Rotational motion• We have learned about translational motion up to this point…• Now, we will venture into ROTATIONAL MOTION of a rigid body – Rigid body: a body with a definite shape that does not change• Rigid bodies can be analyzed as: – The translational motion of its center of mass and the rotational motion about its center of mass.• We will concentrate on purely rotational motion – all points in the body move in circles.
Rotational Motion CLast chapter, we talked about an Bobject in uniform circular motion:• Period• Tangential velocity D r• Centripetal Acceleration m v• Centripetal force ANow, we will talk about an object’smotion as it revolves in acircle, when it is uniform or non-uniform!
Rotational Motion• Now we will describe the circular motion of an object using angles, in terms of its: – Angular displacement θ [radians] – Angular velocity ω [rad/sec, rev/sec] – Angular acceleration α[rad/sec2]
Angular Displacement • Angular Displacement (θ) is the angle that a rotating object goes through. • We measure this in radians – A fraction of a revolution can be measured in degrees, grads or radians • A degree is 1/360 of a revolution • We can convert to radians using: – π radians = 180 degrees – One revolution = 2 radians …and use factor label method!
Example: Angular Displacement• A rubber stopper is twirled over a student’s head in a physics lab. Calculate the angular displacement of the stopper if it travels: – 30 degrees – 0.25 revolutions – 90 degrees – 1700 degrees – 12 revolutions
Angular distance θ in radians We can convert from linear distance (meters) to angular distance (radians) by: • first converting to radians • Use equation θ=s r s = arch length in meters (distance) r = radius of circular path (meters) θ = angle in radians
Example: finding arc length• What is the angular displacement of a rubber stopper that is twirled over a physics teacher’s head at a radius of 0.4 m and it travels 3.0 meters?
Example: arc lengthExample 8-1 from book: A particular bird’s eye can distinguishobjects that subtend an angle no smaller than about 3x10-4 rad.How small of an object can the bird just distinguish when flyingat a height of 100 m?Subtend: The angle formed by an object at a given external point 3 cm
Angular velocity ω In rotational motion, we usually describe the angular velocity as revolutions per second (rev/sec, rps), or radians per second • You will often have to convert this number, since it is usually given as a frequency (revolutions per time frame) • Conversion from linear velocity: ω= v r v = tangential (linear) velocity (m/s) r = radius of circular path (meters) ω = angular velocity (rad/sec)
Angular Velocity Unlike tangential velocity, the angularvelocity is the same atevery point on a rigid body, like a wheel
Example: angular velocityExample 8-3: What is the angular and linear speed of a childsitting 1.2 m from the center of a steadily rotating merry-go-round that makes one complete revolution in 4.0 seconds?
Angular acceleration α Angular acceleration occurs when the angular velocity changes over time. • It acts in the direction of rotation in a circular motion (NOT the same as centripetal acceleration) • In this case, we must also introduce tangential acceleration (at) since the tangential velocity is changing – If there is angular acceleration, there will also be tangential acceleration • We can use the following conversion: α= at r ar= ω2r a = tangential (linear) acceleration (m/s) ar = radial (linear) acceleration (m/s) r = radius of circular path (meters) r = radius of circular path (meters) 2) ω = angular acceleration (rad/sec ω = angular acceleration (rad/sec2)
Example: Angular AccelerationWhat is the tangential and angular acceleration of a child seated1.2 m from the center of a steadily rotating merry-go-round thatmakes one complete revolution in 4.0s?
Linear & rotational motion equivalentsNow let’s re-write the linear motion equations using our rotational motion values!
Examples: kinematic equationsEx 8-5: A centrifuge rotor is accelerated from rest to 20,000 rpmin 5.0 min. What is its average angular acceleration?
Examples: kinematic equationsEx 8-5: A centrifuge rotor is accelerated from rest to 20,000 rpm in5.0 min. (a) What is its average angular acceleration? (b) throughhow many revolutions has the centrifuge rotor turned during itsacceleration period? Assume constant angular acceleration. a. 7.0 rad/s2 b. 50,000 revs