PHYS 2108 Lecture Notes - Lecture 7: Theta, Angular Acceleration, Radian
7-1
LAB 7:
Rotational Motion
The ost asi field of stud i phsis is kieatis: positio, eloit ad aeleatio. It’s the fist
topic and contains the most basic (and often used) equations in physics.
Kieatis does’t alas hae to happe i a staight lie. Rathe tha
travelling across some distance x, it’s possile to oside spinning through
some otatioal distae. We usually choose to represent the angular
displacement as the Greek letter theta, .
We have a special unit of measure for angles that you may have seen in your
calculator called radians. Radians are a more natural unit in the universe
than degrees because radians are the result of ratios. Beause of that the’e
one of the very few units that you can stick in when you need them or
eoe he ou do’t. If ou la a stig as log as a ile’s adius alog
the edge of the circle, the angle it sweeps out is equal to one radian.
If it’s possile to tael though a agle, it’s possile to do it oe soe tie
at soe kid of speed. If e ee to otate the pitued disk fo θ1 to θ2 in
soe tie Δt, it sees e hae ieted a fail logial euatio fo the
angular speed or angular velocity of a rotating system. We use a lower case
omega ω to represent it and the units are in rad/s or more familiarly,
Revolutions Per Minute (RPM). Similarly, if we change the angular speed, we
can make a reasonable argument for an angular acceleration α.
It turns out that every formula we had in kinematics for straight-line motion has an angular analogue.
Do’t let agula poles itiidate ou! You’e done similar problems before with different symbols!
Eqn. Table 7-1: Translational and Rotational Kinematics
Translational (Linear) Equations
Rotational (Angular) Equations
displacement
velocity
the tieless euatio
average velocity
average acceleration
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7-2
If to dots ae paited o a spiig disk, the ust hae the sae agula speed eause the’e
connected to the disk). Do they have the same linear speed? In one revolution, the outer dot travels
2πR while the inner one travels only 2πr. Because the outer dot has to cross a greater distance in the
sae aout of tie, it ust do it oe uikl. Liea o tagetial)
quantities and angular ones can be converted as follows:
Eqn. Table 7-2: Linear-Angular Conversions
Distance (Arc Length)
Tangential Velocity
Tangential Acceleration
Newton’s Second Law Angular
Try to push a door open near the knob, then again near the hinges. Which one is easier?
When thinking about rotation, sheer force alone is not as important as where it’s applied. Forces
applied close to the rotation axis or pivot point are less effective than those applied much further away.
This combination of force and its point of application is called torque in units of meter-Newtons (m-N).
Eqn. 7-3:
Definition of Torque
In the formula, (tau) represents the torque, is the radius between the rotation axis and the location
of the force, and is the component of the force perpendicular to the radius. Any components
parallel to the radius are trying to pull the door off its hinges, not trying to rotate it.
There are apparently three ways to affect the torque on a rotating system:
Change the amount of force. Pushing harder generates
greater torque.
Change the distance from the pivot point. Pushing
farther away generates greater torque.
Change the angle in which the force is applied. Pushing
more directly generates greater torque. Components
parallel to the radius do not contribute to rotation.
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7-3
Try to spin a football on its point (like one thrown
in a good spiral, left diagram), then spin it again on
its equator (right diagram). Which is easier?
Much like the case with force, when trying to
angularly accelerate something not only is its mass
important, but also where the mass is distributed.
Beause oe of the footall’s ass is lose to
the rotation axis in the leftward case, it has a
smaller rotational ass ad the sae toue a
better rotate it than it can in the rightward case.
This rotational ass is alled the moment of inertia and is denoted by a capital . For any point-like
mass some distance R from the axis of rotation:
Eqn. 7-4:
Moment of Inertia (Point Mass)
Most real-life ojets ae’t poit-masses. By using calculus to add up mR2 for every atom in an
extended body, we find moments of inertia for the following shapes.
Eqn. Table 7-5: Moments of Inertia (Extended Bodies)
Knowing that torque acts on a moment of inertia to produce rotational acceleration, we can invent an
agula esio of Neto’s “eod La.
Eqn. 7-6:
Neto’s “eod La Agula
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Document Summary
The (cid:373)ost (cid:271)asi(cid:272) field of stud(cid:455) i(cid:374) ph(cid:455)si(cid:272)s is ki(cid:374)e(cid:373)ati(cid:272)s: positio(cid:374), (cid:448)elo(cid:272)it(cid:455) a(cid:374)d a(cid:272)(cid:272)ele(cid:396)atio(cid:374). It"s the fi(cid:396)st topic and contains the most basic (and often used) equations in physics. We have a special unit of measure for angles that you may have seen in your calculator called radians. Radians are a more natural unit in the universe than degrees because radians are the result of ratios. Be(cid:272)ause of that the(cid:455)"(cid:396)e one of the very few units that you can stick in when you need them or (cid:396)e(cid:373)o(cid:448)e (cid:449)he(cid:374) (cid:455)ou do(cid:374)"t. If (cid:455)ou la(cid:455) a st(cid:396)i(cid:374)g as lo(cid:374)g as a (cid:272)i(cid:396)(cid:272)le"s (cid:396)adius alo(cid:374)g the edge of the circle, the angle it sweeps out is equal to one radian. We use a lower case omega to represent it and the units are in rad/s or more familiarly, Similarly, if we change the angular speed, we can make a reasonable argument for an angular acceleration .
