Volume and density of an object

Question

An ore sample weighs 17.5N in air. When the sample is suspended by a light cord and totally immersed in water, the tension in the cord is 11.2N. How do I find the total volume and the density of the object?

Answer 1

This is an application of Archimedes' Principle. Let's look at the forces of the object when it is submerged:

∑F_y = +Tension of the String + Buoyant Force from the Water - Force of Gravity on the object = 0 (since it is at rest). We are given that the tension is 11.2 N, and the force of gravity acting on the object 17.5 N (what is normally is in air), so the buoyant force from the water has to be the remaining 6.3 N.

The buoyant force is defined as the density of the fluid times g times the displaced volume. The density of the fluid is water in this case (1000 kg/m^3), and the displaced volume is the volume of the ore sample (since it is fully immersed in the water). Therefore:

The Displaced Volume = Buoyant Force / (density of water * g) = 6.3 N / (1000 kg/m³ · 9.8 m/s²)

Displaced Volume = 6.43 · 10^-4 m³.

Density is defined as the mass divided by the volume. To solve this, we need the mass, which we can get by using the force of gravity given in the problem:

Force of Gravity = mg, so

m = Force of gravity / g = 17.5 N / 9.8 m/s² = 1.79 kg

Density = Mass / Volume = 1.79 kg / (6.43 · 10^-4 m³) = 278 kg/m³