A

Step 1: Understand the problem. We have a cylinder filled with water. The radius of the cylinder is denoted as R, and the height of the water column is H.
Step 2: The thrust (or force) on the bottom of the cylinder due to the water can be calculated using the formula:

F_bottom = P_bottom * A_bottom

where:

• P_bottom = Pressure at the bottom = \rho g H
• A_bottom = Area of the bottom = \pi R^2

Therefore,

F_bottom = \rho g H \cdot \pi R^2

Step 3: The thrust on the walls of the cylinder is determined by integrating the forces acting on each infinitesimal ring of height dy, where the radius is R:

dF_wall = P_wall A_wall = \rho g y \cdot 2 \pi R dy

Here, y varies from 0 to H, thus:

F_wall = \int_0^H \rho g y \cdot 2 \pi R dy = \rho g 2\pi R \cdot \frac{H^2}{2} = \rho g \pi R H^2

Step 4: Set the thrust on the bottom equal to the thrust on the walls:

\rho g H \cdot \pi R^2 = \rho g \pi R H^2

Cancelling common terms (\rho, g, \pi) gives us:

H R^2 = H^2 R

This simplifies to:

H = R

Step 5: Therefore, the relationship holds that H = R is necessary for the thrusts to be equal.
Thus, we conclude that when the thrust on the walls is equal to that on the bottom, then H must indeed be equal to R.
Therefore, the statement is proven to be true.