A solid aluminum ingot weighs 89 N in air. What is its volume? The ingot is suspended from a rope and totally immersed in water. What is the tension in the rope?

Weight = 89 N

mass, m = 89 N / g
= 9.08 kg
where g=9.8 m/sec²=acc.due to gravity
Density of aluminium, ρ
= 2700 kg/m³
Volume, V= m / ρ m³

When immersed in water, by the principle of floation, its weight is reduced by the weight of liquid it displaced, namely ρVg.
Thus, the tension in the rope, T is given by:
T = 89 N - 1000 (kg/m³) * V (m³) * g

To find the ingot's volume, we can use Archimedes' principle. According to Archimedes' principle, the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.

Step 1: Determine the weight of the fluid displaced by the ingot.
Let's assume the density of water is ρ, and the density of aluminum is ρ_aluminum. The weight of the fluid displaced by the ingot (W_fluid) is given by:

W_fluid = ρ x V x g, (Equation 1)

where V is the volume of the fluid displaced and g is the acceleration due to gravity.

Step 2: Determine the weight of the ingot in water.
When the ingot is in water, it experiences an upward buoyant force equal to the weight of the fluid displaced. This buoyant force will reduce the net downward force acting on the ingot, resulting in a decrease in its weight. The weight of the ingot in water is given by:

W_water = W_aluminum - W_fluid, (Equation 2)

where W_aluminum is the weight of the ingot in air.

Step 3: Determine the volume of the ingot.
The weight of the ingot in air (W_aluminum) is equal to its actual weight:

W_aluminum = mg, (Equation 3)

where m is the mass of the ingot and g is the acceleration due to gravity. The weight can be converted to mass using the equation:

W_aluminum = m x g = ρ_aluminum x V x g, (Equation 4)

where V is the volume of the aluminum ingot.

Step 4: Calculating the volume.
Combining equations 3 and 4, we can solve for the volume of the ingot (V):

ρ_aluminum x V x g = mg
V = (mg) / (ρ_aluminum x g), (Equation 5)

where m is the mass of the ingot, g is the acceleration due to gravity, and ρ_aluminum is the density of aluminum.

To find the tension in the rope when the ingot is suspended in water, we need to consider the forces acting on the ingot. When the ingot is in equilibrium, the tension in the rope (T_rope) will be equal to the weight of the ingot min​​​​​​​us the buoyant force acting on it.

T_rope = W_aluminum - W_fluid, (Equation 6)

where W_aluminum is the weight of the ingot in air and W_fluid is the weight of the fluid displaced.

Using equations 1, 2, and 6, we can now solve for the volume of the ingot and the tension in the rope.