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Chapter 9: Problem 9

A body is falling in viscous medium. The terminal velocity depends on various factors. Column-l Column-II (a) Terminal velocity depends on acecleraion due (p) Dircctly proportional to square to gravily (b) Terminal velocity depends on radius of sphere (q) Directly proportional falling in viscous medium (c) Greater the density and viscosity of fluid (r) Lesser is terminal velocity (d) Greater the density of solid (s) Smaller is terminal velocity

Short Answer

Expert verified
(a) - (p), (b) - (q), (c) - (r), (d) - (s)

Step by step solution

01

Introduction to Terminal Velocity

Terminal velocity (v_t) is the constant speed that a freely falling object eventually reaches when the resistance of the medium prevents further acceleration.
02

Analysis of Dependency on Gravity

Terminal velocity is directly proportional to the square of acceleration due to gravity since a greater gravitational force will increase the force driving the object through the medium.
03

Dependency on Radius of Sphere

A larger radius increases the terminal velocity; hence it is directly proportional to the radius (q).
04

Effect of Density and Viscosity of Fluid

Denser and more viscous fluids resist motion more, thus decreasing terminal velocity. Therefore, if density and viscosity are greater, the terminal velocity is less (r).
05

Effect of Density of Solid

If the density of the solid is greater, it drives through the fluid more easily, resulting in a larger terminal velocity (s).
06

Matching The Columns

Based on these explanations: (a) Terminal velocity 鈭� g (matches with p), (b) Terminal velocity 鈭� r (matches with q), (c) Increased fluid density and viscosity reduce terminal velocity (matches with r), (d) Greater density of solid results in larger terminal velocity (matches with s).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Viscous Medium
When a body falls through a viscous medium, the medium provides resistance to the motion. Imagine something like oil or honey, which are classic examples of viscous fluids. As the object moves through this medium:
  • The fluid applies a force opposite to the direction of motion.
  • This resistance increases as the speed of the object increases.
  • Eventually, the force from the fluid balances the gravitational force acting on the object.
This balance causes the object to stop accelerating and move at a constant speed. That's when it reaches what we call terminal velocity. In a viscous medium, the body can never fall infinitely fast because the medium always resists its motion with increasing force.
Gravitational Force
Gravitational force plays a pivotal role in determining terminal velocity. It is the force that pulls the object downward towards the Earth. This force depends on two main factors:
  • The mass of the falling object.
  • The acceleration due to gravity (g), which is approximately 9.8 m/s虏 on Earth.
The greater the gravitational pull, the greater the force propelling the object downward through the viscous medium. For terminal velocity:
  • It is directly proportional to the square of the gravitational acceleration.
This means if we increased the value of gravitational acceleration, the terminal velocity would increase significantly. Thus, gravity doesn't just get the object moving; it plays a key role in how fast the object moves through a resisting medium.
Radius Dependency
The size of the falling object, particularly its radius, significantly influences terminal velocity. Larger objects tend to have a higher terminal velocity than smaller ones. Here's why:
  • Larger radius means more surface area in contact with the fluid, potentially increasing drag.
  • However, the increase in mass and gravitational force due to a larger size typically outpaces the added drag, especially for denser objects.
Therefore, when considering the radius of spheres falling through a viscous medium:
  • The terminal velocity is directly proportional to the radius of the sphere.
This relationship shows that simply altering the physical size of an object can change how swiftly it navigates through a fluid.
Fluid Viscosity
Fluid viscosity is essentially a measure of how "thick" or "sticky" a fluid is. It represents the internal friction of the fluid that resists shearing motion. In terms of terminal velocity:
  • Higher viscosity means higher resistance to the motion of objects.
  • This resistance causes objects to achieve a lower terminal velocity.
Imagine trying to push an object through honey versus water:
  • Honey, being more viscous, requires more force for the same speed.
  • Thus, objects moving through honey reach a lower terminal velocity compared to moving through water.
This relationship between viscosity and velocity shows how fluid properties have a direct impact on the dynamics of moving objects within them. Understanding this concept can help predict and manage behaviour in various engineering and scientific applications.

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