Mechanics

Find the law of force to the pole when the path is the cardioid r=a(1cosθ)r=a(1- \cos \theta), and prove that if FF were the force at the apse, and VV the velocity 3V2=4aF3V^2=4aF.

Note by Syed Subhan Siraj
5 years, 8 months ago

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Comments

First we assume that the motion is under a central force. Applying logarithmic differentiation:r=a(1cosθ)1rdrdθ=asinθa(1cosθ)=2sinθ2cosθ22sin2θ2=cotθ2=cotϕϕ=θ2r=a\left( 1-\cos { \theta } \right) \\ \Rightarrow \frac { 1 }{ r } \frac { dr }{ d\theta } =\frac { a\sin { \theta } }{ a\left( 1-\cos { \theta } \right) } =\frac { 2\sin { \frac { \theta }{ 2 } } \cos { \frac { \theta }{ 2 } } }{ 2\sin ^{ 2 }{ \frac { \theta }{ 2 } } } =\cot { \frac { \theta }{ 2 } } =\cot { \phi } \\ \Rightarrow \phi =\frac { \theta }{ 2 } where ϕ\phi is the polar-tangential angle in pedal coordinates. Now we have p=rsinϕ=rsinθ2=r22sin2θ2=r21cosθ=r2ra=rr2a2ap2=r3p=r\sin { \phi } =r\sin { \frac { \theta }{ 2 } } =\frac { r }{ \sqrt { 2 } } \sqrt { 2\sin ^{ 2 }{ \frac { \theta }{ 2 } } } =\frac { r }{ \sqrt { 2 } } \sqrt { 1-\cos { \theta } } =\frac { r }{ \sqrt { 2 } } \sqrt { \frac { r }{ a } } =r\sqrt { \frac { r }{ 2a } } \\ \Rightarrow 2a{ p }^{ 2 }={ r }^{ 3 } Differentiating both sides w.r.t. r:4apdpdr=3r2dpdr=3r24aph2p3dpdr=h2.3r2p3.4ap=3ah2r4=F4ap\frac { dp }{ dr } =3{ r }^{ 2 }\\ \Rightarrow \frac { dp }{ dr } =\frac { 3{ r }^{ 2 } }{ 4ap } \\ \Rightarrow \frac { { h }^{ 2 } }{ { p }^{ 3 } } \frac { dp }{ dr } =\frac { { h }^{ 2 }.3{ r }^{ 2 } }{ { p }^{ 3 }.4ap } =3a\frac { { h }^{ 2 } }{ { r}^{ 4 } } =F Thus force is inversely proportional to fourth power of distance. Now, at an apsedrdθ=0sinθ=0θ=0orπ\frac { dr }{ d\theta } =0\\ \Rightarrow \sin { \theta } =0\\ \Rightarrow \theta =0\quad or\quad \pi But θ=0r=0\theta =0\\ \Rightarrow r=0 which is a cusp of the cardioid. Thus θ=πr=2a=ph=vp=2avF=3a(2av)2(12a)2=3v24a4aF=3v2\theta =\pi \\ \Rightarrow r=2a=p\\ \Rightarrow h=vp=2av\\ \Rightarrow F=3a{ \left( 2av \right) }^{ 2 }{ \left( \frac { 1 }{ 2a } \right) }^{ 2 }=\frac { 3{ v }^{ 2 } }{ 4a } \\ \Rightarrow 4aF=3{ v }^{ 2 }[Q.E.D.]

Kuldeep Guha Mazumder - 5 years, 6 months ago

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thx

Syed Subhan Siraj - 5 years, 5 months ago

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You are welcome..:-)

Kuldeep Guha Mazumder - 5 years, 5 months ago

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@Kuldeep Guha Mazumder sir ap teacher hoo??

Syed Subhan Siraj - 5 years, 5 months ago

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@Syed Subhan Siraj Nahi nahi main to ek student hoon..college mein parta hoon..

Kuldeep Guha Mazumder - 5 years, 5 months ago

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@Kuldeep Guha Mazumder oky me tooo

Syed Subhan Siraj - 5 years, 4 months ago
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