Advanced Mechanics [Phys 3400 Lecture Notes] by E. Poisson

By E. Poisson

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A) What is the speed of the particle as a function of time? (b) What is the position of the particle as a function of time? 7. A particle of mass m rests on top of a sphere of radius R. The particle is then displaced slightly so that it starts to move down the sphere. ) As it moves down the sphere, the particle makes an angle θ with the vertical direction. At some point the particle loses contact with the surface of the sphere, and it proceeds to fall freely. We are interested in the motion of the particle from the initial moment where it is at rest to the final moment where it leaves the sphere.

8 Additional problems 45 (b) Determine the integer n. 20. A two-body system moves under the influence of a central force given by f= b a + 3, 2 r r where a and b are constants. (a) Show that the shape of the orbit is described by r= p , 1 + e cos(kφ) where p, e, and k are constants. Express p and k in terms of a, b, h2 , and µ. ) (b) Plot the orbit in the x-y plane. 99, and let φ range from 0 to 16π. What is happening to the major axis of the ellipse? 8 Additional problems 1. An inclined plane makes an angle α with the horizontal.

Ds 1 − y ′2 , where y ′ = dy/ds. This gives us, We can replace the factor dx/ds by finally, π y A= 0 1 − y ′2 ds. 1) We wish to find the function y(s) that produces the largest possible value for A. Once this function is identified, x(s) can be obtained by integrating the equation x′ = 1 − y ′2 . The maximal curve is then fully determined. 2: A function with a maximum point at x = x¯. Because this is an extremum point, a displacement around x ¯ produces the smallest change in the function. 2 Extremum of a functional To proceed it is helpful to broaden the scope of the preceding discussion and to examine the general structure of the mathematical problem.

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