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Spring 2004
PHY 2053C: College Physics A
Today: v Rotational Motion Āø Kinematics 2: Rotational motion & kinematics Āø Dynamics 2: Torques Moment of Inertia Angular Momentum -- conservation
n Motion, Forces, Energy, Heat, Waves n Dr. David M. Lind n Dr. Kun Yang n Dr. David Van Winkle
L12āCh7Ch
PHY 2053C: College
Physics A
Spring 2004 n Motion, Forces, Energy, Heat, Waves
n Dr. David M. Lind n Dr. Kun Yang n Dr. David Van Winkle
L12āCh7Ch
Today: v Rotational Motion Āø Kinematics 2: Rotational motion & kinematics Āø Dynamics 2: Torques Moment of Inertia Angular Momentum -- conservation
Extended Bodies & Rotation
n Translational motion: Objects move as if all of the extended object's mass was concentrated in the center-of-mass.
n Extended objects can do one thing which point masses can't: they Rotate while they translate!
m m^ m mm m m m m m m m m m m
mm m m m mm m m m mm
m m m mm m
m mm m m m m m m = M
Center of mass
x CM^ m^1 x^1 m^2 x^2 ...^ mN^ xN m 1 m 2 ... mN
y CM
m 1 y 1 m 2 y 2 ... mN yN m 1 m 2 ... mN
Linear and Angular Kinematics
(summary)
n Displacement: x [m] angle? [rad] n Velocity: v [m/s] angular vel.? [rad/s] n acceleration a [m/s^2 ] angular acc. a [rad/s^2 ] n How do the angular kinematics translate to ālinearā kinematics? n Sitting on a merry-go-round, what is your tangential speed? vtan=r? n your tangential acceleration? n atan=ra n your radial (centripetal) acceleration? arad=v^2 /r = r? 2
Rolling Motion
rot. freq. f rev/s => 2 f rad/s period T s/rev => f 1 T rev/s , 2 T rad/s
example:
C
P
C
P
What is the relation between angular velocity? and linear velocity v for a rolling wheel? n The point of contact between wheel and ground P is momentarily at rest: n The axle is moving at velocity v. n Now go into wheel's frame of reference: The point P is moving at velocity -v. Now we see: vtan=r? n? is the same in both frames of ref. n Given
Force
T
c.m.
Torque and Force
n Extended bodies behave like āmass pointsā only when forces acting in the direction toward the center of mass. n If forces act at a radius (tangentially), they produce a āturning forceā , called torque , which produces spin. n Here the torque is defined as:
Force r t
c.m.
tangential component produces torque => spin (+motion)
radial component (toward c.m.) => linear motion
Force acts toward c.m. => only linear motion
example: Striking a billiard ball
Ļ = F r sin Īø
?
Torque: A āTurning Forceā A āForceā sets an object into linear motion. A āTorqueā sets an object into rotation.
n Torque is defined with respect to an axis of rotation:
n units: 1 Newton meter = 1 Nm
example: To turn the door open, you need a larger force acting at the smaller radius.
n To exert the same torque : if r 2 <r 1 => F 1 <F 2
n remember: Lever arm times force = torque
F r sin
F 2 r 2 F 1 r 1
Torque 2
n If the force acts at an angle?? 90?^ with the lever arm, the torque is reduced:
n That is the same as writing n
n You can also use
n with the perpendicular component of the lever arm
F r sin
F r
F r
r
Rotational Dynamics:
Torque and Moment of Inertia Linear Dynamics are based on Newton's laws Rotational Dynamics are similar:
- If there is no net torque acting on a system, it remains at constant angular velocity?.
2) sum of torques analogue to:
= Moment of inertia times angular acceleration
3) Action = Reaction
n Moment of inertia takes the role of (inertial) mass
n We still have not defined Moment of Inertia 0!
F m a
n We have to find I in the formula n We will use a case, where we can look at the same situation in two ways: linear and rotational n Let F act tangentially on m at a radius r. That is equiv. to
n In our equation , the moment of inertia becomes n which we define as the Moment of Inertia of a single mass point.
Moment of Inertia
I
F r
F r and F ma
mar and a r
=> m r
2
I
I mr^2
Moment of Inertia
n For extended bodies, the moment of inertia is calculated by averaging mr^2 within the body:
More complete list fig 8- n The outside mass counts the most!
?
Moment of Inertia 3 n The Moment of Inertia depends on the axis for which it is defined: n Usually, we define the rotational axis to go through the center of mass C of a body:
n This leads to a separation of rotational- from linear motion.
n What does ābalancingā your wheels mean?
- Put the rotational axis through the cm!
- Make the rotational axis a symmetry axis of the masses
example:
Rolling of different Objects
n Depending on the moment of inertia of the bodies, different amounts of rotational energy are taken away from the linear kinetic energy.
n Only the slipping box converts all of the potential energy into linear kinetic energy. (=> and therefore into speed!)
MgH
M vbot^2
I cm bot^2
Stay tuned...
n Friday: CAPA7/Recitation
n Monday: Chapter 9 : Static Equilibrium
n Wednesday : Mini-Exam#
n On chapters 7 & 8 (linear momentum, conservation laws, & rotational motion)
Example: Circular Platform n A horizontal circular platform (M = 62.1 kg, r = 3.37 m) rotates about a frictionless vertical axle. A 79.3 kg student walks slowly from the rim of the platform toward the center. The angular velocity of the system is 3.3 rad/s when the student is at the rim. Find the angular velocity of the system when the student is 2.59 m from the center.
n Angular momentum conservation:
n
Li I i i l f I f f
I (^) i^1 2
M (^) disk R disk^2 M (^) stud R stud i^2
= 0.5 62.1 kg 3.37 m^2 79.3 kg 3.37 m^2 1253.2 kg m^2 I (^) f^1 2
M (^) disk R disk^2 M (^) stud R stud f^2 884 kg m^2
f i
I (^) i I (^) f
3.3 rad s
4.68 rad s
Vector Nature
of Angular Quantities
n So far we have treated rotational quantities as numbers. They are vectors pointing toward the axis of rotation:
n The right hand rule:? points like thumb when the fingers point along rotation.
n When the rotation is ccw in xy-plane, ? points in pos. z direction.
n An? in negative z- direction means a clockwise rotation in x-y.
n The same is true for L , a, t.
z
x
y
