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

130 Aptitude Test Problems in Physics

1.28. In order to describe the motion of the block,

we choose a reference frame fixed to the conveyer

belt. Then the velocity of the block at the initial

moment is vbl = vo v, and the block moves

with a constant acceleration a = —pg. For the in-

stant of time t when the velocity of the block

vanishes, we obtain the equation 0 = vo v —

Rgt. Hence the velocity of the conveyer belt is

v = ttgt — vo = 3 m/s.

1.29. Let us write the equation of motion for the

body over the inclined plane. Let the instanta-

neous coordinate (the displacement of the top of

the inclined plane) be x; then

ma = mg sin a — mg cos a. bx,

where m is the mass of the body, and a is its accel-

eration. The form of the obtained equation of mo-

tion resembles that of the equation of vibratory

motion for a body suspended on a spring of rigid-

ity k = mg cos a. b in the field of the "force of

gravity" mg sin a. This analogy with the vibratory

motion helps solve the problem.

Let us determine the position xo of the body

for which the sum of the forces acting on it is

zero. It will be the "equilibrium position" for the

vibratory motion of the body. Obviously,

mg sin a -- mg cos a. bro = 0. Hence we obtain

xo = (1/b) tan a. At this moment, the body will

have the velocity vo which can be obtained from

the law of conservation of the mechanical energy of

the body:

o^2 kx3

2= g sin a•xo — m

2

mgb cos a 2

2 7o,

vi= 2gx 0 sin a — gbxt cos a= g

sing a

b cos cc •

In its further motion, the body will be dis-

placed again by xo (the "amplitude" value of vibra-

tions, which can easily be obtained from the law

= mg sin a •xo