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Unformatted text preview: 1 Dot Product and Cross Products • For two vectors, the dot product is a number A · B = AB cos( θ ) = A k B = AB k (1) • For two vectors A and B the cross product A × B is a vector. The magnitude of the cross product  A × B  = AB sin( θ ) = A ⊥ B = AB ⊥ (2) The direction of the resulting vector is given by the right hand rule. • A formula for the cross product of two vectors is A × B = ˆ ı ˆ ˆ k A x A y A z B x B y B z (3) 2 Work and Energy • The work done by any force in going from position r A up to position r B is W = Z r B r A F · d r (4) For a constant force the work is simply the dot product of the force with the displacement W = F · Δ r = F Δ r cos( θ ) (5) where θ is the angle between the displacement and the force vector. • The work done by all forces W all forces = Δ K = K f K i (6) where the kinetic energy is K = 1 2 mv 2 (7) • We classify forces as conservative (gravity springs) and nonconservative (friction). For conservative forces we can introduce the potential energy. The change in potential energy is minus the work done by the foce Δ U = U 2 U 1 = Z 2 1 F · d r (8) • The force associated with a given potential energy is F = dU ( x ) dx (9) • Then the fundamental work energy theorem can be written W non consv + W ext = Δ K + Δ U (10) where Δ U is the change in potential energy of the system. • If there are no external or dissipative forces then E = K + U = constant (11) You should understand the logic of how Eq. ?? leads to Eq. ?? and ultimately Eq. ?? . 1 • The potential energy depends on the force that we are considering: – For a constant gravitational force F = mg we have U = mgy (12) where y is the vertical height measured from any agreed upon origin. – For a spring with spring constant k which is displaced from equilibrium by an amount x , we have a potential energy of U = 1 2 kx 2 (13) – For a particle a distance r from the earth the potential energy is U = GMm r (14) • Power is defined as the rate at which work is done or the rate at which energy is transformed from one form to another. P = dW dt = dE dt (15) or P = F · v (16) 3 Momentum • The momentum of an object is p = m v (17) In terms of momentum Newtons Law can X F = d p dt (18) • The total momentum transferred to a particle by a force is the known as the impulse (or simply momentum transfer) Δ p = p f p i = Z t f t i F dt = J (19) If the force last a period Δ t the average force is F ave = Δ p Δ t (20) • For a system of particles with total mass M, we define the center of mass...
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This note was uploaded on 01/25/2012 for the course PHYSICS 141 taught by Professor Teaney during the Fall '10 term at SUNY Stony Brook.
 Fall '10
 teaney
 Physics

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