Question

A cube is floating in a liquid as shown in figure. The incorrect statement is that if

$(1)$ Density of liquid decreases then $x$ will increase.
$(2)$ Size of cube increased then $x$ will increase
$(3)$ Cube is replaced by another cube of same mass and size then $x$ will remain same
$(4)$ Some material of the cube is cut and taken out then $x$ will remain the same.

Answer 1

Here we have to consider different forces acting on the body. We will analyze different situations with the help of Archimedes’ principle. Then we will be able to answer the question, so let’s start answering the question by considering the amount of upthrust which is acting on the body in different situations.
Formula used: $W={{F}_{B}}$

Complete answer:

According to Archimedes principle when a body floats on a liquid i.e the apparent weight of the body is zero then its weight gets completely balanced by the buoyant force acting on it. Thus if $W$ is the weight of a body and ${{F}_{B}}$ is the buoyant force acting on a floating body then we have $W={{F}_{B}}$ . Now from the given options, option $(4)$ says if some material of the cube is cut and taken out then $x$ will remain the same.
But this will not happen because when we cut some material of the body its weight decreases. So a smaller amount of immersed volume of the body will now displace the amount of liquid having the weight of the body. So $x$ will decrease.

So the incorrect statement is (4).

Note:
When a body floats its apparent weight becomes zero. So when we increase the size of a body its mass increases and a greater amount of immersed volume is required to keep its floating and opposite is the case when we decrease the size. As long as material of the body is not changing i.e. the density of the body remains the same,we can solve this question by this method.