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CHENNAI MATHEMATICAL INSTITUTE CLASSICAL MECHANICS I PROBLEM SHEET III 27th August 2012 date due 3rd September 2012 11. An electron moving with an intital velocity enters an electric field E ( t 1 with a velocity V 0 ˆ 1 . The acceleration is given by ~a = eE ( t 1 /m , where m is the mass of the electron. The magnitude of the electric field is given by E ( t ) = 0 for t < 0 ; E(t) = E 0 t τ for 0 < t < τ ; E ( t ) = E 0 τ ( t - τ ) for τ < t < 2 τ ; and so on i.e. E ( t + τ ) = E ( t ) for t > 0 Find the distance travelled in time , n being an integer.. 12. Consider a damped harmonic oscillator described by ( with k > λ/ 2 ) m d 2 x dt 2 + λ dx dt + k x = 0 with x ( t = 0) = A and dx dt | t =0 = 0. Calclate the average value of the total energy < E > nT over a cycle nT t ( n + 1) T , after n cycles.( You can leave the answer in terms of an integral over time) T is the time period. Use the result to show < E > nT = < E > 0 e - 2 γnT for γ << ( k m ) 1 / 2 where γ = λ/ (2 m ). 13. Find the equilibrium position and the frequency of small oscillations
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