Rotational Motion

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 2rad) in a day. Through what angle does earth rotate in 6hours? • 6 hours is 1/4 of a day • =2rad/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

Angular Velocity • Angular velocity = rate of change of angular position • As object rotates its angular displacement is ∆=2-1 • So angular velocity is  =∆/ ∆tmeasured 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

LINEAR a = (vf - vo)/t vf = vo + at s = ½(vf + vo)t s = vot + ½at2 vf2 = vo2 + 2ax ANGULAR α = (ωf - ωo)/t ωf = ωo + αt θ = ½(ωf + ωo)t θ = ωot + ½αt2 ωf2 = ωo2 + 2αθ Equations of Angular Kinematics

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

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?

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

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=2rad • Related to angular velocity: • =2f • Since one revolution = 2r and the time it takes for one revolution = T • Then v= 2r /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= 2r /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: =r2 • 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=ac2+at2

Rolling

Rolling • Rolling= rotation + translation • Static friction between rolling object and ground (point of contact is momentarily at rest so static) v=r a=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 angular velocity of wheels at t=0 total revolutions of each wheel before coming to rest angular acceleration of wheel time it took to stop

Rotational Motion