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Question

A straight tunnel is dug into the Earth as shown in the figure at a distance d from its center. A ball of mass m is dropped from one of its ends. The time it takes to reach the other end is approximately equal to


  1. 42min

  2. 84min

  3. 84dRmin

  4. 42dRmin

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Solution

The correct option is A

42min


Step 1: Given information

Consider the figure given below

Step 2: Calculations required

The gravitational force on mass m at distance r is,

F=GMmrR3

Where, G is the gravitational constant, M is the mass of the Earth, R is the radius of the Earth, and r is the distance from the center O.

From the figure given below,

xr=sinθr=xsinθ

Now,

Fx=-FsinθFx=-GMmrR3xrFx=-GMmxR3

Since Fx-x, its motion is simple harmonic in nature. Further,

max=GMmxR3ax=GMxR3

Hence the time period of oscillation is,

T=2πxg

Where, T is the time period.

T=2πxaxT=2πxGMxR3=2πR3GMg=GMR2=2πRg=2π6.4×1069.8=5077.5seconds=5077.560min=84min

The time take for a particle to go from one end to another is,

t=T2=842=42min

Therefore, option (A) is correct.


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