Download ( ),vx = dx / dt

Useful Equations
Kinematics x = x0 + v0 x t + (1 / 2)ax t 2 , vx = v0 x + ax t , vx2 = v02x + 2ax ( x ! x0 ) , vx = dx / dt ;
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ax = dvx / dt ; v = vx iˆ + vy ĵ + vz k̂ ; a = ax iˆ + ay ĵ + az k̂ ; average speed savg = (total distance)/(total
time); average velocity (x-com) vavg, x = ( x2 ! x1 ) / ( t 2 ! t1 ) , average acceleration (x-com)
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aavg, x = ( v2 x ! v1x ) / ( t 2 ! t1 ) . Newton’s Laws Fnet = ! Fi = 0 (Object in equilibrium); Fnet = ma
(Nonzero net force); Weight: Fg = mg , g = 9.8m / s ; Centripetal acceleration arad
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Friction fs ! µ s FN , fk = µ k FN . Hooke’s Law Fx = !kx . Work and Energy
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W = F • d = ( F cos! ) d = F||d ; W net = !K = (1 / 2)mv 2f " (1 / 2)mvi2 (valid if W net is the net or
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total work done on the object); W grav = !mg y f ! yi (gravitational work),
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W el = ! (1 / 2)kx 2f ! (1 / 2)kxi2 (elastic work)
Conservation of Mechanical Energy (only conservative forces are present) Emech = U + K
W net = !"U = ! (U 2 ! U1 ) = "K = K 2 ! K1 , U1 + K1 = U 2 + K 2 , U grav = mgy , U el = (1 / 2)kx 2
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Also !Emech = !U + !K = U f " U i + K f " K i = 0 # !K = "!U
Non-Conservative Forces Wexternal = !Emech + !Eth ( Wext work done by external forces, and we
set !Eint = 0 ), where !Eth = fk d (thermal energy or negative work done by friction).
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Using !Emech = !U + !K = U f " U i + K f " K i , U f + K f = U i + K i + Wext ! fk d
Work due to variable force 1D: W =
xf
!x
Fx dx " area under Fx vs. x, from x = xi to xf
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t2 !
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Momentum: P = mv , J = ! Fdt = Fav ( t 2 " t1 ) , J = !P = P2 " P1 . Newton’s Law in Terms of
t1
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Momentum Fnet = dp / dt . For Fnet = 0 , dp / dt = 0 gives momentum conservation: P ! constant.
Rotational Kinematics Equations: ! avg = (" 2 # "1 ) / ( t 2 # t1 ) , ! avg = (" 2 # " 1 ) / ( t 2 # t1 )
For ! z = constant, ! = ! 0 + " t , ! = ! 0 + " 0t + (1 / 2)# t 2 , ! 2 = ! 02 + 2" (# $ # 0 )
Linear and angular variables: s = r! , v = r! , atan = R! (tangential) , arad = v 2 / r = ! 2 r (radial)
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Moment of Inertia and Rotational Kinetic Energy I = ! mi ri2 , K rot = (1 / 2)I! 2 . Center of
i=1
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Mass (COM) rcom = ! mi ri / ! mi . Torque and Newton’s Laws of Rotating Body: rigid
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body ! = Fr" , ! net = " ! iext = I# , r! -moment arm about axis; point ! = r " F about origin O.
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Combined Rotation and Translation of a Rigid Body K = (1 / 2)Mvcom
+ (1 / 2)I com! 2 ,
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Fnet = Macom , ! net = I com" . Rolling without slipping s = R! , vcom = R! , acom = R! .
Angular Momentum L = I! (solid object) where I is the moment of inertia about the axis of
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rotation. L = r ! p " L = mvr sin # , valid for point particle about an origin O.
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Newton’s Second Law of rigid body in terms of angular momentum ! net = " ! iext = dL / dt .
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For ! net = 0 , dL / dt = 0 and angular momentum is conserved, L ! constant.
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