Question

A sphere completely submerged in water is tethered to the bottom with a string. The tension in the string is one-third the weight of the sphere. What is the density of the sphere?

Short answer

The density of the sphere if tension within the string is one-third the load of the sphere, ${\rho }_b=750\mathrm{kg}/\mathrm{m}3$

Step-by-step solution

Newton's first law.

The law of inertia, often referred to as Newton's first law, states that if a body is at rest or occupation a line at a relentless speed, it will remain at rest or still move in an exceedingly straight path at a continuing speed until it's acted with by a force.

Derive the Force.

Using the information provided within the exercise and Newton's first law, with the positive direction upwards, we will write:

$$\begin{gathered} F_B-T-F_g=0 \\ \Rightarrow F_B=F_G+T \\ =\frac{4}{3}mg \end{gathered}$$

Find the density.

Knowing the buoyant force formula and expressing the ball's mass as$m={\rho }_{b}V$, we may write:

$$\begin{gathered} {\rho }_{w}gV=\frac{4}{3}{\rho }_{b}Vg \\ \Rightarrow {\rho }_b=\frac{3}{4}{\rho }_w \end{gathered}$$

Because we all know the density of water, we will calculate the numerical solution as,

$$\begin{gathered} {\rho }_b=\frac{3}{4}\times 1000 \\ =750\mathrm{kg}/\mathrm{m}3 \end{gathered}$$