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After drilling a tunnel from surface of ...

After drilling a tunnel from surface of earth to the centre, a body of mass m is dropped into the tunnel. Calculate the speed with which the body strikes the bottom of tunnel. Here, M is the mass of earth and R is radius of earth.

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To solve the problem of calculating the speed with which a body of mass \( m \) strikes the bottom of a tunnel drilled from the surface of the Earth to its center, we can use the principle of conservation of energy. Here’s a step-by-step solution: ### Step 1: Understand the System We have a tunnel drilled from the surface of the Earth to its center. A body of mass \( m \) is dropped into this tunnel. We need to find the speed of the body when it reaches the center of the Earth. ### Step 2: Apply Conservation of Energy According to the conservation of energy, the total mechanical energy of the body remains constant if we ignore air resistance and other non-conservative forces. The total mechanical energy is the sum of kinetic energy (KE) and gravitational potential energy (PE). ...
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Assume that there is a tunnel in the shape of a circular arc through the earth. Wall of the tunnel is smooth. A ball of mass m is projected into the tunnel at A with speed v. The all comes out of the tunnel at B and escapes out of the gravity of the earth. Mass and radius of the earth are M and R respectively and radius of the circle shaped tunnel is also . Find minimum possible value of v (call it v_(0) ) If the ball is projected into the tunnel with speed v_(0) , calculate the normal force applied by the tunnel wall on the ball when it is closest to the centre of the earth. It is given that the closest distance between the ball and the centre of the earth is (R)/(2)

Knowledge Check

  • If a tunnel is dug along the diameter of the earth and a ball is dropped into the tunnel, it will have

    A
    linear motion
    B
    circular motion
    C
    oscillatory motion
    D
    translatory motion

  • If a tunnel is dug along the diameter of earth and a piece of stone is dropped into it, then the stone will

    A
    come out of the another end of earth and will excape out in space
    B
    come to rest at the centre of earth
    C
    start oscillating about the centre of the earth
    D
    stop at another end of earth

  • If body of mass m has to be taken from the surface to the earth to a height h=4R , then the amount of energy required is (R = radius of the earth)

    A
    `mgR`
    B
    `(mgR)/(5)`
    C
    `(4mgR)/(5)`
    D
    `(mgR)/(12)`

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