Transcript Document

Chapter 11 Rotational
Dynamics and
Static Equilibrium
Chapter 11: Rotational Dynamics
and Static Equilibrium
Torque: The ability of a force to rotate a body
about some axis. t = rF
Note: F  r
The torque is larger
if the force is applied
farther from the axis
of rotation.
By convention, the sign of torque is:
t is negative
clockwise (cw)
t is positive
counter-clockwise
(ccw)
General Definition of Torque
Only the component of the force that is
perpendicular to the radius causes a torque.
t = r(Fsinq)
Equivalently, only the perpendicular
distance between the line of force and the
axis of rotation, known as the moment arm
r, can be used to calculate the torque.
t = rF = (rsinq)F
Fsinq
F
q
Each force that acts on an object may cause a torque.
F1
r2
F3
r1
In this figure, the three forces have equal
magnitude.
• Which forces cause a torque?
• Which force causes the biggest
magnitude torque?
F2
pivot point
• Which forces, if any, causes a positive
torque?
When discussing torques, we must identify a pivot point
(or axis of rotation).
The net torque about a point O is the sum of all torques
about O: St = t1 + t2 + ...
HW 11 problem # 1
A person holds a 1.42 N baseball in his hand, a distance of 2L = 34 cm from the
elbow joint, as shown in the figure. The biceps, attached at a distance of d =
2.75 cm from the elbow, exert an upward force of 12.8 N on the forearm.
Consider the forearm and hand to be a uniform rod with a mass of 1.39 kg.
(a) Calculate the magnitude of the
net torque acting on the forearm
and hand. Use the elbow joint as
the axis of rotation. [2.44 N.m]
(b) If the net torque obtained in part
(a) is nonzero, in which direction
will the forearm and hand rotate?
[clockwise]
Moment of Inertia
Recall that mass (inertia) is an object’s resistance to
acceleration. Similarly an object’s resistance to rotation
(angular acceleration) is known as moment of inertia.
For a point mass m:
I = mr2
I = moment of inertia
r = distance from the axis of rotation
For an extended object:
I =Smiri2
Mass near the axis of rotation resists rotation less than
mass far from the axis of rotation.
Solid Sphere
Spherical Shell
Hoop or
Cylindrical Shell
Solid Cylinder
or Thin Disk
Thin Rod or Bar
Thin Rod about its end
Angular Position, q
For circular motion, the distance (arc length) s,
the radius r, and the angle q are related by:
s
q =
r
q > 0 for counterclockwise
rotation from reference line
Note that q is
measured in radians:

q RAD =
q
 DEG
180
1 rev = 360° = 2 rad
Consider a rotating disk:
r
P
q
r
O
t=0
P
O
t>0
s
Angular Velocity, w
Notice that as the disk rotates, q changes. We
define the angular displacement, q, as:
q = qf - qi
which leads to the average angular speed wav
w av
q q f  q i
=
=
t
t f  ti
Period
The period of rotation is the time it takes to
complete one revolution.
2
w=
T
T = period
Rearranging we have
T=
2
w
What is the period of the Earth’s rotation about its
own axis?
What is the angular velocity of the Earth’s rotation
about its own axis?
Angular Acceleration, a
We can also define the average angular
acceleration aav:
a av
and
w w f  wi
=
=
t
t f  ti
w
a = lim
t 0 t
The SI units of a are: rad/s2 = s-2
We will skip any detailed discussion of angular acceleration,
except to note that angular acceleration is the time rate of
change of angular velocity
Torque and Angular Acceleration
Recall Newton’s Second Law:
F = ma
The net force on an object of mass m causes a
(linear) acceleration a.
Similarly, the net torque on an object with moment
of inertia I causes an angular acceleration a.
t = Ia
HW11 - Problem
When a ceiling fan rotating with an angular speed of 2.15
rad/s is turned off, a frictional torque of 0.241 N m slows it
to a stop in 6.25 s. What is the moment of inertia of the
fan?
[0.701] kg m2
Zero Torque and Static Equilibrium
Consider the wheel shown below. Two forces of equal
magnitude are acting on the wheel. Will the wheel
remain at rest?
The net force is zero, so there will be no linear
acceleration.
However, the sum of the torques is not
zero, so there will be an angular
acceleration.
The wheel is not in static equilibrium.
F2
F1
Conditions for Static Equilibrium
For true static equilibrium, two conditions
must be satisfied:
F = 0
t = 0
 Fx = 0
 Fy = 0
For an object in equilibrium, the axis of
rotation is arbitrary (But all torques must be
evaluated about a common axis).
Angular Momentum
For linear momentum:
p = mv
For rotational motion, we define an angular
momentum:
L = Iw
The SI units of angular momentum are kg·m2/s
Angular Momentum - Problem
A 0.013 kg record with a radius of 15 cm rotates with an
angular speed of 29 rpm. Find the angular momentum of
the record.
[4.44E-4] kg m2/s
Kinetic energy of rotation
What is the kinetic energy of a mass m traveling at speed v
in a circle of radius r?
K = (1/2) m v2 = (1/2) mr2 (v/r) 2 = (1/2) I w2
Kinetic energy of rotation = (1/2) I w2
This is not a new form of energy, just a re-labeling (or
alternate formula) for kinetic energy.
Rotational Kinetic Energy - Problem
Calculate the rotational kinetic energy of the Earth as it (a)
orbits the sun (b) rotates about its axis.
Mass of Earth = 5.98E24 kg
Radius of Earth (ave) = 6.38E6 m
Average Earth-Sun distance = 1.50E11 m