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CM_2003a - QM(f j t?<—:23 7 A.3 A Ti" ‘ in“...

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Unformatted text preview: QM}; ' (f j t?» <—-. :23: 7; - , A .3 A Ti" ‘ in“ Classical mechanics L) 7-K ’5 ”-13- \ (.1003) I l. {20 points) A projectile of mass m is fired with velocity v and inclined angle 8 from a point 0 on the surface ofthe Earth. The projectile flies in x-y plane where x axis is the horizontal direction. Neglect all air resistance effects. The gravitation acceleration is g. At the point the projectile reaches one quarter of its horizontal range, find the magnitudes of the following physical quantities: '(a) Torque with respect to point 0 applied to the projectile due to gravity. (b) The linear momentum of the projectile. (c) The angular momentum of the projectile with respect to point 0. 2. {20 points) A particle with mass m moves in a plane under a central force of the —km r2 form F(r)= where k is a positive constant. (a) Prove that the particle can assrnne a stable circular orbit. (b) If the particle moves in a circular orbit with radius p and perturbed slightly in? the radial direction, what is its angular frequency of small oscillation ? 3. (20 points) A uniform rod of mass m and length l is pivoted such that it can rotate freely in a vertical plane. The supporting hinge moves on a horizontal rail with constant velocity v (as shown in the figure). (a) Use the angle 9 as the generalized coordinate, write down the Lagrangian of the rod. (b) Find the equation of motion. (c) Find the angular frequency for small oscillation. 4. (20 points) A particle of mass m is constrained to move on the inner surface of a hemispherical surface with radius R under gravity (as shown in the figure). Assume that all frictions can be neglected. (a) Use angles 9, (p in the spherical coordinates as generalized coordinates, write down the Lagrangian of the particle. (‘0) Find the Hamilton function in terms of pa, pp, (p, 6, where p9, ptp are the generalized momenta. a , . ‘ m ' w (c) Find the canonical equations of motion. ...
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