out to be equal to v. At what frequency of oscillation of the frame
K' will the beat frequency of the point A become equal to 2v?
4.14. A point moves in the plane xy according to the law x
= a sin cot, y = b cos cot, where a, b, and co are positive constants.
Find:
(a) the trajectory equation y (x) of the point and the direction of
its motion along this trajectory;
(b) the acceleration w of the point as a function of its radius vector
r relative to the origin of coordinates.
4.15. Find the trajectory equation y (x) of a point if it moves ac-
cording to the following laws:
(a) x = a sin cot, y = a sin 2cot;
(b) x = a sin cot, y = a cos 2cot.
Plot these trajectories.
4.16. A particle of mass m is located in a unidimensional potential
field where the potential energy of the particle depends on the coor-
dinate x as U (x) = U 0 (1 — cos ax); U 0 and a are constants. Find
the period of small oscillations that the particle performs about the
equilibrium position.
4.17. Solve the foregoing problem if the potential energy has the
form U (x) = alx 2 — blx, where a and b are positive constants.
4.18. Find the period of small oscillations in a vertical plane per-
formed by a ball of mass m = 40 g fixed at the middle of a horizon-
tally stretched string 1 = 1.0 m in length. The tension of the string
is assumed to be constant and equal to F = 10 N.
4.19. Determine the period of small oscillations of a mathematical
pendulum, that is a ball suspended by a thread 1 = 20 cm in length,
if it is located in a liquid whose density is = 3.0 times less than
that of the ball. The resistance of the liquid is to be neglected.
4.20. A ball is suspended by a thread of length l at the point^0 on
the wall, forming a small angle a with the vertical (Fig. 4.1). ThenFig. 4.1, (^) Fig. 4.2.
the thread with the ball was deviated through a small angle 13(l > a)
and set free. Assuming the collision of the ball against the wall to
be perfectly elastic, find the oscillation period of such a pendulum.