Application of Cauchy-Schwarz Inequality in Work Done

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 product, or dot product, of the force and displacement vectors.

Thus,

###W = \vec{F}\cdot\vec{d} = F_xd_x + F_yd_y###

The objective is to use the Cauchy-Schwarz inequality to determine the theoretical upper bound of the work done and to identify the condition under which this maximum value is achieved.

Worked Solution and Step-by-Step Explanation

To determine the maximum possible work, we use the Cauchy-Schwarz inequality. This inequality gives a sharp upper bound for the dot product of two vectors. Since work is itself a dot product, the result applies naturally to this problem.

Step 1: Recall the Cauchy-Schwarz Inequality

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

###(a_1b_1 + a_2b_2)^2 \leq (a_1^2 + a_2^2)(b_1^2 + b_2^2)###

Taking the non-negative square root of both sides gives the absolute-value form:

###|a_1b_1 + a_2b_2| \leq \sqrt{a_1^2 + a_2^2}\,\sqrt{b_1^2 + b_2^2}###

The vector form of the inequality is especially important here:

###|\vec{A}\cdot\vec{B}| \leq |\vec{A}|\,|\vec{B}|###

Step 2: Map the Inequality to Force and Displacement

We now connect the abstract variables in the Cauchy-Schwarz inequality with the physical quantities in the problem.

Substituting these physical variables into the Cauchy-Schwarz inequality gives:

###|F_xd_x + F_yd_y| \leq \sqrt{F_x^2+F_y^2}\,\sqrt{d_x^2+d_y^2}###

Step 3: Identify the Physical Meaning of Each Side

The expression on the left-hand side is the magnitude of the work done because

###W = F_xd_x + F_yd_y = \vec{F}\cdot\vec{d}###

Therefore,

###|W| = |F_xd_x + F_yd_y|###

The two square-root expressions on the right-hand side represent the magnitudes of the force and displacement vectors:

###|\vec{F}| = \sqrt{F_x^2+F_y^2}###

and

###|\vec{d}| = \sqrt{d_x^2+d_y^2}###

Hence, the inequality becomes:

###|W| \leq |\vec{F}|\,|\vec{d}|###

This result has a clear physical meaning: even if the force is large and the displacement is large, the work done also depends on how well the direction of the force aligns with the direction of displacement.

Step 4: Determine the Condition for Equality

Equality in the Cauchy-Schwarz inequality occurs if and only if the two vectors are linearly dependent. In this problem, that means the force vector must be a scalar multiple of the displacement vector:

###\vec{F} = k\vec{d}###

where ##k## is a real scalar. This means that the force and displacement vectors are parallel or antiparallel.

If the components of the displacement vector are non-zero, this proportionality can also be written as:

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

Step 5: Connect with the Standard Formula for Work

The usual formula for work done by a constant force is:

###W = |\vec{F}|\,|\vec{d}|\cos\theta###

Since ##-1 \leq \cos\theta \leq 1##, we immediately get:

###-|\vec{F}|\,|\vec{d}| \leq W \leq |\vec{F}|\,|\vec{d}|###

Therefore, the maximum positive value of work is:

###W_{\max} = |\vec{F}|\,|\vec{d}|###

This maximum occurs when ##\theta = 0^\circ##, meaning the force acts exactly in the direction of displacement.

The most negative value of work is:

###W_{\min} = -|\vec{F}|\,|\vec{d}|###

This occurs when ##\theta = 180^\circ##, meaning the force acts exactly opposite to the displacement.

Thus, the Cauchy-Schwarz inequality confirms a familiar physical principle: the work done is greatest when the force is applied directly along the direction of motion.