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HT2010 First Year Physics: Prelims CP1 Classical Mechanics: Prof. G. Yassin Problems V Lagrangian Dynamics The problems are divided into two sections: (A) Standard and (B) Harder. Section A - Standard Problems Variational Principles 1. Fermat's principle A light beam is propagating in the x y plane in a media whose refraction index n depends only on y. (a) Use Fermat's principle to show that the trajectory of the beam from ( x 0 , y 0 ) to ( x 1 , y 1 ) may be obtained by minimizing the functional S ( y ) = c 1 integraldisplay x 1 x 0 n ( y )[(1 + y 2 )] 1 2 dx where y = dy/dx and c is the speed of light in vacuum. (b) A light ray propagates from ( x 0 , y 0 ) to ( x 1 , y 1 ) by reflection from the surface of a flat mirror located at in the plane y = 0 as shown in Fig 1. Show that the angle of reflection φ r is equal to the angle of incidence φ i 2. The Brachistochrone Problem-Bernoulli's solution The Brachistochrone Problem was launched by Jean Bernoulli in 1696 and may be stated as follows: A particle of mass m is descending under the influence of constant gravity in a vertical plane. Which path should the particle follow in order too move from point A to point B in the shortest possible time? The solution suggested by Bernoulli recognises that the particle will follow the path of a beam of light propagating in a dielectric media of refractive index n(y), inversely proportional to the particle velocity. . (a) Given that the optical path will be minimum if Snell's law is satisfied and that the index of refraction is inversely proportional to the free fall velocity v = 2 gy show that the path equation is given by sin θ 2 gy = 1 2 ga , where θ is the angle between the ray direction (tangent to the curve) and the vertical direction. (b) Eliminate θ to show that the above equation can be written as: dy dx = parenleftbigg a y y parenrightbigg 1 2 (c) Use the substitution y = a sin 2 φ 2 to find that x ( φ ) = a 2 ( φ sin φ ) , hence show that the equation of the path is a cycloid. 1
Euler-Lagrange Equation 3. Atwood's machine The three masses shown below move in a vertical plane under the influence of constant gravity and the tension in the inextensible strings. Assuming that the pulleys are massless and that all friction forces can be neglected, (a) write down the constraints equation that result from the fixed length of the strings, hence show that the motion of the three masses may be described by two generalized coordinates. (b) Use the E-L equation to find the acceleration of each mass. (c) Repeat (b) using Newton second law and compare the two methods. Why does the upper pulley rotate despite the fact that the masses on either side are equal? 4. Motion in two dimensions Consider a particle of mass m moving in the x, y plane under the influence of the potential V ( r ) where r is the postion vector of the particle in an inertial reference frame. Construct the Lagrangian and the Hamiltonian of the particle in polar polar coordinates r, θ , hence find which quantities are constants of motion. Is this consistent with what you expected from Newtonian mechanics? 5. The simple pendulum Use E-L equation to calculate the period of oscillation of a simple pendulum of length l and bob mass m in the small angle approximation. Assume now that the pendulum support is accelerated in the vertical direction at a rate a , find the period of oscillation. For what value of a the pendulum does not oscillate? Comment on this result. 6. A sliding block A block of mass m slides on a frictionless inclined plane of mass M , which itself rests on a horizontal frictionless surface. (a) Choose the displacement of the inclined plane x and the displacement of the block m s relative to the inclined plane as generalized coordinate and find the Lagrangian of the system. (b) Write down the E-L equation for each coordinate and find the ac- celeration of the inclined plane. Compare this solution with the one you obtained using Newton laws. 7. Rotating bead A bead of mass m is constrained to slide on a frictionless wire which is made to rotate about a vertical axis at an angular velocity ω . The wire is tilted away from the vertical by an angle α and the location of the bead is measured by the coordinate r . (a) Write down the equation of motion of the bead using the E-L equa- tion. Test the integrity of your equation by taking extreme values of α . (b)Find the general solution assuming that at t = 0 , r = r 0 , ˙ r = 0 . Based on this solution, show that for r 0 = gcosα/ω 2 sinα , the bead moves in circular motion (as expected!). Describe the motion for r < r 0 and r > r 0 . (c) Which of the following quantities is a constant of the bead motion: angular momentum with respect to the origin, the Hamiltonian, total en- ergy? 2
8. Sliding on a sphere A particle of mass m slides without friction down the surface of a hemisphere of radius R . (a) Construct the Lagrangian of the problem in terms of the polar co- ordinates ( r, ϑ ) , in the range when the constraint r = R is valid. Find the equation of motion. (b) Allow the radius of the sphere to vary by an infinitesimal amount and write the equation of morion with r as a free variable. Include the potential V ( r ) of the rection force applied by the hemisphere on the object. Write the new Lagrangian and find the reaction force for r = R . Compare with the result derived from NII. (c) Assuming that the particle is released from the top of the sphere from rest, show that the particle leaves the surface at an angle cos ϑ max = 2 / 3 . 9. A beed on a rotating hoop A vertical circular hoop of radius R rotates about a vertical axis at an angular velocity ω . A bead of mass m can slide on the hoop without friction and is constrained to stay on the hoop. By taking the angle ϑ between the radius line and the vertical, as a generalized coordinate, (a) Find the Lagrangian and the equation of motion. Using the concept of effective potential or otherwise, find the three equilibrium positions of the bead. (b) Discuss the stabiliy of each equiibrium point and find the frequency of smal oscillations about the stable ones. (b) Find the Hamiltonian and the total energy T+V. Is either of them a constant of motion? Section B - Harder Problems 10. 2-D spring A particle of mass m is attached to the free end of a massless spring of equilibrium length a and spring constant k . The other end of the spring is pivoted to a frictionless horizontal surface and the particle is allowed to move in 2-D under the influence of the spring force which is assumed to obey Hook's law. (a) Write the E-L equations for the polar coordinate ( r, θ ) . Identify the cyclic coordinates and the corresponding conserved quantities. Write down the equation of motion in terms of the variable r . (b) Write the total energy of the system (for a given angular momentum J ) and analyse the motion using the concept of effective potential. Find the radius for circular orbit and show that it is consistent with the value obtained from Newton laws. (c) Use Newton II in Cartesian coordinates to show that if the rest length of the spring is negligible ( a 0 ), the path of the particle is elliptical with r measured from the centre of the ellipse. Use the expression for total energy E to find the major and the minor axes of the ellipse. [Answer: a, b = r 0 / ( 1 ± ǫ ) where r 0 = J 2 /mE and ǫ = radicalbig 1 kJ 2 /mE 2 ) ] 11. The Brachistochrone Problem A particle of mass m is constrained to slide without friction, in a vertical plane, down a cycloidal surface which may be parameterized as: x = a ( φ sinφ ) , y = a (1 + cos φ ) . At t = 0 the partical was at rest at the origin A. (a) Show that the equation of the surface describes the motion of a particle on the rim of a rolling wheel. 3
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