Question #11327

1 Answer
Mar 26, 2017

Let the mass of the boat floating in the tank be #m_b#.

and it carries total #N# steel balls each of mass # m#.

Hence total mass of boat + ball = #m_b+Nm#.

So by the condition of floatation we can say that initially the mass of displaced water will be #m_b+Nm#

So initially the volume of water displaced will be

#V_w=(m_b+Nm)/d_w...[1]#, where #d_w->"density of water"#

When #n# balls are thrown into the water then it will carry #(N-n)# balls .

At this stage the volume of water displaced by the floating boat carrying #(N-n)# balls will be #(m_b+(N-n)m)/d_w#

The volume water displaced by #n# balls thrown in to the tank will be #(nm)/d_s#, where #d_s# represents the density of steel.

Hence the total volume of water displaced in the 2nd case will be
#V'_w=(m_b+(N-n)m)/d_w+(nm)/d_s.....[2]#

Now subtracting [2] from [1] we get

#V_w-V'_w=(m_b+Nm)/d_w-(m_b+(N-n)m)/d_w-(nm)/d_s#

#V_w-V'_w=(nm)/d_w-(nm)/d_s=nm(1/d_w-1/d_s)....[3]#

Now density of steel #d_s# is much greater than that of water
#d_w#

So

#d_s>d_w#

#=>1/d_s<1/d_w#

#=>1/d_w-1/d_s>0.....[4]#

Hence from relation [3] an [4] we get

#V_w-V'_w=nm(1/d_w-1/d_s)>0.....[5]#

This Eq[5] reveals that the difference (decrease) in volume of displaced water continuously increases with the increase in number of steel balls #(n)# thrown into the tank.

This means that the level of water in the tank will gradually fall with increase number of balls #n# thrown into the tank.