Question

You have a ball (volume V, average density ρB) and a glass beaker. The ball is attached to the bottom of the beaker with a light spring (coefficient k). When you fill the beaker with a liquid of density ρL, the ball floats fully submerged under the surface with the spring extended by x.

Derive the expression for x in terms of V, ρB, ρL, and k.

Express your answer in terms of some or all of the variables V, ρB, ρL, and k. Use g for the free-fall acceleration.

Answer 1

The forces acting on  ball will be Force exerted by spring , it's weight and the buoyant force.

From Archimedes principle, when an object is immersed in any fluid, then it exerts an upward force of buoyancy

equal to the weight of displaced fluid.

Buoyant force is given by = ( = density of fluid , = volume of displaced fluid = volume of object immersed)

( = acceleration due to gravity)

As the body is in equilibrium so balancing the forces acting on the body(see fig.)

  

Buoyant force = ρL* V * g  

Mass = m =density of object * volume = ρB * V

Putting these in equation

  

   (Ans.)