I. OBJECTIVE
To study the behaviour of oscillations (Simple Harmonic Motion) in situation where there is resistive force of fluid investigated by study of exponential amplitude decay in the fluid as compared to relatively undamped environment of air.
II. HYPOTHESIS
During Simple Harmonic Motion...
To study the behaviour of oscillations (Simple Harmonic Motion) in situation where there is
resistive force of fluid investigated by study of exponential amplitude decay in the fluid as
compared to relatively undamped environment of air.
II. HYPOTHESIS
During Simple Harmonic Motion (SHM) in damped fluid environment amplitude of
oscillation decreases exponentially. And the rate of amplitude decay depends on viscosity of
fluid medium.
III. THEORY
Simple Harmonic Motion (SHM), is a type of motion in which there is an restoring force
which is always directed towards a unique point. In ideal conditions force equations on a
object undergoing SHM is given by Hooke’s law,
F = -kx
Where, x is displacement from 0 stretch of spring and,
k is the spring constant of the spring.
Upon solving above equation, general form of the displacement, for SHM can be given as,
x = Acos(wt+Φ)
where, A is amplitude,
w is angular frequency and,
Φ is phase of the oscillation.
, w = √(k/m) , k is spring constant and m = mass of attached material to spring
T = 2π/w, T is time period
Thus, velocity of an object undergoing SHM can be written as,
v = -wAsin(wt+Φ) , v is velocity.
Now, if there is a fluid in the system this the resistivity of fluid can be defined as viscosity.
This viscosity provides the damping influence on the SHM and due to resistance dissipates
the energy of SHM making its amplitude smaller, and smaller, and this damping influence
can be given analysed using,
F = -kx-Fdamping
and Fdamping = -bv , b is drag constant, generally representative of viscosity and friction always
acting opposite of the motion.
Upon solving above equation,
we get, x = Ae(-b/2m)cos(wt+Φ)
Also, w = sqrt(wo2-b2/(4m2))
and, wo =sqrt(k/m)
Thus it can be said from above equations that damping reduces both the amplitude and
angular frequency with time for a damped SHM. Also it should be noted that as b in increases
the amplitude decreases by the factor of exponential factor of b. Hence it can be predicted
that for fluids with larger viscosity rate of decay of Amplitude will be larger.
Also, it should be noted that viscosity, b = 2[ps – pl]*ga2/(9v)
where, a is radius of sphere or a2 is surface area on which resistive force of fluid is acting.
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