a hollow metal sphere of mass m has an instantaneous upward acceleration a when released from rest fully submerged in a liquid of density d1.if the density of metal is d2 find the volume of cavity in the sphere.

To solve the problem of finding the volume of the cavity in a hollow metal sphere submerged in a liquid, we need to apply some principles from physics, particularly those related to buoyancy and forces acting on the sphere. Let's break it down step by step.

Understanding the Forces at Play

When the hollow metal sphere is released from rest in the liquid, it experiences two main forces:

• Weight of the Sphere (W): This is the force due to gravity acting on the mass of the sphere, calculated as W = mg, where m is the mass of the sphere.
• Buoyant Force (B): This is the upward force exerted by the liquid, which can be calculated using Archimedes' principle. The buoyant force is equal to the weight of the liquid displaced by the submerged part of the sphere, given by B = d1 * V_submerged * g, where V_submerged is the volume of the sphere that is submerged.

Setting Up the Equation

When the sphere is released, it accelerates upward with an acceleration 'a'. According to Newton's second law, the net force acting on the sphere can be expressed as:

Net Force = Buoyant Force - Weight = ma

Substituting the expressions for the buoyant force and weight, we have:

d1 * V_submerged * g - mg = ma

Relating Volume and Density

Next, we need to express the volume of the sphere and the volume of the cavity. The total volume of the hollow sphere (V_total) can be expressed in terms of its outer radius (R) and inner radius (r) as:

V_total = (4/3)πR³ - (4/3)πr³ = (4/3)π(R³ - r³)

Since the sphere is hollow, the volume of the cavity (V_cavity) is simply:

V_cavity = (4/3)πr³

Substituting and Rearranging

Now, we can substitute the expression for weight (W = mg) into our net force equation:

d1 * V_submerged * g - m * g = ma

Factoring out g gives us:

g(d1 * V_submerged - m) = ma

Dividing both sides by g, we get:

d1 * V_submerged - m = ma/g

Rearranging this gives us:

d1 * V_submerged = m + ma/g

Finding the Volume of the Cavity

Now, we know that the volume of the sphere submerged in the liquid is equal to the total volume of the sphere minus the volume of the cavity:

V_submerged = V_total - V_cavity

Substituting this into our earlier equation, we can express the volume of the cavity in terms of the known quantities:

d1 * (V_total - V_cavity) = m + ma/g

Now, substituting V_total = (4/3)π(R³ - r³) and V_cavity = (4/3)πr³ into the equation allows us to isolate V_cavity and solve for it. This will give us the volume of the cavity in the sphere in terms of the mass of the sphere, the densities, and the acceleration.

Final Thoughts

By carefully analyzing the forces acting on the sphere and applying the principles of buoyancy and volume relationships, we can derive the volume of the cavity. This approach not only helps in solving this specific problem but also reinforces the fundamental concepts of fluid mechanics and dynamics.