Download Rotational Motion - David Brotherton CCCMC

Rotational Movement Notes (sparknotes.com)
Instead of translating, moving in a straight line, an object moves about an axis in a circle.
Frequently, this axis is part of the object that is rotating. Consider a bicycle wheel. When the
wheel is spinning, the axis of rotation is simply a line going through the center of the wheel and
perpendicular to the plane of the wheel.
A rigid body moves in rotational motion if every point of the body moves in a circular path with
a common axis.
The object does not need to be circular
Rotation around A
Rotational Variables
We cannot use our old variables, such as velocity, because some parts of a rotating disk move
at different speeds than others, and a single number for velocity would not describe the motion
of the entire disk. So what is a property of every point on a rotating object? Since every point
rotates in a circle about a common axis, we can say that the angular displacement is the same
for any point on a rotating object. That is, the angle which each point sweeps out in rotating is
the same at any given time for any point on the object:
μ=
Where s is the arc length shown in this figure, r is the
distance from the point to the axis of rotation, and μ is
the measure of the angle.
Torque
Defined as the effect a force will have on rotational motion. Consider point P a distance r away
from an axis of rotation, and a force F applied to P at an angle of θ to the radial direction, as
shown below.
Figure %: A force acting at angle θ to the radius of rotation of point P
If the force is parallel to the radius of the particle ( θ = 0 ), then the force might cause some
translational motion of the particle. But, since the force has no component acting in the
tangential direction, it causes no change in rotational motion. In addition, if the force is close to
the axis of rotation it will cause less change in the rotation of the body than at a farther distance.
Thus we define the torque (denoted by τ ) accordingly:
τ = Fr sinθ
τ = r×F
Or,
torque = force x lever arm
The second equation ( τ = r×F ) expresses the torque in terms of a cross product.The torque
(because it is a cross product) must be perpendicular to both the force applied and the radius of
the particle, implying that it points perpendicular to the plane of rotation of the particle.
The best example of a torque is the force applied to opening a door. The easiest way to open the
door (in other words, the way to provide maximum torque) is to grab a point the furthest away
from the hinges (like the handle), and pull perpendicular to the door itself. In this way, we give a
maximum r, and sinθ = 1 . The closer to the hinges one pulls, the more force must be exerted to
provide the same torque on the door. In addition, the angle with which the torque is applied
changes the force necessary for a given torque. The case of pulling perpendicular to the door
requires the least force.
Torque plays the same role in rotational motion as force plays in translational motion. In fact, we
can restate Newton's First Law to make it apply to rotational motion:
If the net torque acting on a rigid object is zero, it will rotate with a constant angular velocity.
Moment
-Very important—Torque and moment are essentially the same thing.
Moment tends to be used in essentially non-rotational situations, such as
analysis of forces on a beam, while torque when there is an axle or pivot to be
turned around!!!!
- The Principle of Moments, also known as Varignon's theorem (not to be confused with the
geometrical theorem of the same name) states that the sum of torques due to several forces
applied to a single point is equal to the torque due to the sum (resultant) of the forces.
Mathematically, this follows from:
Rotational Inertia
-An object rotating around an axis tends to keep rotating around an axis. The resistance of an
object to changes in its rotational motion is rotational inertia.
-Also called Moment of Inertia, it has the same relationship to angular acceleration as mass has
to linear acceleration.
-The smaller the rotational inertia, the greater the acceleration. See sections11.4 and 11.5 in the
Conceptual Physics book
For a point mass the Rotational Inertia or Moment of Inertia is the mass times the square of
perpendicular distance to the rotation reference axis and can be expressed as
I = m r2
(1)
where
I = moment of inertia (lbm ft2, kg m2)
m = mass (lbm, kg)
r = distance between axis and rotation mass (ft, m)
I = ∑i mi ri2 = m1 r12 + m2 r22 + ..... + mnrn2
(2)
The Principle of Moments, also known as Varignon's theorem (not to be confused with
the geometrical theorem of the same name) states that the sum of torques due to several forces
applied to a single point is equal to the torque due to the sum (resultant) of the forces.
Mathematically, this follows from: