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Unformatted text preview: Classical Dynamics and Fluids P 1 C LASSICAL D YNAMICS AND F LUIDS 20 Lectures Prof. S.F. Gull • NOTES — Provisional hardcopy available in advance. Definitive copies of overheads available on web. Please report all errors and typos. • SUMMARY SHEETS — 1 page summary of each lecture. • EXAMPLES — 2 example sheets — 2 examples per lecture. • WORKED EXAMPLES — Will be available on the web later. • WEB PAGE — For feedback, additional pictures, movies etc. http://www.mrao.cam.ac.uk/ ∼ steve/part1bdyn/ There is a link to it from the Cavendish teaching pages. Classical Dynamics and Fluids P 2 R EVIEW OF N EWTONIAN M ECHANICS • Newtonian Mechanics is:- Non-relativistic i.e. velocities v c (speed of light= 3 × 10 8 m s- 1 .)- Classical i.e. Et ¯ h (Planck’s constant= 1 . 05 × 10- 34 J s.) • Assumptions :- mass independent of velocity, time or frame of reference;- measurements of length and time are independent of the frame of reference;- all parameters can be known precisely. • Mechanics : = Statics (absence of motion); + Kinematics (description of motion, using vectors for position and velocity); + Dynamics (prediction of motion, and involves forces and/or energy). Classical Dynamics and Fluids P 3 B ASIC P RINCIPLES OF N EWTONIAN D YNAMICS • This course contains many applications of Newton’s Second Law. • Masses accelerate if a force is applied. . . • The rate of change of momentum (mass × velocity) is equal to the applied force. • Vectorially ( p = m v ): d p d t = d( m v ) d t = F • Usually m is a constant so m d v d t = F • General case m d v d t + d m d t v = F enables you to do rocket science . v u m d m • Rocket of mass m ( t ) moving with velocity v ( t ) expels a mass d m of exhaust gases backwards at velocity- u relative to the rocket. • In the absence of gravity or other external forces d mu + m d v = 0 where m i , f are the initial and final masses. • Integrating, we find v = u log( m i /m f ) • For a rocket accelerating upwards against gravity m d v d t + d m d t u + mg = 0 . Classical Dynamics and Fluids P 4 I MPORTANT E XAMPLE — S IMPLE H ARMONIC O SCILLATOR • The simple harmonic oscillator (SHO) occurs many times during the course. • Mass m moving in one dimension with coordinate x on a spring with restoring force F =- kx . • The constant k is known as the spring constant . • Newtonian equation of motion: m ¨ x =- kx , where ˙ x denotes d x d t etc. • General solution: x = A cos ωt + B sin ωt where ω 2 = k/m . • Can also write solution as x = < ( Ae iωt ) , where A is complex. • We can integrate the equation of motion to get a conserved quantity — the energy . • Multiplying the equation of motion by ˙ x (a good general trick) we get m ˙ x ¨ x + k ˙ xx = 0 ⇒ 1 2 m ˙ x 2 + 1 2 kx 2 = E = constant • Here the quantity T ≡ 1 2 m ˙ x 2 is the kinetic energy of the mass and V ≡ 1 2 kx 2 is the potential energy stored in the spring....
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- Spring '07
- Force, classical dynamics, Classical Dynamics and Fluids