Exam 3: Equation Summary

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P dp total F P τ S = S,P F P sin θ = F = F Static Equilibium: F total = F + F +... = ; τ total 1 S = τ S,1 + τ S, +... =. total = dl S Rotational dynamics: τ S Angula Velocity: ω = ( dθ )k ˆ Angula Acceleation: α = (d θ )k ˆ Fixed Axis Rotation: τ = I α S S total dω τ S = I S α = IS Moment of Inetia: I S = body Angula Momentum: L S = S, L S = S mv sin θ = p = p,m Angula Impulse: J = t f S τ t S = L = L S S, f L S, Rotation and Tanslation: total obital spin L S = L S + L, obital L S = S p total,, spin L = I ω spin obit obit = dl S, τ spin dl τ = S dm ( ) spin m mv, 1

2 Rotational Enegy: K = 1 I ω dθ Rotational Powe: P ot dw ot = τs ω= τω S = τ S One Dimensional Kinematics: v = d /, a = d v / t = t t = t v () t v x, = a ( t ) x() t x = v ( t ) x x x t = t = Constant Acceleation: x(t) = x + v x, (t t ) + 1 a x ( t t ) vx() t = v x, + a x ( t t ) yt () = y + v y, (t t ) + 1 a y ( t t ) vy () t = v y, + a y ( t t ) whee x, v x,, y, v y, ae the initial position and velocities components at t = t Newton s Second Law: Foce, Mass, Acceleation total F ma F = F + F F total total F total = ma F y = ma = ma Newton s Thid Law: F 1, = F,1 1 x x y z z Foce Laws: 1 Univesal Law of Gavity: F 1, = G mm ˆ1,, attactive 1, Gavity nea suface of eath: F gav = m gav g, towads eath Contact foce: F contact = N +f, depends on applied foces Static 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 tangential velocity: v= Rω ; angula acceleation: α = ω acceleation a θ = Rα. = Rθ ; angula velocity: ω = dθ d d = θ ; tangential

3 π R π 1 ω Peiod: T = = ; fequency: f = =, v ω T π v Radial Acceleation: a = Rω ; = ; a = 4π R 4π R a f ; a = R T i= N total Cente of Mass: R = m i / m i dm / m total ; i=1 body i= N / m total Velocity of Cente of Mass: V = m i v i dm 1 i=1 body 1 Kinetic Enegy: K = mv ; K = mv f 1 mv f Wok: W = F d Powe: P = F v = dk Potential Enegy: U = W ; Wok- Kinetic Enegy: W total = K consevative = F d c B A Potential Enegy Functions with Zeo Points: Constant Gavity: U( y )= mgy with U( y = ) =. v / m total 1 Invese Squae Gavity: U gavity () = Gm m with U gavity ( = ) =. Hooke s Law:U sping ( x ) = 1 kx with U sping (x = ) =. total total Wok- Mechanical Enegy: W = + U = E ) mech = (K f +U total nc K f ) (K +U 3

4 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1T Fall Tem 4 Exam 3 Name Section: Moning Pitchad/Suow 1-1 Aftenoon Doumashkin 1-3 Table and Goup: The following exam consists of fou poblems. Answes without wok shown will not be given any cedit. Good luck! Poblem 1 ( points) Pat a) (5 points) Pat b) (5 points) Pat c) (5 points) Pat d) (5 points) Poblem (5 points) Poblem 3 (3 points) Poblem 4 (5 points) Total (1 points) 4

5 Poblem 1a: (5 points) A physical pendulum consists of a ule of mass m oscillating about a fixed point. The ule is pulled to one side and allowed to oscillate. The pendulum has moment of inetia I P about it s pivot point and the distance fom the pivot point to the cente of mass is l. a) The peiod of the pendulum is equal to π I P / mgl. b) The peiod of the pendulum is geate than π I / mgl. c) The peiod of the pendulum is less than π I / mgl. Explain you easoning: P P 5

6 Poblem 1b: (5 points) A tetheball is attached to a post by a sting. The sting passes though a hole in the cente of the post at the top. The sting is gadually shotened by dawing it though the hole. Ignoe gavity. Until the ball hits the post, a) The enegy and angula momentum about the cente of the post ae constant. b) The enegy of the ball is constant but the angula momentum about the cente of the post changes. c) Both the enegy and the angula momentum about the cente of the post, change. d) The enegy of the ball changes but the angula momentum about the cente of the post is constant. Explain you easoning. 6

7 Poblem 1c: (5 points) A block on a hoizontal table is connected to one end of a sping. The othe end of the sping is attached to a wall. The block is set in motion pependicula to wall so that it oscillates about its equilibium point. A lump of sticky putty is dopped vetically onto the block and lands the instant the block has its maximum velocity. The putty sticks without bouncing. Duing the collision a) the velocity of the block changes; the mechanical enegy of the system consisting of the putty and the block is constant. b) the velocity of the block is constant; the mechanical enegy of the system consisting of the putty and the block is constant. c) the velocity of the block changes; the mechanical enegy of the system consisting of the putty and the block changes. d) the velocity of the block is constant; the mechanical enegy of the system consisting of the putty and the block changes. Explain you easoning. 7

8 Poblem 1d: (5 points) A figue skate is spinning with he ams held close to he body. She elaxes he am muscles and he ams move outwad. Compaed to he initial otational kinetic enegy, he otational kinetic enegy afte she he ams move outwad is a) the same. b) lage. c) smalle. Explain you easoning. 8

9 Poblem : (5 points) In the lab fame, Cat B has mass m B and is moving to the ight with initial velocity v. B, Cat A has mass m A = m B and is at est. Cat B collides elastically with cat A. Afte the collision, Cat A moves up an incline plane that makes an angle θ with espect to the hoizontal. You may assume that the tack is fictionless and the acceleation due to gavity is g. a) What ae the diections and velocities of Cat A and Cat B immediately afte the collision? b) What distance does Cat A move along the incline plane when it comes to est? 9

10 Poblem 3: (3 points) A steel washe, is mounted on a cylindical oto of adius. A massless sting, with a mass m attached to the othe end, is wapped aound the side of the oto and passes ove a massless pulley. Assume that thee is a constant fictional toque about the axis of the oto. The mass is eleased and falls. As the mass falls, the oto undegoes an angula acceleation of magnitude α. Afte the sting detaches fom the oto and the oto 1 coasts to a stop with an angula acceleation of magnitude α. Let g denote the gavitational constant. a) What is the moment of inetia I of the oto assembly (including the washe) R about the otation axis? Show all you wok. Expess you answe in tems of the given quantities, α, α, m, and g. 1 b) Suppose the oto has an angula velocity ω, when a second washe of moment of inetia I is dopped on top of the fist washe. It takes a time t fo the W dopped washe to move at the same angula velocity as the oto. What is the angula velocity ω of the assembly immediately afte the collision is finished f and the dopped washe and oto move at the same angula velocity ω? Expess f you answe in tems of the quantities I, I, α, t, and ω as needed. R W 1

11 Poblem 4: (5 points) The cente of mass of a bicycle wheel of adius R and mass m is initially spinning with angula velocity ω. The wheel is loweed to the gound without bouncing. As soon as the wheel touches the level gound, the wheel stats to move fowad until it begins to oll without slipping with an unknown final angula velocity ω and an unknown velocity of f the cente of mass v f. Assume that all the mass of the wheel is located on the im., a) Daw a fee body diagam of all the foces acting on the bicycle wheel while it is moving fowad. b) What is the elation between the angula velocity of the wheel ω f and the velocity of the cente of mass v f when it begins to oll without slipping?, c) What is the velocity of the cente of mass of the wheel when it begins to oll without slipping? 11

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