Formula sheet

Formula Sheet – Final 06/08/2010 Ph211 Spring 10Multiple (interacting) objectsNewton’s Laws are written separately for each, in their own coordinate systems. The interactionis represented by one of the forces from the Third Newton Law pair in each object’s equations.Work and EnergyThe work-energy relation, with W np being work done by forces without potentials:W = ∆K, or E i + W np = E f . Here E is: E = K + U (U grav , U spr , ...) (W-E)Forces may either not have potentials, like friction, or we may not know them. Note that, for forcesthat have potentials, you either use their potential energy formulas, or calculate their work.Work.W AB =The work of a force ⃗ F(⃗x), along a curve described by ⃗x, is in general calculated as∫ BA⃗F · ⃗dx =∫ BAF dx cos( ⃗ dx, ⃗ F) =∫ BAF x dx(x: parameter along the curve) (W)When a force and its angle to the trajectory are constant, its work over the path of length L is:W AB =∫ BA⃗F · ⃗dx =∫ BAF dx cos( ⃗ dx, ⃗ F) =∫ BNote that the F x above may be positive or negative.Scalar product.A∫ BF x dx = F x dx = F x LThe rule for combining (“multiplying”) vectors so to get a scalar:A(Wc)⃗A · ⃗B = ∑ i A iB i or ⃗ A · ⃗ B = | ⃗ A| | ⃗ B|cos( ⃗ B, ⃗ A) (dot)cos( ⃗ B, ⃗ A) above stands for the angle from ⃗ B to ⃗ A, taken counter-clock wise.Spring.F (x)sprThe restoring force of a spring, along the x direction:= −k ∆x(spr)∆x is the displacement of its free end from the equilibrium, in the coordinate system used.Note that it may be positive or negative, what will determine the sign of the F spr in that frame.Momentum ConservationThe total momentum of a system is conserved when ∑ Fext ⃗ = 0 (for all external forces):⃗P (tot)i= P ⃗ (tot) ∑f, or ⃗pi = ∑ ⃗p f (momenta of all particles in the system) (CoM)In perfectly elastic collisions mechanical energy is conserved too. In inelastic collisions it is not.Collected definitions and formulas. Kinetic and potential energies; and the spring force:Definition of kinetic energy K = 1 2 mv2 (Kin)Gravitational potential energy U gr = mgy (with respect to y = 0 level) (Ugr)Potential energy of the spring U spr = 1 2 k (∆x)2 (Usp)Restoring force of the spring F spr = − k ∆x (Fsp)