Moment of Inertia of Solid Sphere and Thin Spherical Shell
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Questions and Answers

What is the pressure at a point P at depth h below the surface of a liquid of uniform density?

  • hρ/g
  • hρg (correct)
  • h/ρg
  • h^2ρg

    • If the free surface of the liquid is open to the atmosphere, what is the absolute pressure within the liquid at a depth h?

  • p0 / hρg
  • p0 - hρg (correct)
  • p0 + hρg
  • p0 * hρg

    • What is the definition of angular Simple Harmonic Motion (SHM)?

  • The motion of a body in a circular path with uniform speed.
  • The oscillatory motion of a body where restoring torque is directly proportional to angular displacement. (correct)
  • The motion of a body in a straight line with constant acceleration.
  • The oscillatory motion of a body in which force is directly proportional to displacement.

    • What is the differential equation of angular Simple Harmonic Motion (SHM)?

    <p>(I - ct) d^2θ/dt^2 = 0</p> Signup and view all the answers

    What does the moment of inertia of a solid sphere depend on?

    <p>Radius and density</p> Signup and view all the answers

    What is the moment of inertia of a thin spherical shell?

    <p>(2/3)MR^2</p> Signup and view all the answers

    What happens to the moment of inertia of a ballet dancer when she stretches her arms during a spin?

    <p>Increases</p> Signup and view all the answers

    If a hollow cylinder and a solid cylinder of the same radius rotate about their respective cylinder axes, how do their moments of inertia compare?

    <p>Moment of inertia of the hollow cylinder is greater than the solid cylinder</p> Signup and view all the answers

    What is the effect on the duration of a day if the Earth suddenly shrinks while keeping its mass constant?

    <p>Day duration decreases</p> Signup and view all the answers

    In a circular disc and a circular ring with the same radii rotating about a tangential axis in their planes, how do their radii of gyration compare?

    <p>Radii of gyration of the ring is greater than the disc</p> Signup and view all the answers

    If a ballet dancer's angular speed decreases while stretching her arms in a turn, what happens to her linear speed?

    <p>Decreases</p> Signup and view all the answers

    When comparing the moment of inertia of a solid sphere to that of a thin spherical shell, which one is generally larger?

    <p>Moment of inertia of solid sphere is larger</p> Signup and view all the answers

    What is the formula for the moment of inertia of a solid sphere about an axis passing through its centre?

    <p>$\frac{2}{5}MR^2$</p> Signup and view all the answers

    What is the expression for the moment of inertia of a thin spherical shell about its diameter?

    <p>$\frac{2}{3}MR^2$</p> Signup and view all the answers

    Why do a uniform disc and a hollow right circular cone have the same formula for moment of inertia for rotation about their corresponding central symmetry axes?

    <p>The distribution of mass about the respective central symmetry axes is the same.</p> Signup and view all the answers

    How does the moment of inertia of a hollow right circular cone compare to that of a disc?

    <p>Both have equal moments of inertia.</p> Signup and view all the answers

    In the expression for the moment of inertia of a thin spherical shell about its diameter, what does $h$ represent?

    <p>Distance between two axes</p> Signup and view all the answers

    What is used to calculate the moment of inertia about an axis parallel to the diameter for a thin spherical shell?

    <p>$\frac{2}{3}MR^2$</p> Signup and view all the answers

    Study Notes

    Pressure in a Liquid

    • The pressure at a point P at depth h below the surface of a liquid of uniform density is due to the weight of the liquid above it.
    • The absolute pressure within the liquid at a depth h is the sum of the atmospheric pressure and the pressure due to the weight of the liquid above it.

    Angular Simple Harmonic Motion (SHM)

    • Angular SHM is defined as the oscillation of an object about a fixed axis, where the restoring torque is proportional to the angular displacement.
    • The differential equation of angular SHM is τ = -kθ, where τ is the restoring torque, k is the torsion constant, and θ is the angular displacement.

    Moment of Inertia

    • The moment of inertia of a solid sphere depends on its mass and radius.
    • The moment of inertia of a thin spherical shell is 2/3MR^2, where M is the mass and R is the radius.
    • When a ballet dancer stretches her arms during a spin, her moment of inertia increases, causing her angular speed to decrease.
    • The moment of inertia of a hollow cylinder is greater than that of a solid cylinder of the same radius.
    • If the Earth suddenly shrinks while keeping its mass constant, the duration of a day would decrease.

    Rotational Motion

    • The radius of gyration of a circular disc is less than that of a circular ring with the same radii rotating about a tangential axis in their planes.
    • When a ballet dancer's angular speed decreases while stretching her arms in a turn, her linear speed decreases.
    • The moment of inertia of a solid sphere is generally less than that of a thin spherical shell.
    • The formula for the moment of inertia of a solid sphere about an axis passing through its centre is 2/5MR^2, where M is the mass and R is the radius.
    • The expression for the moment of inertia of a thin spherical shell about its diameter is 1/2MR^2, where M is the mass and R is the radius.
    • A uniform disc and a hollow right circular cone have the same formula for moment of inertia for rotation about their corresponding central symmetry axes.
    • The moment of inertia of a hollow right circular cone is greater than that of a disc.
    • In the expression for the moment of inertia of a thin spherical shell about its diameter, h represents the distance from the axis of rotation to the centre of the shell.
    • The parallel axis theorem is used to calculate the moment of inertia about an axis parallel to the diameter for a thin spherical shell.

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    Description

    This quiz covers the formulas for calculating the moment of inertia of a solid sphere about an axis passing through its center and a thin spherical shell about its diameter. Test your understanding of rotational motion concepts with these questions.

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