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Define an ideal simple pendulum. Show th...

Define an ideal simple pendulum. Show that motion of a simple pendulum under certain conditions is simple harmonic. Obtain an expression for its period.
A solid sphere of diameter 50 cm and mass 25 kg rotates about an axis through its centre. Calculate its moment of inertia. If its angular velocity changes from 2 rad/s to 12 rad/s in 5 seconds. calculate the torque applied.

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### Step-by-Step Solution #### Part 1: Ideal Simple Pendulum and Simple Harmonic Motion 1. **Definition of an Ideal Simple Pendulum**: An ideal simple pendulum consists of a massless string of length \( L \) attached to a rigid support, with a point mass \( M \) (the bob) at the other end. The motion occurs in a vertical plane, and we assume no air resistance or friction at the pivot. 2. **Conditions for Simple Harmonic Motion (SHM)**: ...
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A solid sphere of radius 25 cm and mass 25 kg rotates about an axis through its centre. Calculate its moment of inertia if its angular velocity changes from 2 rad/s to 22 rad/s in 5 seconds. Also calculate the torque applied.

Show that for small oscillations the motion of a simple pendulum is simple harmonic. Derive an expression for its time period. Does it depend on the mass of the bob ?

Knowledge Check

  • If a uniform solid sphere of diameter 0.2 m and mass 10 kg is rotated about its diameter with an angular velocity of 2 rad/s, then the its angular momentum in kg m^(2)//s will be

    A
    0.01
    B
    0.02
    C
    0.08
    D
    0.04

  • A uniform solid sphere of radius r = 0.500 m and mass m = 15.0 kg turns counterclockwise about a vertical axis through its centre.Find its vector angular momentum about this axis when its angular speed is 3.00 rad//s .

    A
    `(2.50 kg.m^2//s) hat k`
    B
    `-(4.50 kg.m^2//s) hat k`
    C
    `-(2.50 kg.m^2//s) hat k`
    D
    `(4.50 kg.m^2//s) hat k`

  • A disc of mass 16 kg and radius 25 cm is rotated about its axis. What torque will increase its angular velocity from 0 to 8 pi rad/s in 8 s ?

    A
    `pi Nm`
    B
    `pi//2 Nm`
    C
    `pi//4 Nm`
    D
    `2pi Nm`

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