Rotational Motion

ROTATIONAL MOTION
What can force applied on an object do?
Enduring Understanding 3.F: A force exerted on an object can cause a
torque on that object.
• An object or a rigid system, which can revolve or rotate about a fixed
axis, will change its rotational motion when an external force exerts a
torque on the object.
• The magnitude of the torque due to a given force is the product of the
perpendicular distance from the axis to the line of application of the
force (the lever arm) and the magnitude of the force.
• The rate of change of the rotational motion is most simply expressed
by defining the rotational kinematic quantities of angular
displacement, angular velocity, and angular acceleration, analogous
to the corresponding linear quantities, and defining the rotational
dynamic quantities of torque, rotational inertia, and angular
momentum, analogous to force, mass, and momentum.
• The behaviors of the angular displacement, angular velocity, and
angular acceleration can be understood by analogy with Newton’s
second law for linear motion.
Torque
Essential Knowledge 3.F.1: Only the force component
perpendicular to the line connecting the axis of rotation
and the point of application of the force results in a torque
a. The lever arm is the perpendicular distance from the axis
of rotation or revolution to the line of application of the
force.
b. The magnitude of the torque is the product of the
magnitude of the lever arm and the magnitude of the force.
c. The net torque on a balanced system is zero.
Torque
Torque is the tendency of a force to rotate an object about some
axis.
=   = 
 = torque in Nm
r = lever arm in m
= the perpendicular distance from the axis of rotation to a
line drawn along the direction of the force.
F = force in N
 = angle between F and r
Direction of Torque
Torque is a vector quantity that has
direction as well as magnitude.
Turning the handle of a
screwdriver clockwise and
then counterclockwise will
advance the screw first
inward and then outward.
Sign Convention for Torque
By convention, counterclockwise torques are
positive and clockwise torques are negative.
Positive torque:
Counter-clockwise,
out of page
cw
ccw
Negative torque:
clockwise, into page
Line of Action of a Force
The line of action of a force is an imaginary
line of indefinite length drawn along the
direction of the force.
F2
F1
Line of
action
F3
The Moment Arm
The moment arm of a force is the perpendicular
distance from the line of action of a force to the
axis of rotation.
F1
F2
r
r
r
F3
Calculating Resultant Torque
•Read, draw, and label a rough figure.
•Draw free-body diagram showing all forces,
distances, and axis of rotation.
•Extend lines of action for each force.
•Calculate moment arms if necessary.
•Calculate torques due to EACH individual force
affixing proper sign. CCW (+) and CW (-).
•Resultant torque is sum of individual torques.
Second condition of equilibrium
The second condition for equilibrium states that if an object
if in rotational equilibrium, the net torque acting on it about
any axis must be zero.
 = 0.
Recall: The first condition for equilibrium says that the
summation of all the forces acting on an object in
equilibrium is zero. (F = 0)
Note: A body in static equilibrium must satisfy both
conditions.
Moment of inertia
• Moment of inertia is the mass property of a rigid body
that determines the torque needed for a desired angular
acceleration about an axis of rotation.
• Moment of inertia depends on the shape of the body and
may be different around different axes of rotation.
Moment of inertia
Moment of inertia
A mass m is placed on a rod of length r and negligible mass, and constrained to
rotate about a fixed axis. If the mass is released from a horizontal orientation, it
can be described either in terms of force and acceleration with Newton's
second law for linear motion, or as a pure rotation about the axis with Newton's
second law for rotation.
Rotational Motion
Essential Knowledge 3.F.2: The presence of a net torque
along any axis will cause a rigid system to change its
rotational motion or an object to change its rotational
motion about that axis.
a. Rotational motion can be described in terms of angular
displacement, angular velocity, and angular acceleration
about a fixed axis.
b. Rotational motion of a point can be related to linear
motion of the point using the distance of the point from the
axis of rotation.
c. The angular acceleration of an object or rigid system can
be calculated from the net torque and the rotational inertia
of the object or rigid system.
Linear motion
Rotational motion
Angular Momentum
Essential Knowledge 3.F.3: A torque exerted on an object can
change the angular momentum of an object.
a. Angular momentum is a vector quantity, with its direction
determined by a right-hand rule.
b. The magnitude of angular momentum of a point object about
an axis can be calculated by multiplying the perpendicular
distance from the axis of rotation to the line of motion by the
magnitude of linear momentum.
L = r x mv
c. The magnitude of angular momentum of an extended object
can also be found by multiplying the rotational inertia by the
angular velocity.
L = I
d. The change in angular momentum of an object is given by the
product of the average torque and the time the torque is exerted.
Change in angular momentum
Essential Knowledge 4.D.3: The change in angular
momentum is given by the product of the average torque
and the time interval during which the torque is exerted.
∆ =

Conservation of Angular Momentum
Essential Knowledge 5.E.1: If the net external torque
exerted on the system is zero, the angular momentum of
the system does not change.
I00 = I
Direction of angular motion variables
Essential Knowledge 4.D.1: Torque, angular velocity,
angular acceleration, and angular momentum are vectors
and can be characterized as positive or negative
depending upon whether they give rise to or correspond to
counterclockwise or clockwise rotation with respect to an
axis.
Kepler’s laws of planetary motion
• Kepler’s First Law
• All planets move in elliptical orbits with the Sun at
one focus
• Kepler’s Second Law
• The radius vector drawn from the Sun to a planet
sweeps out equal areas in equal time intervals
• Kepler’s Third Law
• The square of the orbital period of any planet is
proportional to the cube of the semimajor axis of
the elliptical orbit
What is an ellipse?
2 foci
An ellipse is a geometric shape with 2
foci instead of 1 central focus, as in a
circle. The sun is at one focus with
nothing at the other focus.
FIRST LAW OF PLANETARY MOTION
An ellipse also has…
…a major axis
Perihelion
…and a minor axis
Aphelion
Semi-major axis
Perihelion: When Mars or any another planet
is closest to the sun.
Aphelion: When Mars or any other planet is
farthest from the sun.
Kepler also found that Mars changed speed as it orbited around the
sun: faster when closer to the sun, slower when farther from the
sun…
A
B
But, areas A and B, swept out by
a line from the sun to Mars, were
equal over the same amount of
.
time
SECOND LAW OF PLANETARY
MOTION
T1
Kepler found a relationship between the
time it took a planet to go completely
around the sun (T, sidereal year), and the
average distance from the sun (R, semimajor axis)…
R1
T2
R2
T1 2
R1 3
=
T2 2
R2 3
THIRD LAW OF PLANETARY MOTION