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Simple Pendulum

An ideal simple pendulum consists of a heavy point mass body suspended by a weightless, inextensible, and perfectly flexible string from a rigid support about which it is free to oscillate.
Time period, 60038.png
Important Points
  • Time period of simple pendulum is also independent the of mass of the bob. That is why
    • If the solid bob is replaced by a hollow sphere of same radius but different mass, time period remains unchanged.
    • If a girl is swinging in a swing and another sits with her, the time period remains unchanged.
  • Time period, 62827.png, where l is the distance between point of suspension and center of mass of bob and is called effective length.
    • When a sitting girl on a swinging swing stands up, her center of mass will go up and so l and hence T will decrease.
    • If a hole is made at the bottom of a hollow sphere full of water and water comes out slowly through the hole and time period is recorded till the sphere is empty, initially and finally the center of mass will be at the center of the sphere. However, as water drains off the sphere, the center of mass of the system will first move down and then will come up. Due to thisl and hence T first increase, reache a maximum, and then decrease till they become equal to their initial value.
  • If the length of the pendulum is comparable to the radius of earth, then
    • If l << R, then 63023.png. So 63017.png
    • If l >> R(→ ∞) then 1/l < 1/R.
      So 63011.png min
      and it is the maximum time period which an oscillating simple pendulum can have:
    • If l = R, so 63004.png
  • If the bob of simple pendulum is suspended by a wire, then effective length of pendulum will increase with the rise of temperature due to which the time period will increase.
    l = l0(1 + aΔθ) (If Δθ is the rise in temperature, l0 = initial length of wire,l = final length of wire)
    So 62986.png i.e., 62980.png
  • If the bob of a simple pendulum of density ρ is made to oscillate in some fluid of density σ (where σ < ρ), then time period of simple pendulum gets increased.
    As thrust will oppose its weight, therefore, mg = mg –Thrust
    or 63301.png
    i.e., 63295.png
  • A bob of mass m carries a positive charge q and pendulum is placed in a uniform electric field of strength E directed vertically upwards (Fig. 4).
    Fig. 4
    In given condition, net downward acceleration,
    So 63264.png
    If the direction of field is vertically downward, then time period,
  • Pendulum in a lift: A pendulum is suspended from the ceiling of the lift.
    • If the lift is at rest or moving downward / upward with constant velocity, then
      63252.png and 63246.png
    • If the lift is moving up ward with constant acceleration a, then
      63239.png and 63233.png
      Time period decreases and frequency increases.
    • If the lift is moving down ward with constant acceleration a, then
      63531.png and 63525.png
      Time period increases and frequency decreases.
    • If the lift is moving down ward with acceleration a = g, then
      63518.png and 63512.png = 0
      It means there will be no oscillation in a pendulum.
      Similar is the case in a satellite and at the center of Earth where effective acceleration becomes zero and pendulum will stop.
  • The time period of simple pendulum whose point of suspension moving horizontally with acceleration a is
    Fig. 5
    63500.png and θ = tan–1(a/g)
  • A simple pendulum is suspended in a car that is moving with constant speed v around a circle of radius r, then
  • Second’s pendulum: It is a simple pendulum whose time period of vibrations is 2/s.
    Putting T = 2 s and g = 9.8 m/s2 in 63488.png, we get
    63720.pngm = 99.3 cm
    Hence, length of second’s pendulum is 99.3 cm or nearly 1 m on earth surface.
  • Various graphs for simple pendulum are shown in Fig. 6.
(a)63714.png (b)63708.png
(c)63702.png  (d)63696.png
Fig. 6

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