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A piece of stone, with a volume of [tex][tex]$400 \, \text{cm}^3$[/tex][/tex] and a density of [tex][tex]$7.8 \times 10^3 \, \text{kg/m}^3$[/tex][/tex], is immersed totally in water with a density of [tex][tex]$1000 \, \text{kg/m}^3$[/tex][/tex]. Calculate the upthrust of the water.

[Ans: [tex][tex]$1.57 \times 10^4 \, \text{N}$[/tex][/tex]]



Answer :

GinnyAnswer
To calculate the upthrust (buoyant force) experienced by a piece of stone when it is fully submerged in water, we will go through the following steps:

1. Convert the volume of the stone from cm³ to m³:
- The given volume of the stone is [tex]\( 400 \text{ cm}^3 \)[/tex].
- Conversion factor: [tex]\( 1 \text{ cm}^3 = 1 \times 10^{-6} \text{ m}^3 \)[/tex].
- Volume in m³ is [tex]\( 400 \text{ cm}^3 \times 1 \times 10^{-6} \text{ m}^3/\text{cm}^3 \)[/tex].

[tex]\[ \text{Volume of the stone} = 0.0004 \text{ m}^3 \][/tex]

2. Identify the given density of water and the acceleration due to gravity:
- Density of water, [tex]\( \rho_{\text{water}} = 1000 \text{ kg/m}^3 \)[/tex].
- Acceleration due to gravity, [tex]\( g = 9.81 \text{ m/s}^2 \)[/tex].

3. Calculate the upthrust (buoyant force) using the formula:
[tex]\[ \text{Upthrust, } F = \rho \times V \times g \][/tex]
Where:
- [tex]\( \rho \)[/tex] is the density of the fluid (water in this case).
- [tex]\( V \)[/tex] is the volume of the object submerged in the fluid.
- [tex]\( g \)[/tex] is the acceleration due to gravity.

Substitute the values into the formula:

[tex]\[ F = 1000 \text{ kg/m}^3 \times 0.0004 \text{ m}^3 \times 9.81 \text{ m/s}^2 \][/tex]

4. Perform the multiplication to find the upthrust:

[tex]\[ F = 1000 \times 0.0004 \times 9.81 \][/tex]

5. Calculate the numerical result:

[tex]\[ F \approx 3.92 \text{ N} \][/tex]

Thus, the upthrust (buoyant force) acting on the stone when it is fully submerged in water is approximately [tex]\( 3.92 \text{ N} \)[/tex].

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