The rate of decay of oscillations is known as logarithmic decrement.



1. Oscillations:
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. In mechanical systems, oscillation refers to the repetitive motion of a mass or system of masses around a central equilibrium position.

2. Damping:
Damping is the process of reducing the amplitude of oscillations. It is a dissipative force that opposes the motion of the system and converts the mechanical energy into heat energy.

3. Logarithmic Decrement:
Logarithmic decrement is a measure of the rate at which the amplitude of oscillations decreases over time. It is defined as the natural logarithm of the ratio of the amplitude of one oscillation to the amplitude of the next oscillation.

4. Rate of Decay:
The rate of decay of oscillations refers to how quickly the amplitude of oscillations decreases over time. It is a measure of the effectiveness of the damping in reducing the amplitude.

5. Relationship with Logarithmic Decrement:
The logarithmic decrement is directly related to the rate of decay of oscillations. A larger logarithmic decrement indicates a faster rate of decay, meaning that the oscillations will die out more quickly. Conversely, a smaller logarithmic decrement indicates a slower rate of decay, meaning that the oscillations will persist for a longer time.

6. Other Options:
- Option a) Critical damping: Critical damping refers to the condition where the damping is just enough to prevent oscillations from occurring. It is not directly related to the rate of decay of oscillations.
- Option b) Damping factor: The damping factor is a dimensionless measure of the damping in a system. While it is related to the rate of decay, it is not the specific term used to describe the rate of decay.
- Option d) Damping coefficient: The damping coefficient is a parameter that quantifies the damping in a system. It is related to the rate of decay but does not specifically refer to it.

Conclusion:
The rate of decay of oscillations is known as logarithmic decrement. It is a measure of how quickly the amplitude of oscillations decreases over time.