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Chapter 12: Q1P (page 599)

Verify equation (18.3) i.e. $l \frac{d^{2} 胃}{{dl}^{2}} + 2 \frac{诲胃}{dl} + \frac{g}{V^{2}} 胃 = 0$

Short Answer

Expert verified

The resultant answer $l \frac{d^{2} 胃}{{dl}^{2}} + 2 \frac{诲胃}{dl} + \frac{g}{V^{2}} 胃 = 0$ is verified.

Step by step solution

01

Concept of Equation of motion:

The equation of motion:

$\frac{d}{dt} ({ml}^{2} 胃) = 尘驳濒蝉颈苍胃$

02

Consider jn(x)=∑r=0∞(-1)ΓΓ(r+1)Γ(r+1+n)(x2)2Γ+n and simplify it:

Considering the equation of motion:

$\frac{d}{dt} ({ml}^{2} 胃) = mgl 蝉颈苍胃$

For the small value of $胃$ :

$\sin = 胃$

Hence,

${ml}^{2} \frac{d^{2} 胃}{{dl}^{2}} + 2 lm \frac{dl}{dt} 脳 \frac{诲胃}{dt} + 尘驳濒胃 = 0$ 鈥�.. (1)

Consider the following formula.

$l = l_{0} + vt$

Take a derivation with respect to t .

role="math" localid="1659272604614" $\frac{dl}{dt} = v dl = vdt$

Then,

$\frac{诲胃}{dt} = \frac{诲胃}{dl} 脳 \frac{dl}{dt} = v \frac{诲胃}{dt}$ .

And,

$\frac{d^{2} 胃}{{dt}^{2}} = V^{2} \frac{d^{2} 胃}{{dl}^{2}}$ .

03

Put the values into equation (1):

${ml}^{2} v^{2} \frac{d^{2} 胃}{{dl}^{2}} + 2 lmv 脳 v \frac{诲胃}{dt} + 尘驳濒胃 \frac{v^{2}}{v^{2}} = 0 {mlv}^{2} [l \frac{d^{2} 胃}{{dl}^{2}} + 2 \frac{诲胃}{dt} + \frac{g}{v^{2}} 胃] = 0 l \frac{d^{2} 胃}{{dl}^{2}} + 2 \frac{诲胃}{dt} + \frac{g}{v^{2}} 胃 = 0$

Hence, $l \frac{d^{2} 胃}{{dl}^{2}} + 2 \frac{诲胃}{dl} + \frac{g}{V^{2}} 胃 = 0$ is verified.

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