# Lecture 20 Goals: Wrap-up Chapter 14 (oscillatory motion) Start discussion of Ch

## Lecture 20 Goals: Wrap-up Chapter 14 (oscillatory motion) Start discussion of Ch

Strey, Kendra, Features Editor has reference to this Academic Journal, PHwiki organized this Journal Lecture 20 Goals: Wrap-up Chapter 14 (oscillatory motion) Start discussion of Chapter 15 (fluids) Assignment HW-8 due Tuesday, Nov 15 Monday: Read through Chapter 15 The general solution is: x(t) = A cos (wt + f) where A = amplitude = angular frequency = phase constant SHM Solution k m -A A 0(Xeq) k m -A A 0(Xeq) T = 1 s k -1.5A 1.5A 0(Xeq) T is: T > 1 s T < 1 s T=1 s m

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SHM Solution The mechanical energy is conserved: U = ½ k x2 = ½ k A2 cos2(t + ) K = ½ m v2 = ½ k A2 sin2(t+f) U+K = ½ k A2 k m -A A 0 Which mass would have the largest kinetic energy while passing through equilibrium A) 2k 2m -A A 0 B) k -2A 2A 0 m C) SHM So Far For SHM without friction The frequency does not depend on the amplitude ! The oscillation occurs around the equilibrium point where the as long as ce is zero! Mechanical Energy is constant, it transfers between potential in addition to kinetic energies.

Energy in SHM The total energy (K + U) of a system undergoing SHM will always be constant! U = ½ k x2 SHM in addition to quadratic potentials SHM will occur whenever the potential is quadratic. For small oscillations this will be true: For example, the potential between H atoms in an H2 molecule looks something like this: U x What about Vertical Springs k m k equilibrium new equilibrium mg=k L L k m y=0 L y Fnet= -k (y+ L)+mg=-ky Fnet =-ky=ma=m d2y/dt2 Which has the solution y(t) = A cos( t + )

The Simple Pendulum A pendulum is made by suspending a mass m at the end of a string of length L. Find the frequency of oscillation as long as small displacements. S Fy = may = T  mg cos(q) S Fx = max = -mg sin(q) If q small then x L q in addition to sin(q) q dx/dt = L dq/dt ax = d2x/dt2 = L d2q/dt2 so ax = -g q = L d2q / dt2 in addition to q = q0 cos(wt + f) with w = (g/L)½ T = 2 (L/g)½ L m mg z y x T What about friction One way to include friction into the model is to assume velocity dependent drag. Fdrag= -bdragv=-bdrag dx/dt Fnet=-kx-bdrag dx/dt = m d2x/dt2 d2x/dt2=-(k/m)x (bdrag/m) dx/dt a new differential equation as long as x(t) ! Damped oscillations x(t) = A exp(-bt/2m) cos (wt + f) x(t) t For small drag (under-damped) one gets:

Driven oscillations, resonance So far we have considered free oscillations. Oscillations can also be driven by an external as long as ce. wext w0 wext oscillation amplitude w0: natural frequency Chapter 15, Fluids An actual photo of an iceberg At ordinary temperature, matter exists in one of three states Solid – has a shape in addition to as long as ms a surface Liquid – has no shape but as long as ms a surface Gas – has no shape in addition to as long as ms no surface What do we mean by fluids Fluids are substances that flow . substances that take the shape of the container Atoms in addition to molecules are free to move.

Fluids An intrinsic parameter of a fluid Density (mass per unit volume) units : kg/m3 = 10-3 g/cm3 r(water) = 1.000 x 103 kg/m3 = 1.000 g/cm3 r(ice) = 0.917 x 103 kg/m3 = 0.917 g/cm3 r(air) = 1.29 kg/m3 = 1.29 x 10-3 g/cm3 r(Hg) = 13.6 x103 kg/m3 = 13.6 g/cm3 =m/V Fluids Another parameter Pressure ( as long as ce per unit area) P=F/A SI unit as long as pressure is 1 Pascal = 1 N/m2 1 atm = 1.013 x105 Pa = 1013 mbar = 760 Torr = 14.7 lb/ in2 (=PSI) The atmospheric pressure at sea-level is If the pressures were different, fluid would flow in the tube! Pressure vs. Depth For a uni as long as m fluid in an open container pressure same at a given depth independent of the container Fluid level is the same everywhere in a connected container, assuming no surface as long as ces Why is this so Why does the pressure below the surface depend only on depth if it is in equilibrium Imagine a tube that would connect two regions at the same depth.

## Strey, Kendra Features Editor

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