Maharashtra State BoardHSC Science (General) 11th

Answer the following question in detail. Derive an expression for variation in gravitational acceleration of the Earth at with latitude. - Physics

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Answer in Brief

Answer the following question in detail.

Derive an expression for variation in gravitational acceleration of the Earth at with latitude.

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Solution

  1. Latitude is an angle made by the radius vector of any point from the center of the Earth with the equatorial plane.
  2. The Earth rotates about its polar axis from west to east with uniform angular velocity ω as shown in the figure.
    Hence, every point on the surface of the Earth (except the poles) moves in a circle parallel to the equator.
  3. The motion of a mass m at point P on the Earth is shown by the dotted circle with the center at O′.
  4. Let the latitude of P be θ and the radius of the circle be r.
    ∴ PO' = r
    ∠EOP = θ, E being a point on the equator
    ∴ ∠OPO' = θ
    In Δ OPO', cos θ = `"PO'"/"PO" = "r"/"R"`
    ∴ r = R cos θ
  5. The centripetal acceleration for the mass m, directed along PO' is,
    a = rω2
    ∴ a = rωcos θ
    The component of this centripetal acceleration along PO, i.e., towards the centre of the Earth is,
    `"a"_"r" = "a" cos theta`
    ∴ `"a"_"r" = "R"omega^2 cos theta xx cos theta`
    `"a"_"r" = "R"omega^2cos^2theta`
  6. Part of the gravitational force of attraction on P acting towards PO is utilized in providing this component of centripetal acceleration. Thus, the effective force of gravitational attraction on m at P can be written as,
    mg' = mg - mRω2cos2θ
    Thus, the effective acceleration due to gravity at P is given as,
    g' = g - Rω2cos2θ
Concept: Variation in the Acceleration Due to Gravity with Altitude, Depth, Latitude and Shape
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APPEARS IN

Balbharati Physics 11th Standard Maharashtra State Board
Chapter 5 Gravitation
Exercises | Q 3. (x) | Page 98
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