Question

If the time period of undamped oscillation is \(\mathrm{T}\) and that of damped oscillation is \(\mathrm{T}^{\prime}\), then what is the relation between \(\mathrm{T} \& \mathrm{~T}^{\prime}\) (A) \(\mathrm{T}^{\prime}<\mathrm{T}\) (B) nothing can be said, unless the drag force constant is known. (C) \(\mathrm{T}^{\prime}=\mathrm{T}\) (D) \(\mathrm{T}^{\prime}>\mathrm{T}\)

Short answer

The relation between the time period of undamped oscillation (T) and that of damped oscillation (T') cannot be determined without knowing the drag force constant. The correct answer is (B) nothing can be said, unless the drag force constant is known.

Step-by-step solution

Understand oscillation concepts

Undamped oscillation refers to a system that oscillates without any external opposing forces, such as friction or air resistance. This results in the system oscillating with a constant amplitude and time period (T). On the other hand, damped oscillation refers to a system that experiences an external force that resists its motion, causing the amplitude to decrease over time. The damped oscillation has a time period (T') influenced by the I damping force.

Relating undamped and damped oscillation periods

The presence of damping force changes the time period of oscillation. Since the damping force acts as a resistance to the system's natural oscillation, it either shortens or lengthens the period of oscillation depending on the force. Therefore, given the options, we cannot say precisely whether T' is greater than or less than T without knowing the drag force constant. Thus, the correct answer is: (B) nothing can be said unless the drag force constant is known.

Key concepts

Time Period

When we talk about oscillations, understanding the concept of the time period is crucial. The time period of an oscillation is the time it takes for one complete cycle of the wave or oscillation to occur. It is often denoted by the symbol \(T\). In an undamped system, where no external forces such as friction are acting, the time period remains constant. The system oscillates back and forth at a steady rate without any change in the duration of its cycles.

For a damped oscillation, where damping forces like air resistance or friction are present, the time period becomes \(T'\). This damping introduces a resistance that affects the oscillation, potentially altering the time each cycle takes. However, whether it shortens or lengthens the time period heavily depends on the specific damping force acting on the system.

Damping Force

A damping force in an oscillatory system is any force that opposes the motion, usually arising due to friction or resistance from a medium like air. This force is crucial as it causes the amplitude of the oscillation to decrease over time.

As the damping force works against the natural oscillation, energy is lost from the system, and this results in a decay of amplitude. This force can be thought of as a kind of "resistance", and it significantly impacts the motion characteristics of the system. Depending on the magnitude of the damping force, it can also change the time period \(T'\) of the oscillation. A larger damping force might not only reduce the amplitude faster, but it might also alter the time the system takes to complete a full oscillation cycle.

Undamped Oscillation

In undamped oscillation, there are no external forces like friction or air resistance acting on the system. This results in an oscillation that maintains a constant amplitude and time period (denoted as \(T\)). Such oscillations are theoretically ideal because no energy is lost over time.

These oscillations serve as a fundamental benchmark for understanding how real-world forces, such as damping, affect systems. When damping is introduced, the clear-cut pattern of undamped oscillations helps us infer changes in both amplitude and time period. Comparing damped and undamped systems provides insights into how energy dissipation and damping forces influence oscillatory behavior.