Oscillation is the back-and-forth motion of an object between two positions about a fixed point, called the mean or equilibrium position. This motion repeats itself at regular intervals and is therefore a type of periodic motion. One complete oscillation is defined as one full to-and-fro movement about the equilibrium position.
In oscillatory motion,, a body moves repeatedly on either side of a fixed point after a definite time interval. The point about which the body oscillates is known as the equilibrium or mean position. While all oscillatory motions are periodic, not all periodic motions are oscillatory.
Common examples of oscillatory motion include the vibration of a sitar string, a mass suspended from a spring, and the motion of a simple pendulum. Among these, the simple pendulum is one of the most important examples of oscillatory motion and helps in understanding the basic principles of oscillations.
Simple Pendulum
A pendulum is a mechanical setup that shows an oscillatory motion. The simple pendulum is a small bob of mass 'm' suspended by a thin string attached to the free end at an upper edge with length L along its platform. The image of a simple pendulum is added below.
A simple pendulum period can be made longer by elongating the string and taking measurements from the point of suspension to the middle part between two points, which is on a bob. However, it should be emphasized that when the mass of the bob is modified, the period will not vary. This is because the strength of the gravitational field is not constant everywhere, and hence this period depends primarily on the pendulum’s position from Earth.
Oscillation Motion
Oscillatory motion is the back-and-forth movement of an object about a central position, called equilibrium. This type of motion, known as oscillations or cycles, is a fundamental concept in physics and engineering. In oscillatory motion the object moves periodically away from its equilibrium position, reaches a maximum displacement, returns to equilibrium, and then moves to the opposite extreme, repeating this pattern continuously.
Oscillation of Wave:- It is a back-and-forth motion of an object between two points of deformation. Oscillation in any medium creates a wave, which is a disturbance that propagates from where it was created in all possible directions.
The amplitude of oscillation is the maximum distance traveled from its mean position. It is the distance between two crests or two troughs.
Oscillation Types
Various types of oscillations are,
- Damped Oscillations
- Undamped Oscillations
- Free Oscillation
- Resonance
- Coupled Oscillation
A. Damped Oscillation
Damped oscillation is an oscillatory motion in which the amplitude decreases gradually with each cycle due to energy loss. In contrast, an undamped oscillation maintains constant amplitude. Damped oscillations are of two types: underdamped, where oscillations continue with decreasing amplitude, and overdamped, where the system returns to equilibrium without oscillating. Damping is caused by factors like friction or air resistance and is often used to control or stabilize oscillating systems.
Different Types of Damped Oscillations
Various types of damped oscillations are,
- Underdamped Oscillations
- Critically Damped Oscillations
- Overdamped Oscillations
1. Underdamped Oscillations
Underdamping oscillation is a type of the damped system that, after some time, returns back to its own equilibrium position but with very small amplitudes. That is, the force damping applied to such a system does not have sufficient strength to rapidly bring it to rest.
The behavior of the motion in an underdamped system shows a very typical trend. The change in the position of the oscillating object from its resting point obeys a sinusoidal or an exponential decay function. Each subsequent oscillation becomes less, and the system takes a lot of time to return completely to rest.
2. Critically Damped Oscillation
In the critically damped oscillation, the system does not cede any energy. In such an oscillation, the damping is precisely proportional to the displacement, precluding any overshoot or undershoot. The system reaches equilibrium in the shortest possible time with no oscillations. Unlike an underdamped system that does not return to the equilibrium as rapidly and continues in oscillatory mode around the steady state point, this type is a critically damped system. Critically damped systems are widely used in various engineering applications requiring a quick return to equilibrium and minimal vibration.
3. Overdamped Oscillations
Overdamped oscillation is when the oscillations slowly get closer to the equilibrium point without bouncing around it. In the case of overdamped oscillations, the damping constant is greater than one. The energy dissipation in an overdamped oscillating system is larger compared to the other modes of damping.
In the condition of overdamped oscillations, more damping is provided than necessary to bring about a rest or equilibrium point, and therefore it takes a large amount of time as compared with other methods. Under this mechanism, no further shuttling movement will take place beyond the equilibrium point.
B. Undamped Oscillations
The undamped oscillation describes a periodic motion, where being displaced from the point of equilibrium causes a restoring force proportional to its position. Therefore, in the undamped system, the oscillation never disappears, and its magnitude stays at a constant value. Alternating current (AC wave) is an example of an undamped oscillation.
Alternating current varies between two values across the neutral position, and it repeats with no change in magnitude or the time period. The amplitude of the signal does not change over time, and there is no hindering force in the alternating currents.
C. Free Oscillation
A free oscillation occurs when the system vibrates or has an oscillatory motion caused by forces intrinsic to that particular system and which is not disturbed by external actions. The natural frequency of the system refers to free vibration.
An example of free oscillation is when a pendulum moves left and right after initiating its displacement from equilibrium. Had air resistance or friction been nonexistent, the pendulum would have swung back and forth without end at its frequency of resonance.
D. Resonance
Resonance is a phenomenon that takes place when an external force or frequency applied to the system coincides with its own natural frequency. This leads to a substantial growth in the amplitude of the oscillations. Resonance causes vibrations, and boosting is an essential process in many disciplines, including physics and engineering, but also music too.
E. Coupled Oscillation
With regard to coupled oscillation, it means that there is more than one system or an oscillator that interacts with each other. The movement of one system can affect the other and vice versa, resulting in a coupled or linked behavior. This coupling may involve the physical links or energy transfer between the oscillators.
Oscillation Formula
1. Oscillation Period Formula
The time period in oscillation is calculated using the formula,
where,
- L is the length of the pendulum.
- g is acceleration due to gravity.
2. Frequency Formula Oscillation
Frequency is the inverse of time period and is calculated by the formula,
where,
- f is the frequency of oscillation.
- T is the time period of oscillation.
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Solved Examples
Question 1: A simple pendulum has a length of 1.0 m. Calculate its time period. (Take g = 9.8 m/s²)
Solution:
T = 2\pi \sqrt{\frac{L}{g}}
T = 2\pi \sqrt{\frac{1.0}{9.8}}
T = 2\pi \sqrt{0.102}
T = 2\pi (0.319)
T \approx 2.0\,\text{s}
Question 2: A pendulum oscillates 40 times in 80 seconds. Calculate the time period and frequency.
Solution: Time Period
T = \frac{Total Time}{Number of oscillation}
T = \frac{80}{40}
T = 2s Frequency
f = \frac{1}{T} = \frac{1}{2}
= 0.5 Hz Therefore Time period = 2 s & Frequency = 0.5 Hz
Question 3: The length of a simple pendulum is increased from 0.5 m to 2 m. Find the ratio of their time periods.
Solution:
T \propto \sqrt{L}
\frac{T_1}{T_2} = \sqrt{\frac{L_1}{L_2}}
\frac{T_1}{T_2} = \sqrt{\frac{0.5}{2}}
\frac{T_1}{T_2} = \sqrt{\frac{1}{4}}
\frac{T_1}{T_2} = \frac{1}{2} Ratio of time periods = 1 : 2
Question 4: A wave completes 120 oscillations in 3 minutes. Find the frequency and time period.
Solution: Convert time into seconds: 3 minutes = 180 s
Frequency
f = \frac{120}{180}
f = 0.67 Hz Time Period
T = \frac{1}{f}
T = \frac{1}{0.67}
T ≈ 1.5s Therefore Frequency = 0.67 Hz & Time period ≈ 1.5 s
Unsolved Problems
Question 1: A simple pendulum has a length of 0.8 m. Calculate its time period. (g=9.8 m/s2)
Question 2: A pendulum completes 60 oscillations in 120 s. Find its frequency and time period.
Question 3: The length of a simple pendulum is changed from 0.25 m to 1.0 m. Find the ratio of their time periods.
Question 4: A simple pendulum has a time period of 3 s. Calculate the length of the pendulum.
(g=9.8 m/s2).
Question 5: A pendulum oscillates with a frequency of 0.5 Hz. Find its time period and the number of oscillations in five minutes.