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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 non-conservative (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