Rotational Motion and Angular Momentum

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Rotational Motion and Angular Momentum

• Unit 6

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Lesson 1 Angular Position, Velocity, and
Acceleration
When a rigid object rotates about its axis, at
any given time different parts of the object have
different linear velocities and linear
accelerations.
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P is at polar coordinate (r, q)
s rq
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q is measured in radians (rad)
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Converting from Degrees to Radians
90o p/2 rad
60o p/3 rad
45o p/4 rad
270o 3p/2 rad
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Angular Displacement (Dq)
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Average Angular Speed (w)
Ratio of the angular displacement of a rigid
object to the time interval Dt.
The rad/s is the unit for angular speed.
w is positive when q increases (counterclockwise
motion)
w is negative when q decreases (clockwise motion)
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Instantaneous Angular Speed (w)
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Instantaneous Angular Acceleration (a)
The rad/s2 is the unit for angular speed.
a is positive when object rotates
counterclockwise and speeds up
OR
when object rotates clockwise and slows down
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Direction of Angular Velocity and Acceleration
Vectors
w and a are vector quantities with magnitude and
direction
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Right-Hand Rule
Wrap four fingers of the right hand in the
direction of rotation.
Thumb will point in the direction of angular
velocity vector (w).
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Example 1
A rigid object is rotating with an angular speed
w lt 0. The angular velocity vector w and the
angular acceleration vector a are antiparallel.
The angular speed of the rigid object is
a) clockwise and increasing
b) clockwise and decreasing
c) counterclockwise and increasing
d) counterclockwise and decreasing
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Example 2
During a certain period of time, the angular
position of a swinging door is described by q
5.00 10.0 t 2.00 t2, where q is in radians
and t is in seconds. Determine the angular
position, angular speed, and angular acceleration
of the door at
a) at t 0
b) at t 3.00 s
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Lesson 2 Rotational Kinematics with Constant
Angular Acceleration
dw a dt
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Eliminating t from previous two equations,
Eliminating a from previous two equations,
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x
Position
q
v
Velocity
w
a
Acceleration
a
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Example 1
A wheel rotates with a constant angular
acceleration of 3.50 rad/s2.
a) If the angular speed of the wheel is 2.00
rad/s at ti 0, through what angular
displacement does the wheel rotate in 2.00 s ?
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b) Through how many revolutions has the wheel
turned during this time interval ?
c) What is the angular speed of the wheel at t
2.00 s ?
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Example 2
A rotating wheel requires 3.00 s to rotate
through 37.0 revolutions. Its angular speed at
the end of the 3.00 s interval is 98.0 rad/s.
What is the constant angular acceleration of the
wheel ?
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Lesson 3 Angular and Linear Quantities
Since s rq,
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Tangential Speed
v rw
Tangential speed depends on distance from axis of
rotation
Angular speed is the same for all points
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Tangential Acceleration
Since v rw,
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Centripetal Acceleration in terms of Angular Speed
Since v rw,
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Total Linear Acceleration
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Example 1
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a) Find the angular speed of the disc in rev/min
when information is being read from the
innermost first track (r 23 mm) and the
outermost final track (r 58 mm).
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b) The maximum playing time of a standard music
CD is 74 min 33 s. How many revolutions does
the disc make during that time ?
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c) What total length of track moves past the lens
during this time ?
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d) What is the angular acceleration of the CD
over the 4,473 s time interval ? Assume that a
is constant.
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Example 2
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a) Calculate the speed of a link of the chain
relative to the bicycle frame.
b) Calculate the angular speed of the bicycle
wheels.
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c) Calculate the speed of the bicycle relative
to the road.
d) What pieces of data, if any, are not
necessary for the calculations ?
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Lesson 4 Rotational Kinetic Energy
The total kinetic energy of a rotating rigid
object is the sum of the kinetic energies of its
individual particles.
KErot SKEi S ½mivi2
Since v rw,
KErot ½ S miri2wi2
Factoring out w2,
KErot ½ (S miri2) w2
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KErot ½ (S miri2) w2
The kg . m2 is the SI unit for moment of inertia.
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Substituting I,
I
m
w
v
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Example 1
Consider an oxygen molecule (O2) rotating in the
x-y plane about the z-axis. The rotation axis
passes through the center of the molecule,
perpendicular to its length. The mass of each
oxygen atom is 2.66 x 10-26 kg, and at room
temperature the average separation between the
two atoms is d 1.21 x 10-10 m. (The atoms are
modeled as particles.)
a) Calculate the moment of inertia of the
molecule about the z-axis.
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b) If the angular speed of the molecule about the
z-axis is 4.60 x 1012 rad/s, what is its
rotational kinetic energy ?
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Lesson 5 Calculation of Moments of Inertia
Moment of inertia of a rigid object is evaluated
by dividing the object into many small volume
elements, each with mass Dmi.
Since r m/V,
dm r dV
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Example 1
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Example 2
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Example 3
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Parallel-Axis Theorem
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Example 4
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Moment of Inertia of a Thin Cylindrical Shell
(Hoop)
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Moment of Inertia of a Hollow Cylinder
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Moment of Inertia of a Solid Cylinder (Disk)
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Moment of Inertia of a Rectangular Plate
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Moment of Inertia of a Long Thin Rod
(Axis Through Center)
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Moment of Inertia of a Long Thin Rod
(Axis Through End)
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Moment of Inertia of a Solid Sphere
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Moment of Inertia of a Thin Spherical Shell
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Lesson 6 Torque
The tendency of a force to rotate an object about
some axis is measured by a vector quantity called
torque.
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t r Fsinf Fd
Torque has units of force x length or N.m. (Same
as work but not called Joules.)
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F1 tends to rotate counterclockwise (t)
F2 tends to rotate clockwise (-t)
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Example 1
a) What is the net torque acting on the cylinder
about the rotation axis (z-axis) ?
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b) Suppose T1 5.0 N, R1 1.0 m, T2 15.0 N,
and R2 0.50 m. What is the net torque about
the rotation axis, and which way does the
cylinder rotate starting from rest ?
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Lesson 7 Relationship between Torque and
Angular Acceleration
Ft mat
t Ftr (mat)r
t (mra)r (mr2)a
Since I mr2,
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Example 1
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Example 2
a) Calculate the angular acceleration of the
wheel.
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b) Calculate the linear acceleration of the
object.
c) Calculate the tension in the cord.
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Lesson 8 Work, Power, and Energy in Rotational
Motion
dW F.ds (Fsinf)r dq
(The radial component of F does no work because
it is perpendicular to the displacement.)
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Since t rFsinf,
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St dq Iw dw
Since dW St dq,
dW Iw dw
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Integrating to find total work done,
The net work done by external forces in rotating
a symmetric rigid object about a fixed axis
equals the change in the objects rotational
energy.
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Example 1
a) What is its angular speed when it reaches its
lowest position ?
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b) Determine the tangential speed of the center
of mass and the tangential speed of the lowest
point on the rod when it is in the vertical
position.
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Example 2
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Example 3 AP 2001 3
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a) Determine the rotational inertia of the
rod-and- block apparatus attached to the top of
the pole.
b) Determine the downward acceleration of the
large block.
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c) When the large block has descended a distance
D, how does the instantaneous total kinetic
energy of the three blocks compare with the
value 4mgD ? Check the appropriate space below.
____ Greater than 4mgD
____ Equal to 4mgD
____ Less than 4mgD
Justify your answer.
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The system is now reset. The string is rewound
around the pole to bring the large block back to
its original location. The small blocks are
detached from the rod and then suspended from
each end of the rod, using strings of length l.
The system is again released from rest so that as
the large block descends and the apparatus
rotates, the small blocks swing outward, as shown
in Experiment B above. This time the downward
acceleration of the block decreases with time
after the system is released.
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d) When the large block has descended a distance
D, how does the instantaneous total kinetic
energy of the three blocks compare to that in
part c) ? Check the appropriate space below.
____ Greater
____ Equal
____ Less
Justify your answer.
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Lesson 9 Rolling Motion of a Rigid Object
Center moves in a straight line (green line).
A point on the rim moves in a path called a
cycloid (red curve).
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Speed of CM of Cylinder Rolling without Slipping
Since s Rq,
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Acceleration of CM of Cylinder Rolling without
Slipping
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Total Kinetic Energy of a Rolling Cylinder
Since IP ICM MR2, (parallel-axis theorem)
KErot ½ (ICM MR2)w2
Since vCM Rw,
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KEtotal ½ ICMw2 ½ MvCM2
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Example 1
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Example 2
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Example 3 AP 1986 2
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a) Determine the following for the sphere when it
is at the bottom of the plane.
i. Its translational kinetic energy
ii. Its rotational kinetic energy
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b) Determine the following for the sphere when it
is on the plane.
i. Its linear acceleration
ii. The magnitude of the frictional force acting
on it
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The solid sphere is replaced by a hollow sphere
of identical radius R and mass M. The hollow
sphere, which is released from the same location
as the solid sphere, rolls down the incline
without slipping.
c) What is the total kinetic energy of the hollow
sphere at the bottom of the plane
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d) State whether the rotational kinetic energy of
the hollow sphere is greater than, less than,
or equal to that of the solid sphere at the
bottom of the plane. Justify your answer.
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Lesson 10 Angular Momentum
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dr
dr
x p is zero since v,
dt
dt
and v and p are parallel.
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Instantaneous Angular Momentum
The instantaneous angular momentum L of a
particle relative to the origin O is defined by
the cross product of the particles instantaneous
position vector r and its instantaneous linear
momentum p.
The SI unit of angular momentum is kg . m2/s.
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Since L r x p,
The torque acting on a particle is equal to the
time rate of change of the particles angular
momentum.
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The direction of L is always perpendicular to the
plane formed by r and p.
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Example 1
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Angular Momentum of a System of Particles
The total angular momentum of a system of
particles about some point is defined as the
vector sum of the angular momenta of the
individual particles.
Differentiating with respect to time
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The net external torque acting on a system about
some axis passing through an origin in an
inertial frame equals the time rate of change of
the total angular momentum of the system about
that origin.
This theorem applies even if the center of
mass is accelerating, as long as t and L are
evaluated relative to the center of mass.
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Example 2
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Lesson 11 Angular Momentum of a Rotating Rigid
Object
Since v rw,
L mr2w
Since I mr2,
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L Iw
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Example 1
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Example 2
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a) Find an expression for the magnitude of the
systems angular momentum.
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b) Find an expression for the magnitude of the
angular acceleration of the system when the
seesaw makes an angle q with the horizontal.
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Example 3 AP 1983 2
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a) On the diagram below draw and identify all of
the forces acting on the cylinder and on the
block.
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b) In terms of m1, m2, R, and g, determine each
of the following.
i) The acceleration of the block.
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ii) The tension in the cord.
iii) The angular momentum of the disk as a
function of t.
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Example 4 AP 1982 3
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a) The initial angular momentum of the system
about the axis through P.
b) The frictional torque acting on the system
about the axis through P.
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c) The time T at which the system will come to
rest.
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Example 5 AP 1996 3
a) Show that the rotational inertia of the rod
about an axis through its center and
perpendicular to its length is Ml 2/12.
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b) What is the rotational inertia of the rod-hoop
assembly about the axle ?
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Several turns of string are wrapped tightly
around the circumference of the hoop. The system
is at rest when a cat, also of mass M, grabs the
free end of the string and hangs vertically from
it without swinging as it unwinds, causing the
rod-hoop assembly to rotate. Neglect friction and
the mass of the string.
c) Determine the tension T in the string.
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d) Determine the angular acceleration a of the
rod-hoop assembly.
e) Determine the linear acceleration of the cat.
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f) After descending a distance H 5 l /3, the
cat lets go of the string. At that instant, what
is the angular momentum of the cat about point P
?
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Lesson 12 Conservation of Angular Momentum
The total angular momentum of a system is
constant in both magnitude and direction if the
resultant external torque acting on the system is
zero, that is, if the system is isolated.
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Lbefore Lafter
Since L Iw,
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Example 1
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Example 2
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Example 3
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Example 4 AP 1987 3
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Immediately after the collision the object moves
with speed v at an angle q relative to its
original direction. The bar swings freely, and
after the collision reaches a maximum angle of
90o with respect to the vertical. The moment of
inertia of the bar about the pivot is Ibar ml
2/3. Ignore all friction.
a) Determine the angular velocity of the bar
immediately after the collision.
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b) Determine the speed v of the 1 kg object
immediately after the collision.
c) Determine the magnitude of the angular
momentum of the object about the pivot just
before the collision.
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d) Determine the angle q.
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Example 5 AP 1992 2
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a) Determine the torque about the axis
immediately after the bug lands on the sphere.
b) Determine the angular acceleration of the
rod- spheres-bug system immediately after the bug
lands.
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The rod-spheres-bug system swings about the axis.
At the instant that the rod is vertical, as shown
above, determine each of the following.
c) The angular speed of the bug.
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d) The angular momentum of the system.
e) The magnitude and direction of the force that
must be exerted on the bug by the sphere to
keep the bug from being thrown off the sphere.
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Lesson 13 Rotational Equilibrium
A system is in rotational equilibrium if the net
torque on it is zero about any axis.
Since St Ia,
St 0 does not mean an absence of rotational
motion. Object can be rotating at a constant
angular speed.
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Example 1
____ force equilibrium but not torque equilibrium.
____ torque equilibrium but not force equilibrium.
____ both force and torque equilibrium.
____ neither force nor torque equilibrium.
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Example 2
____ force equilibrium but not torque equilibrium.
____ torque equilibrium but not force equilibrium.
____ both force and torque equilibrium.
____ neither force nor torque equilibrium.
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Center of Gravity
To compute the torque due to the gravitational
force on an object of mass M, we need only
consider the force Mg acting at the center of
gravity of the object.
Center of gravity center of mass if g is
constant over the object.
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Example 3
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a) Determine the magnitude of the upward force n
exerted by the support on the board.
b) Determine where the father should sit to
balance the system.
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Example 4