Question

A 25 kg circular disk has a diameter of 2.5 feet and a thickness of 2.5 cm. Find the density of the disk in kg/m3. Next, find the weight of the object. Then calculate the buoyant force on the disk if it is submerged under water. Finally, will the object sink or float?

Answer 1

Answer:

Assume that $\rm g= 9.81\; N\cdot kg^{-1}$; $\rho(\text{Water}) = \rm 1000\;kg\cdot m^{-3}$.

Density of the disk: approximately $\rm 2.19\times 10^{3}\; kg\cdot m^{-3}$.

Weight of the disk: approximately $\rm 245\;N$.

Buoyant force on the disk if it is submerged under water: approximately $\rm 112\; N$.

The disk will sink when placed in water.

Explanation:

Convert the dimensions of this disk to SI units:

• Diameter: $d = \rm 25\; inches = (25\times 0.3048)\; m = 0.762\;m$.
• Thickness $h = \rm 2.5\; cm = (2.5\times 0.01)\; m = 0.025\;m$.

The radius of a circle is 1/2 its diameter:

$\displaystyle r = \rm \frac{1}{2}\times 0.762\;m = 0.381\; m$.

Volume of this disk:

$V(\text{disk}) = \pi\cdot r^{2}\cdot h = \pi\times 0.381^{2}\times 0.025 \approx 0.0114009\; m^{3}$.

Density of this disk:

$\displaystyle \rho(\text{disk}) = \frac{m}{V} = \rm \frac{25\; kg}{0.0114009\; m^{3}} = 2.19\times 10^{3}\;kg\cdot m^{-3}$.

$\rho(\text{disk}) >\rho(\text{water})$ indicates that the disk will sink when placed in water.

Weight of the object:

$W(\text{disk}) = m\cdot g = \rm 25\times 9.81 = 245.25\; N$.

The buoyant force on an object in water is equal to the weight of water that this object displaces. When this disk is submerged under water, it will displace approximately $\rm 0.0114009\; m^{3}$ of water. The buoyant force on the disk will be:

$\begin{aligned}F(\text{buoyant force}) &= W(\text{Water Displaced}) \\& = \rho\cdot V(\text{Water Displaced})\cdot g\\ & = \rm 1\times 10^{3}\; kg\cdot m^{-3}\times 0.0114009\; m^{3}\times 9.81\; N\cdot kg^{-1}\\ &\approx \rm 112\; N\end{aligned}$.

The size of this disk's weight is greater than the size of the buoyant force on it when submerged under water. As a result, the disk will sink when placed in water.