In this illustration we'll be analyzing a
ball balancing over a jet. we are given that
a jet of water is having a speed v from a
nozzle of area ay, and a ball of mass m is
balanced in air by impact of water on its
underside we are required to find the height
of the ball above the jet nozzle. here, we
can first draw the figure to understand it
better. we are given that their is, water
jet, and this jet nozzle is of, area, ay,
from which say, fluid comes out with the velocity
v. and when as the water jet moves up, we
know its velocity decreases and area of cross
section increases, and say at a height h above
the nozzle, it is balancing a ball. then in
this situation this water jet is continuously
striking the ball say with a velocity v 1.
if it is coming out with a velocity v, v 1
can be directly written as it is under free
fall. so this is root of v square minus 2
gee h. now in this situation. we can say,
if. area of water jet, bellow the ball. is
ay 1. we use, here, v 1 ay 1 should be equal
to. v ay, if we consider v to be, the velocity
with which, water is coming out from the nozzle
which is already given to us. so from here
we are getting the value of. area. ay 1, which
is given as, v, ay divided by, the value of
v 1 which we have calculated as root of v
square minus, 2 gee h. and now, if we talk
about, the balancing, the weight of ball,
then we can say, the force. exerted. by collision,
of, water jet. on ball, is balancing. the
weight of, ball. and, this force we have already
studied as this force is given as, we already
studied the force due to collision of a water
jet we calculate as, ro ay 1, v 1 square if
ay 1 is the area of cross section, and v 1
is the speed with which water jet is colliding.
so this will be balancing the weight of, this
ball which is m g. so if we substitute the
values here. here we can write this as ro.
area we can write as, v, ay divided by root
of, v square minus 2 gee h. and v 1 square
is, v square minus 2 gee h. is equal to, m
g. so in this situation if we simplify the
relation, here we can see 1 root of v square
minus 2 gee h gets cancelled out. and we get
v square minus 2 gee h is equal to. here,
you can take this term on other side ro v
ay, and we square it. then this will give
us, a value, m g. divided by, ro v, ay, whole
square. so in this situation we can simply
calculate the value of h. which we are required
to find the height of the ball above the jet
nozzle. so in this situation, this gives us
the value of h is equal to. v square. minus.
m g by ro v, ay whole square. whole, divided
by, 2 gee, that is the result of this problem.
