`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). Then

Fig. 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.