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?

Answer 1

Step1: Given data

The 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 used

Whenever 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

$l=3m,b=2m,h=1cm=0.01m$

$$\begin{gathered} v=l\times b\times h[v=volume,l=length,b=breadth,h=height] \\ v=3\times 2\times 0.01 \\ v=0.06m^3 \end{gathered}$$

Step4: Calculating buoyant force experienced by the boat

Density of water ${\rho }_w-100kg/m^3$

Now we'll substitute the value of volume and density in the buoyant force formula

$$\begin{gathered} F_b={\rho }_{w}vg=1000\times 0.06\times g \\ {F}_b=60g..........\left(i\right) \end{gathered}$$

Step5: Calculating the mass of the man

Let 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,

$$\begin{gathered} 60g=mg \\ m=60kg \end{gathered}$$

Hence, the mass of the man is 60kg.