Buoyancy without density of fluid

[Vanessa23]

Homework Statement

A ball of mass m_b and volume V is lowered on a string into a fluid of density rho_f. Assume that the object would sink to the bottom if it were not supported by the string.
What is the tension T in the string when the ball is fully submerged but not touching the bottom?
It claims that rho_f is not needed in the part representing buoyancy

Homework Equations

I think it should be:
T=Fg - B
where T is the force of tension and B is the buoyancy force.
Fg=rho(object)*V*g
B=rho_f*V*g
therefore, T=mg - rho_f*V*g

how can you not use rho_f in calculating the buoyancy force?

The Attempt at a Solution

 

[mgb_phys]

You need the density, unles you are going to quote the answer in terms of density.
Imagine the fluid is air and a metal ball, if you change the fluid to water - then the tension in the string is going to change.

 

[thomate1]

I agree with your attempt at a solution. The equation T=mg-rho_f*V*g is the correct equation for calculating the tension in the string when the ball is fully submerged but not touching the bottom. It is important to note that the buoyancy force, B, is equal to the weight of the fluid displaced by the object, which is why the density of the fluid, rho_f, is necessary in the calculation. Without knowing the density of the fluid, it would be impossible to accurately calculate the buoyancy force and therefore the tension in the string. So, it is not possible to have buoyancy without considering the density of the fluid.