/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 93 An open tank is filled to the to... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 93

An open tank is filled to the top with water. A steel cylindrical container, wall thickness \(\delta=1 \mathrm{mm},\) outside diameter \(D=100 \mathrm{mm},\) and height \(H=1 \mathrm{m},\) with an open top, is gently placed in the water. What is the volume of water that overflows from the tank? How many 1 kg weights must be placed in the container to make it sink? Neglect surface tension effects.

Short Answer

Expert verified
The volume of water that overflows from the tank can be calculated using the formula \(V_{overflow} = \pi R^2 H - \pi (R-\delta)^2 H\). The number of 1 kg weights needed to sink the container can be found by first calculating the buoyant force and then subtracting the weight of the cylinder and rounding up to the nearest whole number.

Step by step solution

01

Calculate the volume of the steel cylinder

First determine the external radius \(R\) of the cylinder using the equation \(R = D/2\), where \(D\) is the given diameter of the cylinder. Once you have the radius, calculate the volume \(V_c\) of the steel cylinder using the formula \(V_c = \pi R^2 H\). Subtract from this the volume of the steel material to get the volume of the space inside the container, or \(V_{container} = V_c - \pi (R-\delta)^2 H\).
02

Determine the volume of the overflowed water

The volume of the water that overflows when the cylinder is placed in the water is the same as the volume of the cylinder. Therefore, calculate this using the formula obtained from the previous step, \(V_{overflow} = \pi R^2 H - \pi (R-\delta)^2 H\)
03

Calculate the buoyant force acting on the cylinder

The buoyant force can be determined using Archimedes' principle, which states that the buoyant force is equal to the weight of the volume of the displaced water. The displaced water is the overflowed water. The buoyant force \(F_B\) can be calculated as \(F_B = \rho_{water} g V_{overflow}\), where \(\rho_{water}\) is the density of water, \(g\) the acceleration due to gravity, \(V_{overflow}\) the volume of the overflowed water.
04

Determine the number of 1 kg weights

To sink the cylinder, the weight inside the cylinder must be greater than the buoyant force. Thus, the number of 1 kg weights can be determined using the formula \(n = ceil(F_B - m_{cylinder})\), where \(m_{cylinder}\) is the mass of the cylinder, calculated by the volume of the cylinder times the density of steel, \(F_B\) is the buoyant force calculated from Step 3, and using the function \(ceil()\) helps us round up to the nearest whole number as weights cannot be fractions.

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