(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.rnFig. 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 con-
stant, 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 v 0 directed along the unit vector j. Find the law
of motion x (t) , y (t) of the particle, and also the equation of its tra-
jectory.
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 iden-
tical threads 1 = 90 cm in length (Fig. 4.13) was turned through a