Exam 3: Equation Summary

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1 MAACHUETT INTITUTE OF TECHNOLOGY Depatment of Physics Physics 8. TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t = Exam 3: Equation ummay = Impulse: I F( t ) = p Toque: τ =,P dp F P τ =,P F P sin θ = F = F tatic Equilibium: F = F + F +... = ; τ = τ, + τ, +... =. = dl Rotational dynamics: τ Angula Velocity: ω = ( dθ )kˆ Angula Acceleation: α = (d θ )k ˆ Fixed Axis Rotation: τ = I α dω τ = I α = I Moment of Inetia: I = body Angula Momentum: L = L = mv sin θ = p = p dm ( ), m,m Angula Impulse: J = t f τ t = L = L, f L, Rotation and Tanslation: obital spin L = L + L cm, obital L =, cm p, spin L cm = I cm ω spin obit obit = dl, τ spin dl τ = cm cm spin mv,

2 Rotational Enegy: K = I ω cm cm cm dθ Rotational Powe: P ot dw ot = τ ω= τω = τ One Dimensional Kinematics: v = d /, a = d v / t = t t = t v () t v = ) () x = v (t ) x, a (t x t x x x t = t = Constant Acceleation: x(t) = x + v x, (t t ) + a x (t t ) v () t = v x, + a (t t ) x x yt () = y + v y, (t t ) + a (t t y ) v () y t = v + a (t t ) y, y whee x, v x,, y, v y, ae the initial position and velocities components at t = t Newton s econd Law: Foce, Mass, Acceleation F ma F = F + F F F = ma F y = ma = ma Newton s Thid Law: F, = F, x x y z z Foce Laws: Univesal Law of Gavity: F, = G mm ˆ,, attactive, Gavity nea suface of eath: F gav = m gav g, towads eath Contact foce: F contact = N +f, depends on applied foces tatic Fiction: f f s s,max s Kinetic Fiction: f k = µ N opposes motion k Hooke s Law: F = k x, estoing Kinematics Cicula Motion: ac length: s = µ N diection depends on applied foces = Rθ ; angula velocity: ω = dθ tangential velocity: v= Rω ; angula acceleation: α = dω = d θ ; tangential acceleation a θ = Rα.

3 π R π ω Peiod: T = = ; fequency: f = =, v ω T π v Radial Acceleation: a = R ω ; = ; a = 4π R 4π R a f ; a = R T i= N Cente of Mass: R cm = m i / m i dm / m ; i= body i= N / m Velocity of Cente of Mass: V cm = m i v i dm i= body Toque: τ =,P F P τ =,P F P sin θ = F = F v / m tatic Equilibium: F = F + F +... = ; τ = τ, + τ, +... =. Kinetic Enegy: K = mv ; K = mv mv f f Wok: W = F d Powe: P = F v = dk Potential Enegy: U = W ; Wok- Kinetic Enegy: W = K consevative = F d c B A Potential Enegy Functions with Zeo Points: Constant Gavity: U( y )= mgy with U( y = ) =. Invese quae Gavity: U gavity () = Gm m with U gavity ( = ) =. Hooke s Law:U sping ( x ) = kx with U sping (x = ) =. Wok- Mechanical Enegy: W = + U = E ) mech = (K f +U nc K f ) (K +U 3

4 Poblem 3: Toque, Rotation and Tanslation A Yo-Yo of mass m has an axle of adius b and a spool of adius R. It s moment of inetia can be taken to be I = ( ) mr and the thickness of the sting can be neglected. The Yo-Yo is eleased fom est. a) What is the tension in the cod as the Yo-Yo descends? b) Use consevation of enegy to find the angula velocity of the Yo-Yo when it eaches the bottom of the sting. c) What happens to the Yo-Yo at the bottom of the sting? Poblem 4: Rolling Cylinde A solid cylinde of mass m and adius R is initially thown along a wooden floo hallway with an initial speed v and zeo initial angula velocity ω = as shown in the figue below. a) What is the moment of inetia of the cylinde about the axis of otation of the cylinde? b) What effects does fiction have on the cylinde befoe it eaches its final speed? Is the fiction static o kinetic? Which diection does it point? c) What is the angula speed, ω, of the cylinde when it just stats to oll without f slipping? Expess you answe in tems of m, R, and v. 7

5 Poblem 4: Rolling Cylinde A solid cylinde of mass m and adius R is initially thown along a wooden floo hallway with an initial speed v and zeo initial angula velocity ω = as shown in the figue below. a) What is the moment of inetia of the cylinde about the axis of otation of the cylinde? b) What effects does fiction have on the cylinde befoe it eaches its final speed? Is the fiction static o kinetic? Which diection does it point? c) What is the angula speed, ω, of the cylinde when it just stats to oll without f slipping? Expess you answe in tems of m, R, and v.

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Exam 3: Equation Summary

Exam 3: Equation Summary MAACHUETT INTITUTE OF TECHNOLOGY Depatment of Physics Physics 8. TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t = Exam 3: Equation ummay = Impulse: I F( t ) = p Toque: τ =,P dp F P τ =,P