Consider a force F₁ = $A/\sqrt{x}$ which acts on an object during its journey along the x axis from x = 0.0 to x = 1.0m, where A = 3.0 Nm¹⸍². Show that during this journey, even though F₁ is infinite at x = 0.0, the work W done on the object by this force is finite, and determine W.

Verified step by step guidance

1. Step 1: Recall the formula for work done by a force along a path. The work done by a variable force F(x) along the x-axis is given by the integral: $W_x={\int }_x^{f}F\left(x\right)\mathrm{dx}$, where x_i and x_f are the initial and final positions, respectively.
2. Step 2: Substitute the given force $F_1=\frac{A}{\sqrt{x}}$ into the work integral. The work becomes: $W={\int }_0^1\frac{A}{\sqrt{x}}\mathrm{dx}$, where A = 3.0 N·m^1/2.
3. Step 3: Simplify the integral. Rewrite $\frac{A}{\sqrt{x}}$ as $A{x}^{-}$. The integral becomes: $W=A{\int }_0^1{x}^{-}\mathrm{dx}$.
4. Step 4: Evaluate the integral. Use the power rule for integration: ${\int }_a^b{x}^n\mathrm{dx}=\frac{b^n}{-}/(n+1)$, where n ≠ -1. Here, n = -1/2, so the integral becomes: $W=A\left[\frac{x^{\frac{1}{2}}}{+}\right]{|}_{0}1$.
5. Step 5: Simplify the result. After evaluating the definite integral, you will find that the work done is finite because the singularity at x = 0.0 is integrable. Substitute the limits x = 1.0 and x = 0.0 into the expression to determine the final value of W.

Key Concepts

Force and Its Mathematical Representation

In physics, force is defined as an interaction that causes an object to change its velocity, and it can be represented mathematically. In this case, the force F₁ is given as A/√x, where A is a constant. This representation indicates that as x approaches zero, the force becomes infinitely large, which is crucial for understanding the behavior of the force over the specified range.

Work Done by a Force

Work is defined as the integral of force over a distance, mathematically expressed as W = ∫ F dx. In this scenario, even though the force F₁ becomes infinite at x = 0, the work done can still be finite if the force decreases rapidly enough as x increases. This concept is essential for evaluating the total work done on the object as it moves from x = 0 to x = 1.0 m.

Improper Integrals

An improper integral is used when integrating functions that have infinite limits or discontinuities. In this case, the integral for work involves evaluating the limit as x approaches zero, which requires careful handling to determine if the integral converges to a finite value. Understanding improper integrals is key to solving the problem and confirming that the work done is indeed finite despite the infinite force at the starting point.