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Chapter 4: Problem 447

A force \(F=A y^{2}+B y+C\) acts on a body in the \(y\) -direction. The work done by this force during a displacement from \(\mathrm{y}=-\mathrm{a}\) to \(\mathrm{y}=\mathrm{a}\) is (A) \(\left[\left\\{2 \mathrm{Aa}^{3}\right\\} / 3\right]\) (B) \(\left[\left\\{2 \mathrm{Aa}^{3}\right\\} / 3\right]+2 \mathrm{ca}\) (C) $\left[\left\\{2 \mathrm{Aa}^{3}\right\\} / 3\right]+\left[\left\\{\mathrm{Ba}^{2}\right\\} / 2\right]+\mathrm{ca}$ (D) None of these.

Short Answer

Expert verified
The short answer for the work done is: (B) \(\left[\left\\{2 \mathrm{Aa}^{3}\right\\} / 3\right]+2 \mathrm{ca}\)

Step by step solution

01

Identify the given information

The force acting on the body in the y-direction is given by F = Ay² + By + C, and we need to find the work done during the displacement from y = -a to y = a.
02

Write the formula for work done

The work done (W) by a force acting in the y-direction can be calculated by integrating the force F with respect to y over the given interval [-a, a]: \[W = \int_{-a}^{a} F(y) dy\]
03

Substitute the expression for F

Replace F in the integral expression with Ay² + By + C: \[W = \int_{-a}^a (Ay^2 + By + C) dy\]
04

Integrate the expression

Now, integrate each term with respect to y: \[W = \int_{-a}^a Ay^2 dy + \int_{-a}^a By dy + \int_{-a}^a C dy\] \[W = A\int_{-a}^a y^2 dy + B\int_{-a}^a y dy + C\int_{-a}^a dy\] For each integral: \[\int y^2 dy = \frac{1}{3} y^3 + D_1\] \[\int y dy = \frac{1}{2} y^2 + D_2\] \[\int dy = y + D_3\] Integration constants D1, D2, and D3 are not necessary, so the results are: \[W = A\left[\frac{1}{3} y^3\right]_{-a}^a + B\left[\frac{1}{2} y^2\right]_{-a}^a + C[y]_{-a}^a\]
05

Evaluate the integral at the endpoints

Now, evaluate the expression at the endpoints y = -a and y = a: \[W = A\left(\frac{1}{3}(a^3) - \frac{1}{3}(-a)^3\right) + B\left(\frac{1}{2}(a^2) - \frac{1}{2}(-a)^2\right) + C(a - (-a))\]
06

Simplify the expression

Simplify the expression for the work done: \[W = A\left(\frac{2a^3}{3}\right) + B\left(\frac{a^2}{2} - \frac{a^2}{2}\right) + C(2a)\] \[W = \frac{2Aa^3}{3} + 0 + 2Ca\]
07

Choose the correct answer

The simplified expression for the work done is: \[W = \frac{2Aa^3}{3} + 2Ca\] Comparing this with the given choices, we find that the correct answer is (B) \(\left[\left\\{2 \mathrm{Aa}^{3}\right\\} / 3\right]+2 \mathrm{ca}\).

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