Tanslation and Rotation Kinematics
Oveview: Rotation and Tanslation of Rigid Body Thown Rigid Rod Tanslational Motion: the gavitational extenal foce acts on cente-of-mass F ext = dp sy s dt dv total cm total = m = m A dt Rotational Motion: object otates about cente-ofmass. Note that the cente-of-mass may be acceleating cm
Oveview: Rotation about the Cente-of-Mass of a Rigid Body I cm The total extenal toque poduces an angula acceleation about the cente-of-mass ext dl " = I! = cm cm cm cm dt is the moment of inetial about the cente-of-mass! cm L cm is the angula acceleation about the cente-of-mass is the angula momentum about the cente-of-mass
Fixed Axis Rotation CD is otating about axis passing though the cente of the disc and is pependicula to the plane of the disc. Fo staight line motion, bicycle wheel otates about fixed diection and cente of mass is tanslating
Review: Relatively Inetial Refeence Fames Two efeence fames. Oigins need not coincide. One moving object has diffeent position vectos in diffeent fames 1 = R + 2 Relative velocity between the two efeence fames V = dr dt is constant since the elative acceleation is zeo A = dv dt = 0
Review: Law of Addition of Velocities Suppose the object is moving; then, obseves in diffeent efeence fames will measue diffeent velocities Velocity of the object in Fame 1: v 1 = d 1 dt Velocity of the object in Fame 2: v 2 = d 2 dt Velocity of an object in two diffeent efeence fames d 1 R 2 = + dt dt dt v 1 = V + v 2
Cente of Mass Refeence Fame Fame O: At est with espect to gound Fame O cm : Oigin located at cente of mass Position vectos in diffeent fames: = +R i cm,i =!R i cm,i cm cm Relative velocity between the two efeence fames A = d V / dt = 0 V = dr / dt cm cm cm cm Law of addition of velocities: v = v V v = v!v + i cm,i cm cm,i i cm
Rolling Bicycle Wheel Refeence fame fixed to gound Cente of mass efeence fame Motion of point P on im of olling bicycle wheel Relative velocity of point P on im: v = v + V P cm, P cm
Rolling Bicycle Wheel Distance taveled in cente of mass efeence fame of point P on im in time Δt:!s = R!" = R# cm!t Distance taveled in gound fixed efeence fame of point P on im in time Δt:!X cm =V cm!t
Rolling Bicycle Wheel: Constaint elations Rolling without slipping:!s =!X cm R! cm = V cm Rolling and Skidding!s <!X cm R! cm < V cm Rolling and Slipping!s >!X cm R! cm > V cm
Rolling Without Slipping: velocity of points on the im in efeence fame fixed to gound The velocity of the point on the im that is in contact with the gound is zeo in the efeence fame fixed to the gound.
Rotational Wok-Kinetic Enegy Theoem Change in kinetic enegy of otation about cente-of-mass 1 2 1 2!K " K # K = I $ # I $ ot ot, f ot,i cm cm, f cm cm,i 2 2 Change in otational and tanslational kinetic enegy!k =!K +!K tans ot # 1 1 2 & # 1 1 &!K =!K +!K = mv 2 " mv I ) 2 " I ) 2 tans ot cm, f cm,i ( + cm cm, f cm cm,i $ % 2 2 ' $ % 2 2 ( '
Checkpoint Poblem: Cylinde on Inclined Plane Enegy Method A hollow cylinde of oute adius R and mass m with moment of inetia I cm about the cente of mass stats fom est and moves down an incline tilted at an angle θ fom the hoizontal. The cente of mass of the cylinde has dopped a vetical distance h when it eaches the bottom of the incline. Let g denote the gavitational constant. The coefficient of static fiction between the cylinde and the suface is µ s. The cylinde olls without slipping down the incline. Using enegy techniques calculate the velocity of the cente of mass of the cylinde when it eaches the bottom of the incline.
Checkpoint Poblem: Descending Yo-Yo A Yo-Yo of mass m has an axle of adius b and a spool of adius R. It s moment of inetia about the cente of mass can be taken to be I = (1/2)mR 2 and the thickness of the sting can be neglected. The Yo-Yo is eleased fom est. What is the angula speed of the Yo-Yo at the bottom of its descent.
Demo B107: Descending and Ascending Yo-Yo M = wheel+axle 435 g R! oute R! inne 6.3 cm 4.9 cm 1 I! M R + R 2 = 1.385 " 10 g # cm ( 2 2 ) cm oute inne 4 2
Angula Momentum fo Rotation and Tanslation
Angula Momentum fo 2-Dim Rotation and Tanslation The angula momentum fo a otating and tanslating object is given by (see next two slides fo details of deivation) i=n sys L S = R S,cm!p + " cm,i! m i v cm,i i=1 The fist tem in the expession fo angula momentum about S aises fom teating the body as a point mass located at the cente-of-mass moving with a velocity equal to the cente-of-mass velocity, sys LS,cm = R S,cm!p The second tem is the angula momentum about the cente-of mass, i=n L cm = " cm,i! m i v cm,i i=1
Deivation: Angula Momentum fo 2-Dim Rotation and Tanslation The angula momentum fo a otating and tanslating object is given by i=n! " L = $( S m i i # m i vi % & i=1 ' The position and velocity with espect to the cente-of-mass efeence fame of each mass element is given by = R + v = V + v i S,cm cm,i i cm cm,i So the angula momentum can be expessed as i=n i=n i=n i=n! "! " L = R # m V + R # m v + m # V + # m v & i=1 ' i=1 & i=1 ' i=1 S S,cm $( i % cm S,cm ( i cm,i $( i cm,i % cm ( cm,i i cm,i
Deivation: Angula Momentum fo 2-Dim Rotation and Tanslation i=n i=n i=n i=n! "! " L S = R S,cm # $( m i % V cm + R S,cm #( m i v cm,i + $( m i cm,i %# V cm + ( cm,i # m i vcm,i & i=1 ' i=1 & i=1 ' i=1 The two middle tems in the above expession vanish because in the cente-of-mass fame, the position of the cente-of-mass is at the oigin, and the total momentum in the cente-of-mass fame is zeo, i=n i=n 1! m i v cm,i = 0 total! m i cm,i = 0 m i=1 Then then angula momentum about S becomes i=n i=n! " L S = R S,cm # $( m i % V cm + ( cm,i # m i vcm,i & i=1 ' i=1 i=n sys! " The momentum of system is p = # ' m i $ V cm % i=1 & So the angula momentum about S is i=n L S = R S,cm!p sys + " cm,i! m i v cm,i i=1 i=1
Eath s Motion about Sun: Obital Angula Momentum Fo a body undegoing obital motion like the eath obiting the sun, the two tems can be thought of as an obital angula momentum about the cente-of-mass of the eath-sun system, denoted by S, L = R!p sys S,cm S,cm = s, e mev cm kˆ Spin angula momentum about cente-of-mass of eath spin 2 L = I! = m R 2 cm cm spin e e! spin nˆ 5 Total angula momentum about S total ˆ 2 2 L S = s, e me v cm k + mer e! spin nˆs 5
Eath s Motion Obital Angula Momentum about Sun Obital angula momentum about cente of sun!p obital L total S = S,cm = m v s,e e cm kˆ Cente of mass velocity and v = angula velocity cm s,e! obit 2" $1 Peiod and angula velocity! = = 2.0 # 10 $7 obit ad % s Tobit m 2 2" obital e s,e obital #1 = m k ˆ = kˆ L ˆ S = 2.67! 10 40 kg " m 2 " s k Magnitude L S e s,e 2! obit Tobit
Eath s Motion Spin Angula Momentum Spin angula momentum about cente of mass of eath L spin cm 2 = I! 2 spin = m R! spin nˆ cm 5 e e Peiod and angula velocity 2" = = 7.29 # 10 $5 ad % s $1! spin Tspin Magnitude L spin cm = 7.09! 10 33 kg " m 2 " s #1ˆ n
Checkpoint Poblem: Angula Momentum fo Eath What is the atio of the spin angula momentum to the obital angula momentum of the Eath? What is the vecto expession fo the total angula momentum of the Eath about the cente of its obit aound the sun (you may assume the obit is cicula and centeed at the sun)?
MIT OpenCouseWae http://ocw.mit.edu 8.01SC Physics I: Classical Mechanics Fo infomation about citing these mateials o ou Tems of Use, visit: http://ocw.mit.edu/tems.