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,
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.
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
Poblem : Exploding Puck A hockey playe shoots a tick hockey puck along the ice towads the cente of the goal fom a position d diectly in font of the goal. The initial speed of the puck is v and the puck has a mass m. Half way to the goal the puck explodes into two fagments. One piece of mass m = (3 5) m comes back towads the playe and passes 3d 8 to the side of the spot it was initially shot fom with a speed v, f = (5 ) v. The othe piece of the puck with mass m = ( 5) m continues on towads the goal with a speed v, f. Assume that thee is no fiction as the puck slides along the ice and that the mass of explosive in the puck is negligible. a) Wite down the equations fo consevation of momentum of the puck and fagments in tems of the quantities shown in the figue above. b) By what distance, y, does the piece that continues towads the goal miss the cente of the goal? Expess you answe in tems of d. 6
Poblem : Exploding Puck A hockey playe shoots a tick hockey puck along the ice towads the cente of the goal fom a position d diectly in font of the goal. The initial speed of the puck is v and the puck has a mass m. Half way to the goal the puck explodes into two fagments. One piece of mass m = (3 5) m comes back towads the playe and passes 3d 8 to the side of the spot it was initially shot fom with a speed v, f = (5 ) v. The othe piece of the puck with mass m = ( 5) m continues on towads the goal with a speed v, f. Assume that thee is no fiction as the puck slides along the ice and that the mass of explosive in the puck is negligible. a) Wite down the equations fo consevation of momentum of the puck and fagments in tems of the quantities shown in the figue above. b) By what distance, y, does the piece that continues towads the goal miss the cente of the goal? Expess you answe in tems of d. 6
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