot turned to the right the plane
Tended to go into dive, while a left turn caused the plane to climb
steeply. These tendencies caused inexperienced pilots to crash
or stall during takeoff.
Table Problem: Sopwith Camel
From the perspective of the pilot, who sat behind the engine,
did the engine rotate clockwise or counter clockwise?
Odd tables: Argue on the basis of torque (and third law pairs).
Even tables: Argue on the basis of conservation of angular
momentum in the horizontal plane.
Concept Question: Stabilizing a Turning Car
When making a turn every car
has a tendency to roll over
because its center of mass is
above the plane where the
wheels contact the road.
Imagine a race car going counter
clockwise on a circular track. It
could mitigate this effect by
mounting a gyroscope on the car.
To be effective the angular
velocity vector of the gyro should
point
1) ahead
2) behind
3) to the left
4) to the right
5) up
6) down
Concept Question: Stabilizing a Turning Car
Answe r 4.
Note that the same orientation of the gyro will stabilize the car when it turns
to the right as well. The torque applied to the car by the gyro will change
direction, pointing toward the rear of the car instead of the front. But now the
result of the curved path is tending to roll the car to the left instead of to the
right. The gyro still acts to prevent a roll-over.