Transcript Rotational Mechanics

Rotational Mechanics
Rotary Motion
Rotation about internal axis (spinning)
Rate of rotation can be constant or variable
Use angular variables to describe rotation
All parts of a rigid rotating object have same angular
displacement, velocity and acceleration
Linear displacement and velocity depend on distance
from rotation axis.
Radian Measure of Angles
Radian is ratio of arc distance subtended by
angle to the radius: q = Ddarc/r
Radian is dimensionless (meters/meters)
One complete rotation equals 2p radians
One radian equals 57.3 degrees
Angular Displacement
Angular displacement (q), angle of rotation
measured in radians
Linear displacement equals angular
displacement times the radius
All parts of rotating object have same angular
displacement
Angular Velocity
Change in angular displacement with respect to time
w = Dq/Dt; fundamental units are rad/s, but usually
measured in rev/s, or rev/min (rpm)
Vector with direction found using right hand rule: Curl
fingers of right hand in direction of rotation. Extended
thumb points in direction of vector
Angular Velocity
Linear velocity = angular velocity times radius, v
= wr
All parts of rotating object have same angular
velocity
Angular Acceleration
Change in angular velocity with respect to time:
a = Dw / Dt; units are rad/s2
Vector whose direction is found using right hand
rule
If angular acceleration is constant, constant
acceleration equations are used.
Comparing Linear and Angular
Variables
Quantity
Linear
Displacement d (x, y)
Velocity
v
Acceleration
a
Angular Conversion
q
d=qr
w
v=wr
a
a=ar
Constant Acceleration Equations
Linear
vf = vi + aDt
Dd = viDt + 1/2(aDt2)
vf2 = vi2 + 2aDd
Angular
wf = wi + aDt
D q = wiDt + 1/2(aDt2)
wf2 = wi2 + 2aDq
Center of Gravity
The point at which all object’s weight can be
considered to be concentrated.
For symmetrical bodies with uniform density, c.o.g.
will be at geometric center.
May be located outside the body of some objects.
Bodies or systems rotate about their center of
gravity.
Similar to center of mass but not always the same
Parallel Forces
Forces acting in the same or opposite directions
at different points on an object
Can produce rotation
Concurrent forces act at the same point (often
the center of gravity) at the same time on an
object
Weight Vectors
Drawn from center of gravity of object
Actually are the sum of an infinite number of
parallel weight vectors from an infinite number of
mass units
The effect is as if all the weight was
concentrated at the center of gravity
Torque
The result of a force that produces rotation, a
vector
The product of the force and its lever arm,
Lever arm (or moment arm) is a vector whose
magnitude is the distance from the point of
rotation to the point of application of the force
t=rxF
A product of two vectors that produces a third
perpendicular vector
Torque
units are meters x newtons
signs: ccw torques are considered +, cw torques
are direction of net torque is direction of resulting
rotation
Rotational Inertia
Resistance of an object to any change in angular
velocity
Depends on mass and its orientation with
respect to axis of rotation.
Is rotational analogue to mass; symbol I, units
kg m2
Sometimes called moment of inertia
Rotational Inertia
For an object rotating about an external point,
I = mr2
For objects rotating about an internal axis, inertia
must be calculated using calculus
Use rotational inertia equations for general type
of regularly shaped solid bodies
Newton’s Second Law for Rotation
Substitute angular variables for linear
F = ma becomes t = Ia where t is the net
torque and I is the rotational inertia of the body.
Work in Rotary Motion
Work done by torque
W = tDq = FrDq
q is angular displacement in radians
Assumes force is perpendicular to radius
Power in Rotary Motion
Power is rate of doing work
P = tDq/Dt
Dq/Dt = w, so P = tw
Kinetic Energy in Rotary Motion
Energy possessed by rotating object
KErot = 1/2(Iw2)
Rolling objects have both linear and rotational
kinetic energy
Kinetic Energy in Rotary Motion
When object rolls downhill, potential energy is
converted to both types of kinetic energy;
amount of each depends on rotational inertia of
object.
Angular Momentum
The tendency of a rotating object to
continue rotating
A combination of the rotational inertia and
angular velocity
For a rotating object, L = Iw
A vector
Angular Momentum
Objects in circular motion also have
angular momentum: L = mvr
Angular momentum can be applied to any
moving object with respect to an external
point
Radial distance is perpendicular distance
form path of object to the point
Conservation of Angular Momentum
External net torque is required to change angular
momentum
If no net external torque is present, angular
momentum of a system will remain constant
Total angular momentum before the interaction
equals total angular momentum after the
interaction as long as no net external torque acts
on the system
Conservation of Angular Momentum
Always true, from atomic to galactic interactions
If rotational inertia changes, angular velocity
must change to conserve angular momentum
Precession
A secondary rotation of the axis of rotation
Due to torque produced by weight of rotating
object
Causes angular acceleration that changes
direction of angular velocity of rotating object
Earth precesses on its axis with a secondary
rotation period of 26,000 years