A small sphere (D=6 mm) is observed to fall through castor...

A small sphere $(D=6 \mathrm{mm})$ is observed to fall through castor oil at a terminal speed of $60 \mathrm{mm} / \mathrm{s}$. The temperature is $20^{\circ} \mathrm{C} .$ Compute the drag coefficient for the sphere. Determine the density of the sphere. If dropped in water, would the sphere fall slower or faster? Why?

Key Concepts

Density Determination of Solid Objects

The density of an object can be determined by analyzing the balance of forces acting on it while it moves through a fluid. By equating gravitational forces modified by buoyancy with the drag force at terminal velocity, one can solve for the object's density. This approach helps predict how an object will behave in different fluids.

Impact of Fluid Properties on Motion

The properties of the fluid, particularly its density and viscosity, significantly affect the drag and buoyant forces acting on a moving object. A change in fluid—such as from a more viscous fluid like castor oil to a less viscous one like water—alters these forces, influencing the object's terminal velocity. These differences determine whether an object will fall faster or slower in a different medium.

Buoyancy Force (Archimedes' Principle)

Buoyancy is the upward force exerted by a fluid on an immersed object, which is equal to the weight of the fluid displaced by the object. This principle is critical in computing the net force on an object in a fluid and hence its terminal velocity and apparent density.

Terminal Velocity

Terminal velocity is the constant speed reached by a falling object when the net force acting on it becomes zero. At this point, the downward gravitational force is exactly balanced by the sum of the upward buoyant force and drag force. This concept is fundamental in analyzing the motion of objects in fluids.

Drag Coefficient

The drag coefficient is a dimensionless number that characterizes the resistance an object experiences as it moves through a fluid. It relates the drag force acting on the object to the fluid density, the square of the object's velocity, and its reference area. Its value is determined by the shape of the object and the nature of the fluid flow (laminar or turbulent).

Force Equilibrium in Fluids

When an object attains terminal velocity while falling through a fluid, the forces on the object are in equilibrium. This balance involves the gravitational force, the buoyant force (due to the displaced fluid), and the drag force. Understanding this equilibrium is key to solving problems related to the motion of objects in fluids.

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