Question

# A boat having a length of 3m and a breadth of 2m is floating on a lake. The boat sinks by 1cm when a man gets on it. What is the mass of the man?

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Solution

## Step1: Given dataThe boat has a length of 3 m and a breadth of 2 m and sinks by 1cm when the man gets on it.Step2: Formula usedWhenever an object has some of its volume inside the water, this object will experience an upward force called buoyant force which is given by the formula, ${F}_{b}={\rho }_{w}vg$[ where ${\rho }_{w}=$density of water, $g=$acceleration due to gravity, $v=$volume of the object that is inside the water. ]Step3: Calculating volume$l=3m,b=2m,h=1cm=0.01m$$v=l×b×h\left[v=volume,l=length,b=breadth,h=height\right]\phantom{\rule{0ex}{0ex}}v=3×2×0.01\phantom{\rule{0ex}{0ex}}v=0.06{m}^{3}$Step4: Calculating buoyant force experienced by the boatDensity of water ${\rho }_{w}-100kg/{m}^{3}$Now we'll substitute the value of volume and density in the buoyant force formula${F}_{b}={\rho }_{w}vg=1000×0.06×g\phantom{\rule{0ex}{0ex}}{F}_{b}=60g..........\left(i\right)$Step5: Calculating the mass of the manLet us assume the mass of the man is m kg.Since man is inside the boat, the boat will experience a downward force. This downward force is equal to,$F=mg.................\left(ii\right)$Since the boat is in equilibrium, we can equate the upward force in equation $\left(i\right)$ and the downward force in equation $\left(ii\right)$. So, we get, $60g=mg\phantom{\rule{0ex}{0ex}}m=60kg$Hence, the mass of the man is 60kg.

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