2. Newtonian Mechanics Newton’s law Motion under constant forces Position dependent forces Velocity dependent forces SOONGSIL UNIVERSITY CLASSICAL MECHANICS

2.2 Motion under a constant force SOONGSIL UNIVERSITY CLASSICAL MECHANICS

Bottom 에서의 속도는? 마찰이 있을 때와 없을 때 마찰이 있을 때 마찰이 없을 때 SOONGSIL UNIVERSITY CLASSICAL MECHANICS

2.3 Position dependent force Potential E can be defined as SOONGSIL UNIVERSITY CLASSICAL MECHANICS

v0 Free fall Maximum height of a ball thrown upward 2.3 Position dependent force continued Free fall v0 h Maximum height of a ball thrown upward For a given initial velocity, there always exists maximum height. Is it correct? SOONGSIL UNIVERSITY CLASSICAL MECHANICS

Variation of Gravity with height 2.3 Position dependent force continued Variation of Gravity with height Definition of the gravitational acceleration Escape speed SOONGSIL UNIVERSITY CLASSICAL MECHANICS

= + Morse function V(x) : potential energy of a diatomic molecule x 2.3 Position dependent force continued Morse function V(x) : potential energy of a diatomic molecule = + x Therefore, x=x0 is the equilibrium position. separation of the atoms and –V0 is the binding energy. SOONGSIL UNIVERSITY CLASSICAL MECHANICS

Near the equilibrium position V(x) can be expanded 2.3 Position dependent force continued Near the equilibrium position V(x) can be expanded The potential becomes parabolic in the leading order, And it can be considered as a simple harmonic oscillator problem. SOONGSIL UNIVERSITY CLASSICAL MECHANICS

Maximum separation at room temp.? 2.3 Position dependent force continued Hydrogen molecule Maximum separation at room temp.? Binding energy = -4.52 eV Equilibrium separation = 0.074 nm Delta = 0.036 nm Room temp. ~ 1/40 eV X = 0.074 +- 0.0027 nm ~3.6% change in size which means thermal expansion SOONGSIL UNIVERSITY CLASSICAL MECHANICS

2.4 velocity dependent force in SI unit, The ratio SOONGSIL UNIVERSITY CLASSICAL MECHANICS

As time goes to infinity, it converges to a point. 2.3 velocity dependent force continued linear resistance (Dominant in low speed) quadratic resistance (Dominant in high speed) As time goes to infinity, it converges to a point. As time goes to infinity, does it converge??? It shows logarithmic divergence!!! SOONGSIL UNIVERSITY CLASSICAL MECHANICS

vertical fall through a fluid (linear resistance) mg -C1v As time goes to infinity, the velocity approaches to –mg/c1. Terminal velocity where the terminal speed = , and = It occurs when F = 0. If an object is dropped, after 5 , v = 0.993 vt SOONGSIL UNIVERSITY CLASSICAL MECHANICS

vertical fall through a fluid (quadratic resistance) If an object is dropped, after 5 , v = 0.99991 vt SOONGSIL UNIVERSITY CLASSICAL MECHANICS

Terminal speed of raindrops and basketballs Raindrop ~ 0.1 mm Basketball ~ 0.25 m The ratio raindrops basketballs The ratio = 0.14 v v = 7.1m/s linear term dominates. The ratio = 350 v v = 0.0029m/s quadratic term dominates. SOONGSIL UNIVERSITY CLASSICAL MECHANICS

The end 2. Newtonian Mechanics Newton’s law Motion under constant forces Position dependent forces Velocity dependent forces The end SOONGSIL UNIVERSITY CLASSICAL MECHANICS