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Its a tunnel in a planet

Consider a planet of radius \(r\) having density \(\lambda\).A tunnel is dig inside it it at a distance \(\frac{r}{2}\) from its centre as shown. An object of mass m is left in the tunnel at the surface at \(t=0\)

Find the time taken by the object to reach the mid of the tunnel and the normal reaction applied by wall of the tunnel on the object.

Note by Tanishq Varshney
2 years, 1 month ago

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Hint, it doesn't matter how far the tunnel is from the center of the Earth. Josh Silverman Staff · 2 years, 1 month ago

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Can I have your views over here Kushal Patankar · 2 years, 1 month ago

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First of all, thanks for mentioning me @Tanishq Varshney !! Is the answer for time period = \(\frac { \pi }{ 2 } \sqrt { \frac { 3 }{ 4\pi \lambda G } } \) ??? Abhineet Nayyar · 2 years, 1 month ago

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@Abhineet Nayyar I have same guess. Kushal Patankar · 2 years, 1 month ago

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@Kushal Patankar Can u provide a hint or some expression to get the answer Tanishq Varshney · 2 years, 1 month ago

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@Tanishq Varshney See, first of all, i found the restoring force, in this case, a component of the gravitational force between the Earth and the object.

Tunnel In a Planet

Tunnel In a Planet

Let's consider that the object starts from the centre and moves towards the surface. Consider a very small displacement \('x'\) from the centre such that the restoring force remains constant. Now, the motion of the body through this \('x'\) is an SHM which i will prove later. According to shell theorem, the Earth around this motion won't apply any force. So, the force is applied by a small sphere with radius \(R=\sqrt { \frac { { r }^{ 2 } }{ 2 } +{ x }^{ 2 } } \) which is equal to \(\frac { Gm\frac { 4 }{ 3 } \pi \lambda { R }^{ 3 } }{ { R }^{ 2 } } \). But the restoring force will be the vertical component of this force = \(\frac { G(\frac { 4 }{ 3 } \pi { R }^{ 3 }\lambda )m }{ { R }^{ 2 } } \frac { x }{ R } \) = \(G\frac { 4 }{ 3 } \pi \lambda m(x)\)

Clearly this force is directly proportional to \(x\) and so, the motion is a simple harmonic one. So, the spring factor in the case is =\(G\frac { 4 }{ 3 } \pi \lambda m\) and the time period is given by = \(2\pi \sqrt { \frac { Mass\quad factor }{ Spring\quad factor } } \) = \(2\pi \sqrt { \frac { m }{ \frac { 4 }{ 3 } \pi \lambda Gm } } =2\pi \sqrt { \frac { 3 }{ 4\pi \lambda G } } \)

But since, you have asked about the time to move from extreme to mean, it is equal to \(Time period/4\) = \(\frac { \pi }{ 2 } \sqrt { \frac { 3 }{ 4\pi \lambda G } } \)

And the normal reaction (I guess :P) will be given by the other component of the attractive force = \(\frac { Gm\frac { 4 }{ 3 } \pi \lambda { R }^{ 3 } }{ { R }^{ 2 } } (\frac { r/2 }{ R } )\quad =\quad \frac { 2 }{ 3 } Gm\pi \lambda r\)

So, how did i do?? Lol:D Abhineet Nayyar · 2 years, 1 month ago

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@Abhineet Nayyar Thats it. Kushal Patankar · 2 years, 1 month ago

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@Kushal Patankar Yayy! lol Abhineet Nayyar · 2 years, 1 month ago

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@Tanishq Varshney Since the motion inside is SHM , we know that force at amplitude is \(F=A \omega^2\) here A= √3r/2 and you know what F is. You will get \omega there. And the rest is piece of cake. Kushal Patankar · 2 years, 1 month ago

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@Tanishq Varshney

Use dimensional analysis(during exam).

The actual way is to use restoring force and stuff, with it you will get that the motion of the object in the tunnel is S.H.M. Raghav Vaidyanathan · 2 years, 1 month ago

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@Raghav Vaidyanathan Hi Raghav

I need your view on this problem Kushal Patankar · 2 years, 1 month ago

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@Raghav Vaidyanathan I want that stuff @Raghav Vaidyanathan Tanishq Varshney · 2 years, 1 month ago

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@Tanishq Varshney How much are you getting in AIITS-8.

@Tanish Ronak Agarwal · 2 years, 1 month ago

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@Ronak Agarwal I totally messed up with the paper. Prakhar Gupta · 2 years, 1 month ago

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@Ronak Agarwal The answer key webpage is showing error so i couldn't match my answers Tanishq Varshney · 2 years, 1 month ago

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