Finding Tension in a Submerged Ball Using Archimedes' Principle

[Soaring Crane]

A ball of mass m_b and volume V is lowered on a string into a fluid of density p_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? Express your answer (T) in terms of the given quantities and g , the acceleration due to gravity.


Although the fact may be obscured by the presence of a liquid, the basic condition for equilibrium still holds: The net force on the ball must be zero.

Here are the steps that I went through:
Compute the mass m_f of the fluid displaced by the object when it is entirely submerged.
Express your answer in terms of p_f, v, m_b, and g.

m_f = p_f*V

f_buoyant = p_f* V * g

How do I get from here to finding the tension?

Thanks.

 

[Doc Al]

As the problem states: The net force on the ball must be zero. So what are the forces acting on the ball? (Hint: Three forces act on the ball.)

 

[Soaring Crane]

Tension of string, m_b*g, and F_buoyancy?

 

[Doc Al]

Right. Now write the equation that relates them and solve for Tension.

 

[Soaring Crane]

So these 3 forces must add up to 0?
I don't think this is right (both the signs and equation I'm unsure of), but would it be

F_T + m_b*g + p_f* V * g (F_buo) = 0, where p_f = m_f/V

Then isolate F_T to solve for the answer?

 

[Doc Al]

Yes, the forces add to 0. But don't forget that forces are vectors: Direction matters! (Which way do the forces act?) Be sure to use a consistent sign convention: For example, let forces acting upward be positive; downward, negative. Then rewrite your equation.

 

[Soaring Crane]

The answer for F_T does not depend on m_f. What am I supposed to do now?

What I did

F_t - (m_b*g) + f_buo = 0

F_T = (m_b*g) - F_buo , F_buo = g*m_f

F_T = g(m_b - m_f)

but this is wrong

 

[Soaring Crane]

Never mind; got it! Thanks.