Question

Water is filled in a rectangular tank of size 3 m x 2 m x 1 m. 1 m is the height. (a) Find the total force exerted by the water on the bottom surface of the tank. (b) Consider a vertical side of area 2 m x 1 m. Take a horizontal strip of width $\delta$x metre in this side, situated at a depth of x metre from the surface of water. Find the force by the water on this strip. (c) Find the torque of the force calculated in part (b) about the bottom edge of this side. (d) Find the total force by the water on this side. (e) Find the total torque by the water on the side about the bottom edge. Neglect the atmospheric pressure and take g = 10 m/s².

Answer 1

(a) 60000 N (b) $20000 x \, \delta x$ N (c) $20000 x(1 - x) \, \delta x$ N-m (d) 10000 N (e) $\frac{10000}{3}$ N-m

Answer 2

The density of water is $\rho = 1000 \, \text{kg/m}^3$ and $g = 10 \, \text{m/s}^2$.

(a) Force on the bottom: The pressure at the bottom is $P_{bottom} = \rho g H = 1000 \times 10 \times 1 = 10000 \, \text{Pa}$. The area of the bottom is $A_{bottom} = 3 \, \text{m} \times 2 \, \text{m} = 6 \, \text{m}^2$. The total force is $F_{bottom} = P_{bottom} \times A_{bottom} = 10000 \times 6 = 60000 \, \text{N}$.

(b) Force on a horizontal strip: The pressure at depth $x$ is $P_{strip} = \rho g x = 1000 \times 10 \times x = 10000x \, \text{Pa}$. The area of the strip is $dA = 2 \, \delta x \, \text{m}^2$. The force on the strip is $dF = P_{strip} \times dA = 10000x \times 2 \, \delta x = 20000x \, \delta x \, \text{N}$.

(c) Torque of the force on the strip about the bottom edge: The distance of the strip from the bottom edge is $(1 - x)$ m. The torque is $d\tau = dF \times (1 - x) = (20000x \, \delta x)(1 - x) = 20000x(1 - x) \, \delta x \, \text{N-m}$.

(d) Total force on the side (2 m x 1 m): Integrate $dF$ from $x=0$ to $x=1$: $F_{total\_side} = \int_{0}^{1} 20000x \, dx = 20000 \left[ \frac{x^2}{2} \right]_{0}^{1} = 20000 \times \frac{1}{2} = 10000 \, \text{N}$.

(e) Total torque on the side about the bottom edge: Integrate $d\tau$ from $x=0$ to $x=1$: $\tau_{total\_side} = \int_{0}^{1} 20000x(1 - x) \, dx = 20000 \int_{0}^{1} (x - x^2) \, dx = 20000 \left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_{0}^{1} = 20000 \left( \frac{1}{2} - \frac{1}{3} \right) = 20000 \times \frac{1}{6} = \frac{10000}{3} \, \text{N-m}$.