Question

A ferry boat has internal volume 1 m³ and weight 50 kg. (a) Neglecting the thickness of the wood, find the fraction of the volume of the boat immersed in water. (b) If a leak develops in the bottom and water starts coming in, what fraction of the boat's volume will be filled with water before water starts coming in from the sides ?

Answer 1

(a) 0.05 (b) 0.95

Answer 2

(a) For a floating object, the buoyant force equals its weight. The buoyant force is given by $F_B = \rho_{water} V_{immersed} g$, and the weight of the boat is $W_{boat} = m_{boat} g$. Setting these equal: $m_{boat}g = \rho_{water} V_{immersed} g$. Given $m_{boat} = 50$ kg, $\rho_{water} = 1000$ kg/m³, and $V_{boat} = 1$ m³ (assuming external volume is approximately internal volume as thickness is neglected), we have $50g = 1000 \times V_{immersed} \times g$. Solving for $V_{immersed}$, we get $V_{immersed} = \frac{50}{1000} = 0.05$ m³. The fraction immersed is $\frac{V_{immersed}}{V_{boat}} = \frac{0.05}{1} = 0.05$.

(b) The boat is on the verge of sinking when the total weight (boat + water inside) equals the maximum buoyant force, which occurs when the boat is fully submerged. The maximum buoyant force is $F_{B,max} = \rho_{water} V_{boat} g$. The weight of water inside the boat is $W_{water\_in} = \rho_{water} V_{water\_in} g$. The total weight is $W_{total} = W_{boat} + W_{water\_in} = m_{boat}g + \rho_{water} V_{water\_in} g$. At the point of sinking: $m_{boat}g + \rho_{water} V_{water\_in} g = \rho_{water} V_{boat} g$. Substituting values: $50g + 1000 V_{water\_in} g = 1000 \times 1 \times g$. Cancelling $g$: $50 + 1000 V_{water\_in} = 1000$. Solving for $V_{water\_in}$: $1000 V_{water\_in} = 950$, so $V_{water\_in} = 0.95$ m³. The fraction of the boat's internal volume filled with water is $\frac{V_{water\_in}}{V_{internal}} = \frac{0.95}{1} = 0.95$.