Oscillations

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Problem solving tips

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Tip-1: A pendulum clock is running slow or fast
The time period of a pendulum clock gives the time. When this time period is changed, the clock does not give correct time. Following are some causes behind a pendulum clock running slow or fast.

Tip-2: A pendulum is attached to the top of a lift
When a pendulum is attached to the top of a lift, its time period changes with the motion of the lift. It is because as the lift is moving up or down the acceleration of the bob of the pendulum changes.

Tip-3: Series and parallel connection of identical springs
When a few identical springs are connected in series or parallel then we can always replace the combination of springs with a single equivalent spring. In order to do that, we have to find the equivalent spring constant. Following tips will help you to find the equivalent spring constant in series and parallel combination.

Tip-4: Horizontal spring-mass system
In case of horizontal spring-mass system, a mass is connected by multiple springs in different fashion. In order to find the time period of oscillation of the mass, we need to find the net restoring force (F) on the mass and the net extension (x) of the springs. Then we apply the Hooke's law to calculate the equivalent spring constant. In this case, one need to be very careful about the direction of the restoring force exerted by different springs while calculating the net restoring force. The following two systems will make the point clear.

Tip-5: Finding the decay constant of a damped oscillation
In a weakly damped oscillator, the amplitude decreases exponentially with time. But overall it executes some sort of oscillatory motion. The motion does not repeat itself and is, therefore, not periodic in the usual sense. The expression of amplitude with time is

A (t) = A_{0} e^{- \frac{γ t}{2}}

,where

γ

is the decay constant.

Tip-6: Various systems executing simple harmonic motion
Like spring-mass system, there are several other systems that executes simple harmonic motion. Some of these systems are torsional pendulum, liquid in a U-tube, floating object etc. To find the time period of oscillations for this kind of systems, one need to find the expression of restoring force. This expression will directly tell us the angular frequency of oscillation. Once we know the angular frequency, we can easily calculate the time period.