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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 (Plancks 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 Newtons 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 t + B sin t where 2 = k/m . Can also write solution as x = < ( Ae it ) , 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