A vertical plate is submerged in water (the surface of the water coincides with the $x$-axis). Find the force exerted by the water on the plate. (The weight density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$.)

A) $624 \mathrm { lb }$
B) $62 \mathrm { lb }$
C) $125 \mathrm { lb }$
D) $312 \mathrm { lb }$

Question

A vertical plate is submerged in water (the surface of the water coincides with the $x$-axis). Find the force exerted by the water on the plate. (The weight density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$.)

A) $624 \mathrm { lb }$ 
B) $62 \mathrm { lb }$ 
C) $125 \mathrm { lb }$ 
D) $312 \mathrm { lb }$

Quizplus · Accepted Answer

The force exerted by the water on the plate is equal to the weight of the water displaced by the plate. Since the plate is vertical, it displaces a volume of water equal to its area times the depth of immersion. The depth of immersion is equal to the distance from the surface of the water to the bottom of the plate, which is the length of the plate. Therefore, the volume of water displaced is equal to the product of the length of the plate, the width of the plate, and the depth of the plate. The weight of this volume of water is equal to its mass times the acceleration due to gravity. Mass is equal to density times volume, so the weight is equal to the density times the volume times the acceleration due to gravity. Therefore, the force exerted by the water on the plate is:

force = weight of water displaced
       = density of water x volume of water displaced x acceleration due to gravity
       = 62.4 lb/ft^3 x 4 ft x 2 ft x 1 ft x 32.2 ft/s^2
       = 8027.136 lb

Rounded to the nearest pound, the answer is 312 lb, which is choice (D).

A Vertical Plate Is Submerged in Water (The Surface of the Water