Chapter 12: Q1P (page 599)

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

Short Answer

Expert verified

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

Step by step solution

Concept of Equation of motion:

The equation of motion:

$\frac{d}{dt} ({ml}^{2} θ) = mglsinθ$

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

Considering the equation of motion:

$\frac{d}{dt} ({ml}^{2} θ) = mgl sinθ$

For the small value of $θ$ :

$\sin = θ$

Hence,

${ml}^{2} \frac{d^{2} θ}{{dl}^{2}} + 2 lm \frac{dl}{dt} \times \frac{dθ}{dt} + mglθ = 0$ ….. (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,

$\frac{dθ}{dt} = \frac{dθ}{dl} \times \frac{dl}{dt} = v \frac{dθ}{dt}$ .

And,

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

Put the values into equation (1):

${ml}^{2} v^{2} \frac{d^{2} θ}{{dl}^{2}} + 2 lmv \times v \frac{dθ}{dt} + mglθ \frac{v^{2}}{v^{2}} = 0 {mlv}^{2} [l \frac{d^{2} θ}{{dl}^{2}} + 2 \frac{dθ}{dt} + \frac{g}{v^{2}} θ] = 0 l \frac{d^{2} θ}{{dl}^{2}} + 2 \frac{dθ}{dt} + \frac{g}{v^{2}} θ = 0$

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

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

Short Answer

Step by step solution

Concept of Equation of motion:

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

Put the values into equation (1):

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Verify equation (18.3) i.e.ld2θdl2+2dθdl+gV2θ=0

Short Answer

Step by step solution

Concept of Equation of motion:

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

Put the values into equation (1):

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

Electric Charge Field and Potential

Modern Physics

Conservation of Energy and Momentum

Oscillations

Electromagnetism

Study anywhere. Anytime. Across all devices.

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