Aptitude Test Problems in Physics Science for Everyone by S Krotov ( PDFDrive.com )

258 Aptitude Test Problems in Physics

3.19. From the equality of the electric and elastic
forces acting on the free ball,

q2 = kl

4ne 0 •41 2

we obtain the following expression for the length 1

of the unstretched spring:

is —

16neok '

where k is the rigidity of the spring, and q are the

charges of the balls.

Let us suppose that the free ball is deflected

from the equilibrium position by a distance x

which is small in comparison with 1. The potential

energy of the system depends on x as follows:

1

—

q2 (^5)
W k (1- x) 2 + (^) kl 2 + hrs.
2 4ne, (21—x) T
Here we have taken into account the relation be-
tween k, and 1 obtained above and retain the
terms of the order of [x/(21)] 2 in the expression
1 1
21—x^ 2/ (1 — xI21)
1 x

• --N- 1 1+( -2/)+( 2i )^2 + ]*

Thus, it is as if the stretched spring has double the

rigidity, and the ratio of the frequencies of har-

monic vibrations of the system is

vo jig

vi

3.20. Let us consider the two charged balls to be

a single mechanical system. The Coulomb inter-

action between the balls is internal, and hence it

does not affect the motion of the centre of mass.

The only external force acting on the system is

the force of gravity. It is only this force that will

q2