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

step 1 We have problem number 33 in this question says a spherical solid ball of volume v is made of ma

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 $\rho_{2}\left(\rho_{2}<\rho_{1}\right) .$ Assume that the liquid applies a viscous force on the ball that is proportional to the square of its speed $v$, i.e., $F_{\text {viscous }}=-\mathrm{k} v^{2}(\mathrm{k}>0)$. The terminal speed of the ball is: (a) $\sqrt{\frac{\operatorname{Vg}\left(\rho_{1}-\rho_{2}\right)}{\mathrm{k}}}$ (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 $\rho_{2}\left(\rho_{2}<\rho_{1}\right) .$ Assume that the liquid applies a viscous force on the ball that is proportional to the square of its speed $v$, i.e., $F_{\text {viscous }}=-\mathrm{k} v^{2}(\mathrm{k}>0)$. The terminal speed of the ball is:
(a) $\sqrt{\frac{\operatorname{Vg}\left(\rho_{1}-\rho_{2}\right)}{\mathrm{k}}}$
(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

Gravitational Force

This is the force exerted by the Earth on any mass, acting downward. In dynamics, it is given by the product of the mass of the object and the acceleration due to gravity. It forms one of the primary forces that influence the motion of freely falling bodies.

Buoyancy Force

This force results from the displaced fluid when an object is immersed in it, acting upward. According to Archimedes’ principle, the buoyancy force equals the weight of the fluid displaced by the object. It plays a critical role in determining whether an object will sink, float, or remain suspended in a fluid.

Viscous Drag Force (Quadratic Drag)

This is a resistive force that acts opposite to the direction of motion of an object moving through a fluid, and when the drag is proportional to the square of the object's speed, it is known as quadratic drag. This form of drag becomes significant at higher speeds and is dependent on factors like the shape of the object and the properties of the fluid.

Terminal Velocity

This is the constant speed reached by an object when the net force acting on it becomes zero. At terminal velocity, the downward gravitational force, reduced by the upward buoyant force, is exactly balanced by the viscous drag force, resulting in no further acceleration. This equilibrium condition is central in analyzing the motion of bodies moving through resistive media.

Transcript

Please give Ace some feedback