A spherical solid ball of volume V is made of a material of density ρ1. It is falling through a

step 1 During terminal velocity, the force is acting on a body are MG in the downward direction and the

A spherical solid ball of volume V is made of a material of...

A spherical solid ball of volume $V$ is made of a material of density $\rho_{1}$. It is falling through a liquid of density $\rho_{2}\left(\rho_{2}<\rho_{1}\right.$ ) [Assuming that the liquid applies a viscous force on the ball that is proportional to the square of its speed $v$, i.e., $\left.F_{\text {viscous }}=-k v^{2}(k>0)\right]$. The terminal speed of the ball is [2008] (A) $\sqrt{\frac{V g\left(\rho_{1}-\rho_{2}\right)}{k}}$ (B) $\frac{V g \rho_{1}}{k}$ (C) $\sqrt{\frac{V g \rho_{1}}{k}}$ (D) $\frac{V g\left(\rho_{1}-\rho_{2}\right)}{k}$

A spherical solid ball of volume $V$ is made of a material of density $\rho_{1}$. It is falling through a liquid of density $\rho_{2}\left(\rho_{2}<\rho_{1}\right.$ ) [Assuming that the liquid applies a viscous force on the ball that is proportional to the square of its speed $v$, i.e., $\left.F_{\text {viscous }}=-k v^{2}(k>0)\right]$. The terminal speed of the ball is [2008]
(A) $\sqrt{\frac{V g\left(\rho_{1}-\rho_{2}\right)}{k}}$
(B) $\frac{V g \rho_{1}}{k}$
(C) $\sqrt{\frac{V g \rho_{1}}{k}}$
(D) $\frac{V g\left(\rho_{1}-\rho_{2}\right)}{k}$

Key Concepts

Quadratic Drag Force

Quadratic drag force is a type of resistive force that is proportional to the square of the object's speed. It becomes significant in scenarios involving high speeds or large objects moving through a fluid, and it plays a critical role in balancing forces to achieve terminal velocity.

Buoyant Force

Buoyant force 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 force counteracts part of the gravitational force and is essential in determining the net force on objects submerged in a fluid.

Terminal Velocity

Terminal velocity refers to the constant speed attained by an object when the net forces acting on it become balanced. In a falling object, this occurs when the upward resistive forces, such as drag and buoyancy, exactly counterbalance the downward gravitational force, resulting in zero net acceleration.

Force Equilibrium

Force equilibrium is the state where all forces acting on an object cancel each other out, resulting in no net force. In dynamics, this principle is used to determine conditions such as terminal velocity, where the sum of forces like gravity, buoyancy, and drag become equal and opposite.

Gravitational Force

Gravitational force is the force that attracts an object toward the center of a massive body, typically calculated as the product of the object's mass and the acceleration due to gravity. It provides the primary driving force for the motion of falling objects.

Please give Ace some feedback