Question

A body floats in water with one-third of its volume above the surface of water. It is placed in oil it floats with half of: Its volume above the surface of the oil. The specific gravity of the oil is. (A) \((5 / 3)\) (B) \((4 / 3)\) (C) \((3 / 2)\) (D) 1

Short answer

The specific gravity of the oil is \(\frac{4}{3}\) (Option B).

Step-by-step solution

Understanding the concept and defining the variables

Let the volume of the body be \(V\). When the body is in water, \(\frac{2}{3}\) of its volume is submerged which means the body has displaced \(\frac{2}{3}V\) of water. Similarly, when the body is in oil it has displaced \(\frac{1}{2}V\) of oil. The weight of the body is the same in both cases(because weight is a property of the body). According to Archimedes' principle, this weight is equal to the weight of the fluid displaced.

Formulating the equations

We have two situations thus we can define two equations. 1) For the instance when the body is in water: The weight of the body = Weight of water displaced. Weight of water displaced is given by: volume displaced * density of water * g (acceleration due to gravity) So, we have: \(Volume_{body} * Density_{body} * g = \frac{2}{3}V * Density_{water} * g\), where \(V\) is the volume of the body. The g on both sides will cancel out. We call this Equation 1. 2) For the instance when the body is in oil: Again, from Archimedes' principle, the weight of the body = weight of oil displaced So, we get: \(Volume_{body} * Density_{body} * g = \frac{1}{2}V * Density_{oil} * g\). The g on both sides will cancel out. We will call this Equation 2.

Solving the equations

From Equation 1, we can infer that, \(Density_{body} = \frac{2}{3} * Density_{water}\) Similarly, from Equation 2, we can infer that, \(Density_{body} = \frac{1}{2} * Density_{oil}\) Using these two equations, we can derive, \(\frac{2}{3} * Density_{water} = \frac{1}{2} * Density_{oil}\) Therefore, \(Density_{oil} = \frac{4}{3} * Density_{water}\) As specific gravity of oil is the ratio of density of oil to the density of water, we get: Specific gravity of oil = \(\frac{Density_{oil}}{Density_{water}} = \frac{4}{3}\) So, the answer to the problem is (B) \(\frac{4}{3}\).

Key concepts

Specific Gravity

Specific gravity is a dimensionless quantity that helps us understand how dense a material is compared to a reference substance, typically water for liquids and solids. It is a measure that indicates how heavy a substance is relative to water. Here’s how it is useful:

• Specific gravity can quickly tell us if an object will float or sink in water. If an object's specific gravity is less than 1, it floats. If it's more than 1, it sinks.
• It serves as a useful indicator of purity or concentration in certain solutions.

In the exercise problem, the specific gravity of the oil is calculated by comparing its density to that of water. We found that the specific gravity of oil is \(\frac{4}{3}\), which indicates the oil is denser than water.

Density

Density is defined as mass per unit volume of a substance, usually expressed in kilograms per cubic meter (kg/m³). It is a key concept in understanding how substances interact with each other in a fluid environment. Density affects whether objects will float or sink.

• High-density substances have more mass in a given volume than low-density substances.
• Knowing the density of a substance allows for the calculation of other properties, such as specific gravity.

In the problem, we derive the densities of the body and the oil in relation to water. This helps us determine how much volume of the body is submerged in each fluid, thus providing insights into the specific gravity of the oil.

Floating Bodies

Floating bodies connect to the principle of buoyancy. An object will float if the upward buoyant force exerted by the fluid is equal to the downward force of gravity on the object. This happens when the weight of the fluid displaced by the object is equal to the object's weight.

• The percentage of the body that is submerged is determined by its density relative to the fluid’s density.
• In the exercise, the body floats differently in water and oil due to the differing densities of these fluids.

By observing how much of the body remains above the surface in each case, we calculate the fluid's relative density—or specific gravity—against water.

Buoyancy

Buoyancy is the force that allows objects to float. According to Archimedes' principle, an object submerged in a fluid experiences an upward force equal to the weight of the fluid displaced by the object. Buoyancy makes it possible for ships to stay afloat and balloons to rise in the air.

• Buoyant force opposes the weight of an object, reducing the net force acting on it.
• The degree of buoyancy determines if an object sinks, floats, or rises, and it is influenced by both the density of the fluid and that of the object.

In our exercise, the floating nature of the body in oil and water is an application of buoyancy. The calculation and understanding of buoyancy help explain why the body behaves differently in oil and water.