JUPITER SCIENCE
Where Exploration Meets Excellence
Explore
Advertisement

PHYSICS

Application of Cauchy-Schwarz Inequality in Work Done

A 3D coordinate system grid showing an orange Force Vector F and a blue Displacement Vector d, with an angle marked between them.
The geometric relationship between force and displacement vectors as represented in three-dimensional space.

On This Page

  1. Problem: Application of Cauchy-Schwarz Inequality in Work Done

Advertisement

Problem: Application of Cauchy-Schwarz Inequality in Work Done

In classical mechanics, the work done ##W## by a constant force ##\vec{F} = F_x \hat{i} + F_y \hat{j}## acting on a particle undergoing a displacement ##\vec{d} = d_x \hat{i} + d_y \hat{j}## is defined as the scalar (dot) product ##W = \vec{F} \cdot \vec{d} = F_x d_x + F_y d_y##.

Quantity Mathematical Representation Physical Meaning
Force Vector ##\vec{F} = F_x \hat{i} + F_y \hat{j}## The push or pull acting on a particle
Displacement Vector ##\vec{d} = d_x \hat{i} + d_y \hat{j}## The change in position of the particle
Work Done ##W = \vec{F} \cdot \vec{d}## Energy transferred by the force over displacement
Using the Cauchy-Schwarz inequality for vectors in ##\mathbb{R}^2##, establish the mathematical upper bound for the magnitude of the work done ## W ## in terms of the magnitudes of the force and displacement vectors. Furthermore, determine the exact physical condition under which this upper bound is achieved.

Worked Solution & Step-by-Step Explanation

To determine the upper bound of the work done, we rely on the Cauchy-Schwarz inequality. This inequality is a fundamental theorem in linear algebra and vector calculus, providing a tight bound on the inner product of two vectors.

Step 1: The Cauchy-Schwarz Inequality Definition

For any two sequences of real numbers ##(a_1, a_2)## and ##(b_1, b_2)##, the Cauchy-Schwarz inequality states:

###(a_1 b_1 + a_2 b_2)^2 \le (a_1^2 + a_2^2)(b_1^2 + b_2^2)###

By taking the non-negative square root of both sides, we obtain the expression for the absolute value:

### a_1 b_1 + a_2 b_2 \le \sqrt{a_1^2 + a_2^2} \cdot \sqrt{b_1^2 + b_2^2}###
Form Mathematical Expression Context
Absolute Value Form
## a_1 b_1 + a_2 b_2 \le \sqrt{a_1^2 + a_2^2} \cdot \sqrt{b_1^2 + b_2^2}##
Squared Form ##(a_1 b_1 + a_2 b_2)^2 \le (a_1^2 + a_2^2)(b_1^2 + b_2^2)## Original inequality for sequences

Derived by taking non-negative square root

Vector Dot Product Equivalent

## \vec{A} \cdot \vec{B} \le \ \vec{A}\ \ \vec{B}\ ##

General vector form (implied by inequality)

Step 2: Mapping to Physical Variables

We define our force vector components as ##a_1 = F_x## and ##a_2 = F_y##, and our displacement vector components as ##b_1 = d_x## and ##b_2 = d_y##. Substituting these into the inequality yields:

Cauchy-Schwarz Variable Physical Variable Description
##a_1## ##F_x## X-component of the Force Vector
##a_2## ##F_y## Y-component of the Force Vector
##b_1## ##d_x## X-component of the Displacement Vector
##b_2## ##d_y## Y-component of the Displacement Vector
### F_x d_x + F_y d_y \le \sqrt{F_x^2 + F_y^2} \cdot \sqrt{d_x^2 + d_y^2}###

Step 3: Identification of Physical Quantities

The left-hand side of the inequality is the definition of the work done, ##W = \vec{F} \cdot \vec{d}##. Thus, we represent the absolute value as ## W ##. On the right-hand side, we recognize the Euclidean norms (magnitudes) of the force and displacement vectors:
###\ \vec{F}\ = \sqrt{F_x^2 + F_y^2}###
###\ \vec{d}\ = \sqrt{d_x^2 + d_y^2}###

Consequently, the inequality simplifies to:

### W \le \ \vec{F}\ \ \vec{d}\ ###

This establishes that the magnitude of the work done can never exceed the product of the magnitudes of the force and the displacement.

Step 4: Determining the Condition for Equality

The equality holds if and only if the vectors are linearly dependent (i.e., proportional). Mathematically, this occurs when:

###\dfrac{F_x}{d_x} = \dfrac{F_y}{d_y} = k###

where ##k## is a non-zero scalar constant. In physical terms, this implies that the force vector ##\vec{F}## is parallel or antiparallel to the displacement vector ##\vec{d}##.

Condition Mathematical Meaning Physical Interpretation

:--- :--- :---

##k > 0## ##\vec{F} = k\vec{d}## Force is in the same direction as displacement; ##\theta = 0^\circ##

##k < 0## ##\vec{F} = k\vec{d}## Force is in the opposite direction; ##\theta = 180^\circ##

## W = \ \vec{F}\
Since the standard definition of work is ##W = \ \vec{F}\ \ \vec{d}\ \cos\theta##, the Cauchy-Schwarz inequality confirms that ## W ## reaches its maximum possible value when ##\cos\theta = \pm 1##. This rigorous approach confirms that the most efficient transfer of energy (maximum work) occurs when the force is applied directly along the line of motion.

RESOURCES

This article is tagged in
MATHEMATICS SPACE EXPLORATION MATHS SPACE SCIENCE ALGEBRA PROBABILITY FUNCTIONS BIOTECHNOLOGY SPACE TECHNOLOGY WIKI GENETICS Q&A EXAM+ MEDICAL TECHNOLOGY LIMITS

Comments

What do you think?

0 Comments

Submit a Comment

Your email address will not be published. Required fields are marked *

Recommended Next for You

  1. >
    Upper Bound on Error Propagation using Triangle InequalityGENERAL
  2. >
    The Triangle Inequality in Vector AdditionVECTOR AND 3D GEOMETRY
  3. >
    AM-GM Inequality for Minimum Power DissipationGENERAL
  4. >
    Weierstrass Product Inequality in Cascaded EfficiencyMATHEMATICS
  5. >
    Jensen's Inequality in Statistical Thermodynamics of Ideal GasesPHYSICS
  6. >
    Cauchy-Schwarz Inequality for Multi-Component Error AnalysisGENERAL
  7. >
    Advanced Speed Conversion and Unit Analysis TechniquesPHYSICS
  8. >
    Predicting Distance Over Time: Solving Linear Kinematic EquationsKINEMATICS
  9. >
    Understanding Initial Values in Linear EquationsMATHEMATICS
  10. >
    Understanding Why |x| = |-x| for All Real NumbersMATHEMATICS
  11. >
    Do Real Numbers Always Have a Decimal Expansion?MATHEMATICS
  12. >
    NEET-UG 2026 Cancelled: The Impact of Guess Paper Leaks and Future StepsGENERAL
  13. >
    NEET-UG 2026 Cancelled: The Impact of Guess Paper Leaks and Future StepsGENERAL
  14. >
    Elastic and Inelastic Collisions for IIT JEE: Momentum, Restitution, and Impact StrategyMECHANICS
  15. >
    Signum Function: Direction, Piecewise Values, and CodeMATHEMATICS
  1. 01
    Kinematics Online Practice Quiz for Physics Students
  2. 02
    Simple Physics Practice Quiz
  3. 03
    Calculating Monthly Energy Consumption for a 3W Bulb: A Technical Guide
  4. 04
    10 Physics Numerical Problems with Solutions for IIT JEE Preparation
  5. 05
    Free Fall Time Calculation: Solving the 80m Stone Drop Problem
  6. 06
    Final Velocity after Braking: Solving Kinematic Equations
  7. 07
    Average Speed Calculation: A Technical Guide to Kinematics
  8. 08
    ISRO Aditya-L1 Discovery: Validating the Parker Transport Equation
  9. 09
    The Azure Veil: Unraveling the Physics of Why the Sky is Blue
  10. 10
    Unveiling Chemical Equilibrium: The Interplay of Gibbs Free Energy and Equilibrium Constant