Title: Experiment 5: Rotational Dynamics and Angular Momentum 8.01 W10D1 Young and Freedman: 10.5-10.6
1Experiment 5 Rotational DynamicsandAngular
Momentum8.01W10D1Young and Freedman
10.5-10.6
2Announcements
Vote Tomorrow Math Review Night Tuesday from
9-11 pm Pset 9 Due Nov 8 at 9 pm No Class
Friday Nov 11 Exam 3 Tuesday Nov 22 730-930
pm W010D2 Reading Assignment Young and Freedman
10.5-10.6
3Rotor Moment of Inertia
4Review Table Problem Moment of Inertia Wheel
- A steel washer is mounted on a cylindrical
rotor . The inner radius of the washer is R. A
massless string, with an object of mass m
attached to the other end, is wrapped around the
side of the rotor and passes over a massless
pulley. Assume that there is a constant
frictional torque about the axis of the rotor.
The object is released and falls. As the mass
falls, the rotor undergoes an angular
acceleration of magnitude a1. After the string
detaches from the rotor, the rotor coasts to a
stop with an angular acceleration of magnitude
a2. Let g denote the gravitational constant. -
- What is the moment of inertia of the rotor
assembly (including the washer) about the
rotation axis?
5Review Solution Moment of Inertia of Rotor
- Force and rotational equations while weight is
descending -
-
-
- Constraint
- Rotational equation while slowing down
- Solve for moment of inertia
Speeding up
Slowing down
6Review Worked Example Change in Rotational
Energy and Work
- While the rotor is slowing down, use work-energy
techniques to find frictional torque on the
rotor. -
7Experiment 5 Rotational Dynamics
8Review Cross Product
- Magnitude equal to the area of the parallelogram
defined by the two vectors - Direction determined by
- the Right-Hand-Rule
9Angular Momentum of a Point Particle
- Point particle of mass m moving with a velocity
- Momentum
- Fix a point S
- Vector from the point S to the location
of the object - Angular momentum about the point S
- SI Unit
10Cross Product Angular Momentum of a Point
Particle
- Magnitude
- moment arm
- perpendicular momentum
11Angular Momentum of a Point Particle Direction
- Direction Right Hand Rule
12Concept Question
- In the above situation where a particle is moving
in the x-y plane - with a constant velocity, the magnitude of the
angular momentum - about the point S (the origin)
- decreases then increases
- increases then decreases
- is constant
- is zero because this is not circular motion
-
13Concept Question
Solution 3. As the particle moves in the positive
x-direction, the perpendicular distance from the
origin to the line of motion does not change and
so the magnitude of the angular momentum about
the origin is constant.
14Table Problem Angular Momentum and Cross Product
A particle of mass m 2 kg moves with a uniform
velocity At time t, the position vector of the
particle with respect ot the point S is Find
the direction and the magnitude of the angular
momentum about the origin, (the point S) at time
t.
15Angular Momentum and Circular Motion of a Point
Particle
- Fixed axis of rotation z-axis
- Angular velocity
- Velocity
- Angular momentum about the point S
16Concept Question
- A particle of mass m moves in a circle of radius
R at an angular speed ? about the z axis in a
plane parallel to but above the x-y plane.
Relative to the origin - 1. is constant.
- 2. is constant but is
not. - 3. is constant but is not.
- 4. has no z-component. .
17Concept Question
Answer 2. The vector represents
a unit vector in the direction of . The
angular momentum about the origin is shown in the
figure to the right The magnitude is constant.
As the particle moves in a circle, the angular
momentum sweeps out a cone (shown in the figure
below) and so the direction of is
changing and hence is not a constant unit
vector.
18Worked Example Changing Direction of Angular
Momentum
- A particle of mass m moves in a circle of radius
R at an angular speed ? about the z axis in a
plane parallel to but a distance h above the x-y
plane. -
- Find the magnitude and the direction of the
angular momentum relative to the origin. - Also find the z component of .
19Solution Changing Direction of Angular Momentum
20Solution Changing Direction of Angular Momentum
21Table Problem Angular Momentum of a Two
Particles About Different Points
Two point like particles 1 and 2, each of mass m,
are rotating at a constant angular speed about
point A. How does the angular momentum about the
point B compare to the angular momentum about
point A? What about at a later time when the
particles have rotated by 90 degrees?
22Table Problem Angular Momentum of Two Particles
- Two identical particles of mass m move in a
circle of radius R, 180º out of phase at an
angular speed ? about the z axis in a plane
parallel to but a distance h above the x-y plane.
-
- a) Find the magnitude and the direction of the
angular momentum relative to the origin. - b) Is this angular momentum relative to the
origin constant? If yes, why? If no, why is it
not constant?
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24Table Problem Angular Momentum of a Ring
- A circular ring of radius R and mass dm rotates
at an angular speed ? about the z-axis in a plane
parallel to but a distance h above the x-y plane.
- a) Find the magnitude and the direction of the
angular momentum relative to the origin. - b) Is this angular momentum relative to the
origin constant? If yes, why? If no, why is it
not constant?
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26Concept Question
- A non-symmetric body rotates with constant
angular speed ? about the z axis. Relative to the
origin - 1. is constant.
- 2. is constant but is
not. - 3. is constant but is not.
- 4. has no z-component.
-
-
27Concept Question
- Answer 2. For this non-symmetric rigid body,
the angular momentum about the origin has time
varying components in the x-y plane, - (where is a radial unit vector pointing
outward from the origin) The magnitude of is
constant because both and are
constant. The direction of in the x-y
plane depends on the instantaneous orientation of
the body and so the direction of is
changing. -
-
28Concept Question
- A rigid body with rotational symmetry body
rotates at a constant angular speed ? about it
symmetry (z) axis. In this case - 1. is constant.
- 2. is constant but is
not. - 3. is constant but is not.
- 4. has no z-component.
- 5. Two of the above are true.
-
-
29Concept Question
- Answer 1. For a symmetric body, all the non-z
components of the angular momentum about any
point along the z-axis cancel in pairs leaving
only a constant non-zero z-component of the
angular momentum so is constant. -
-
30Angular Momentum of Cylindrically Symmetric Body
- A cylindrically symmetric rigid body rotating
about its symmetry axis at a constant angular
velocity with
has angular momentum about any point on its
axis
31Angular Momentumfor Fixed Axis Rotation
- Angular Momentum about the point S
- Tangential component of momentum
- z-component of angular momentum
- about S
32Concept Question Angular Momentum of Disk
A disk with mass M and radius R is spinning with
angular speed ? about an axis that passes through
the rim of the disk perpendicular to its plane.
The magnitude of its angular momentum is
33Concept Question Angular Momentum of Disk
Answer 6. The moment of inertia of the disk
about an axis that passes through the rim of the
disk perpendicular to its plane is I Icm MR2
(3/2)MR2 . So the magnitude of its angular
momentum is L (3/2)MR2 ?.