Rotational Motion

Rotation of rigid objects- object with definite shape

A brief lesson in Greek

theta tau omega alpha

Rotational Motion

• All points on object move in circles

• Center of these circles is a line=axis of rotation

• What are some examples of rotational motion?

Radians• Angular position of

object in degrees=• More useful is

radians• 1 Radian= angle

subtended by arc whose length = radius

=l/r

Converting to Radians

• If l=r then =1rad

• Complete circle = 360º so…in a full circle 360==l/r=2πr/r=2πrad

So 1 rad=360/2π=57.3

*** CONVERSIONS*** 1rad=57.3 360=2πrad

Example: A ferris wheel rotates 5.5 revolutions. How many radians has it rotated?

• 1 rev=360=2πrad=6.28rad

• 5.5rev=(5.5rev)(2πrad/rev)=

• 34.5rad

Example: Earth makes 1 complete revolution (or 2rad) in a day. Through what angle does earth rotate in 6hours?

• 6 hours is 1/4 of a day=2rad/4=rad/2

Practice

• What is the angular displacement of each of the following hands of a clock in 1hr?– Second hand– Minute hand– Hour hand

Hands of a Clock

• Second: -377rad

• Minute: -6.28rad

• Hour: -0.524rad

Velocity and Acceleration

• Velocity is tangential to circle- in direction of motion

• Acceleration is towards center and axis of rotation

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Angular Velocity

• Angular velocity = rate of change of angular position

• As object rotates its angular displacement is ∆=2-1

• So angular velocity is

=∆/ ∆t measured in rad/sec

Angular Velocity

• All points in rigid object rotate with same angular velocity (move through same angle in same amount of time)

• Direction: right hand rule- turn your fingers in direction of rotation and if thumb points up=+– clockwise is -– counterclockwise is +

Angular Acceleration

• If angular velocity is changing, object would undergo angular acceleration

= angular acceleration

=/t

Rad/s2

• Since is same for all points on rotating object, so is so radius does not matter

Equations of Angular Kinematics

• LINEAR

• a = (vf - vo)/t

• vf = vo + at

• s = ½(vf + vo)t

• s = vot + ½at2

• vf2 = vo

2 + 2ax

• ANGULAR

• α = (ωf - ωo)/t

• ωf = ωo + αt

• θ = ½(ωf + ωo)t

• θ = ωot + ½αt2

• ωf2 = ωo

2 + 2αθ

Linear vs Angular

They are related!!!

Velocity:Linear vs Angular

• Each point on rotating object also has linear velocity and acceleration

• Direction of linear velocity is tangent to circle at that point

• “the hammer throw”

Velocity:Linear vs Angular

• Even though angular velocity is same for any point, linear velocity depends on how far away from axis of rotation

• Think of a merry-go-round

Velocity:Linear vs Angular

• v= l/t=r/t

• v=r

Linear and Angular Measures

Quantity Linear Angular Relationship

Displacement d(m)

Velocity v(m/s)

Acceleration a(m/s2)

Linear and Angular Measures

Quantity Linear Angular Relationship

Displacement d(m) (rad) d=r

Velocity v(m/s) (rad/s) v=r

Acceleration a(m/s2) (rad/s2) a=r

Practice

• If a truck has a linear acceleration of 1.85m/s2 and the wheels have an angular acceleration of 5.23rad/s2, what is the diameter of the truck’s wheels?

Truck

• Diameter=0.707m • Now say the truck is towing a trailer with wheels that have a diameter of 46cm

• How does linear acceleration of trailer compare with that of the truck?

• How does angular acceleration of trailer wheels compare with the truck wheels?

Truck

• Linear acceleration is the same• Angular acceleration is increased because

the radius of the wheel is smaller

Frequency

• Frequency= f= revolutions per second (Hz)

• Period=T=time to make one complete revolution

• T= 1/f

Frequency and Period example

• After closing a deal with a client, Kent leans back in his swivel chair and spins around with a frequency of 0.5Hz. What is Kent’s period of spin?

T=1/f=1/0.5Hz=2s

Period and Frequency relate to linear and angular acceleration

• Angle of 1 revolution=2rad

• Related to angular velocity:=2f

• Since one revolution = 2r and the time it takes for one revolution = T

• Then v= 2r /T

Try it…

• Joe’s favorite ride at the 50th State Fair is the “Rotor.” The ride has a radius of 4.0m and takes 2.0s to make one full revolution. What is Joe’s linear velocity on the ride?

V= 2r /T= 2(4.0m)/2.0s=13m/s

Now put it together with centripetal acceleration: what is Joe’s centripetal acceleration?

And the answer is…

• A=v2/r=(13m/s2)/4.0m=42m/s2

Centripetal Acceleration

• acceleration= change in velocity (speed and direction) in circular motion you are always changing direction- acceleration is towards the axis of rotation

• The farther away you are from the axis of rotation, the greater the centripetal acceleration

• Demo- crack the whip• http://www.glenbrook.k12.il.us/gbssci/phys/

mmedia/circmot/ucm.gif

Centripetal examples

• Wet towel

• Bucket of water

• Beware….inertia is often misinterpreted as a force.

The “f” word• When you turn quickly- say in a car or roller

coaster- you experience that feeling of leaning outward

• You’ve heard it described before as centrifugal force

• Arghh……the “f” word• When you are in circular motion, the force is

inward- towards the axis= centripetal• So why does it feel like you are pushed

out???INERTIA

Centripetal acceleration and force

• Centripetal acceleration=v2/r• Or: =r2

– Towards axis of rotation

• Centripetal force=macentripetal

• If object is not in uniform circular motion, need to add the 2 vectors of tangential and centripetal acceleration (perpendicular to each other) so: a2=ac

2+at2

Rolling

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Rolling

• Rolling= rotation + translation• Static friction between rolling object and

ground (point of contact is momentarily at rest so static)

v=ra=r

Example p. 202

A bike slows down uniformly from v=8.40m/s to rest over a distance of 115m. Wheel diameter = 68.0cm. Determine

(a) angular velocity of wheels at t=0

(b) total revolutions of each wheel before coming to rest

(c) angular acceleration of wheel

(d) time it took to stop