Question

A plastic body of density $ \rho $ is to be held in air of density $ {\rho _a} $ first and then in water of density $ {\rho _w} $ . For holding, we need to apply a force on it of:
A) It is easier to hold in water if $ \rho > {\rho _a} $ and $ \rho < \dfrac{{{\rho _w} + {\rho _a}}}{2} $
B)It is easier to hold in water if $ \rho > {\rho _w} $
C)It is easier to hold in air if $ \rho < {\rho _w} $ and $ \rho < \dfrac{{{\rho _w} + {\rho _a}}}{2} $
D) It is easier to hold in water if $ \rho < {\rho _w} $ and $ \rho > \dfrac{{{\rho _w} + {\rho _a}}}{2} $

Answer 1

Hint: In this solution, we will use the concepts of buoyant force acting on the body. To hold the object in a medium, it must sink in the medium and it must be submerged enough to stay stable in the medium.

Formula used
In this solution, we will use the following formula
-Buoyant force $ {F_B} = \rho Vg $ where $ \rho $ is the density of the plastic body, $ V $ is the volume of the object, and $ g $ is the gravitational acceleration.

Complete step by step answer:
When the plastic body is to be held, the weight of the body must be less than the buoyant force on the object otherwise the object will sink.
So, $ W < {F_B} $
The weight of the object $ W = mg = {\rho _{object}}{V_1}g $ where $ {V_1} $ is the object’s total volume has to be less than the buoyant force $ {F_B} = {\rho _{water}}{V_2}g $ where $ {V_2} $ is the volume of the object immersed in the object.
Now, for the object to be held stable, it must have sufficiently less density so that it doesn’t sink in the water which implies $ {\rho _{object}} < {\rho _{water}} $ (As the object will have submerged completely, $ {V_1} = {V_2} $ ).
Again, to make sure that the object stays submerged enough and it doesn’t tip over, the buoyant force must be enough that at least half the volume of the object will be submerged in water which implies that $ {\rho _{object}} > \dfrac{{{\rho _w} + {\rho _a}}}{2} $
Hence the correct choice is option (D).

Note:
To hold the object in the air, it must have a density less than the density of air which is not given in any of the options so we should only focus on holding the object in the water. The precise density of the object depends on the shape of the object and how much of it is submerged in the water.