Q39 of 40 Page 103

A tunnel is dug through the centre of the Earth. Show that a body of mass ‘m’ when dropped from rest from one end of the tunnel will execute simple harmonic motion.

When the body is made to fall through the tunnel it falls with some acceleration, and on reaching the centre it has some velocity to move ahead. ∴ the velocity vector points away from the centre. But gravity of the earth tries to pull it back due to which it experiences force in opposite direction of motion. All happens when it crosses the centre for the first time.


And acceleration due to gravity inside the earth is given as,


,


where d=depth from the surface of the earth


and R=radius of the earth


∴ net force


Now let , which the distance from the centre of the earth.



∴ we see that is proportional to -


Hence it is Simple Harmonic Motion.


Comparing the above equation with ,where k is the spring constant,


We get,



And time period


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36

A body of mass m is attached to one end of a massless spring which is suspended vertically from a fixed point. The mass is held in hand so that the spring is neither stretched nor compressed. Suddenly the support of the hand is removed. The lowest position attained by the mass during oscillation is 4cm below the point, where it was held in hand.

(a) What is the amplitude of oscillation?


(b) Find the frequency of oscillation?


 37

A cylindrical log of wood of height h and area of cross-section A floats in water. It is pressed and then released. Show that the log would execute S.H.M. with a time period.

T = 2π √


where m is mass of the body and ρ is density of the liquid.


 38

One end of a V-tube containing mercury is connected to a suction pump and the other end to atmosphere. The two arms of the tube are inclined to horizontal at an angle of 45° each. A small pressure difference is created between two columns when the suction pump is removed. Will the column of mercury in V-tube execute simple harmonic motion? Neglect capillary and viscous forces. Find the time period of oscillation.

 40

A simple pendulum of time period 1s and length l is hung from a fixed support at O, such that the bob is at a distance H vertically above A on the ground (Fig. 14.11). The amplitude is θ0. The string snaps at θ = θ0 /2. Find the time taken by the bob to hit the ground. Also find distance from A where bob hits the ground. Assume θ0 to be small so that sine and cosine is equal to 1.