In this illustration, we are analyze a floating
sphere with a cavity. we are given that a
cemented sphere of radius r has a cavity of
radius small r, which is packed with sawdust,
and the relative density of cement and sawdust
are 2 point 4 and zero point 3 respectively,
we are required to find the ratio of mass
of cement to the mass of sawdust for which
the sphere will float in water in completely
submerged state. so, here first we can write,
the mass of cement, in the, sphere will be,
4 by 3 pie, r cube, minus small r cube because.
we have subtracted the cavity volume from
the total volume, multiplied by 2 point 4
ro of water that is a density of cement. similarly
we calculate the mass of, sawdust, in this
sphere this will be 4 by 3 pie small r cube
multiplied by zero point 3 times, ro of water,
and we know for floating. we can write, the,
weight of water displaced by the whole sphere
as we are given that it is in completely submerged
state. that would be balancing the total weight
of, the sphere. so here we can write the sphere,
weight m g will be balancing v, ro water g.
so in this situation mass we can substitute
as 4 by 3 pie, r cube minus small r cube multiplied
by, 2 point 4 ro water, here g we can cancel
out here only. plus. this is 4 by 3 pie small
r cube multiplied by zero point 3 ro water.
is equal to total volume is 4 by 3 pie r cube
multiplied by, ro water. if we simplify this
relation here 4 by 3 pie, and ro w gets cancelled
out, on simplifying this we are getting 1
point, 4 r cube is equal to 2 point 1 small
r cube. which gives us the value of r cube
by small r cube is equal to 3 by 2. and, in
this situation if we calculate, the ratio
of masses here we can continue. the ratio
of mass of cement to mass of sphere, we can
divide these 2 values. so that'll be 4 by
3 pie, r cube minus small r cube multiplied
by 2 point 4 ro water. divided by this is
4 by 3 pie, small r cube multiplied by zero
point 3 ro water, here 4 by 3 pie ro water
all gets cancelled out, so this will give
us m c by m s mass of cement to that of sawdust.
is equal to this r cube by r cube here, we
are having is 3 by 2. so this is 3 by 2 minus
1. multiplied by, this 2 point 4 by point
3 is 8. which comes out to be, 4 that is a
result of this problem.
