(a) w = const; (b) w = at, where a is a constant. 4.39. A body of mass m = 0.50 kg is suspended from a rubber cord with elasticity coefficient k = 50 Is17m. Find the maximum distance over which the body can be pulled down for the body's oscillations to remain harmonic. What is the energy of oscillation in this case? 4.40. A body of mass m fell from a height h onto the pan of a spring balance (Fig. 4.10). The masses of the pan and the spring are negligible, the stiffness of the latter is x. Having stuck to the pan, the body starts performing harmonic oscillations in the vertical direction. Find the amplitude and the energy of these oscillations.
rn
Fig. 4.10.
Fig. 4.11.
4.41. Solve the foregoing problem for the case of the pan having a mass M. Find the oscillation amplitude in this case. 4.42. A particle of mass m moves in the plane xy due to the force varying with velocity as F = a (yi — xj), where a is a positive constant, i and j are the unit vectors of the x and y axes. At the initial moment t = 0 the particle was located at the point x = y = 0 and possessed a velocity v0 directed along the unit vector j. Find the law of motion x (t) , y (t) of the particle, and also the equation of its trajectory. 4.43. A pendulum is constructed as a light thin-walled sphere of radius R filled up with water and suspended at the point 0 from a light rigid rod (Fig. 4.1.1). The distance between the point 0 and the centre of the sphere is equal to 1. How many times will the small oscillations of such a pendulum change after the water freezes? The viscosity of water and the change of its volume on freezing are to be neglected. 4.44. Find the frequency of small oscillations of a thin uniform vertical rod of mass m and length 1 hinged at the point 0 (Fig. 4.12). The combined stiffness of the springs is equal to x. The mass of the springs is negligible. 4.45. A uniform rod of mass m = 1.5 kg suspended by two identical threads 1 = 90 cm in length (Fig. 4.13) was turned through a