Question

For periodic motion of small amplitude \(A\), the time period \(T\) of this particle is proportional to A) \(\mathrm{A} \sqrt{\frac{\mathrm{m}}{\alpha}}\) B) \(\frac{1}{\mathrm{~A}} \sqrt{\frac{\mathrm{m}}{\alpha}}\) C) \(\mathrm{A} \sqrt{\frac{\alpha}{\mathrm{m}}}\) D) \(\frac{1}{\mathrm{~A}} \sqrt{\frac{\alpha}{\mathrm{m}}}\)

Short answer

Option B is the closest to the true formula for the time period of small amplitude periodic motion, considering the given options, but note that it incorrectly suggests an inverse dependence on amplitude.

Step-by-step solution

Understand the Concept of Time Period in Periodic Motion

In periodic motion, small amplitude means that the motion can be considered linear, and the restoring force is proportional to the displacement from the mean position, often modeled by Hooke's law. The time period (T) in such a system is independent of the amplitude (A) of the motion, as for small oscillations, the time period remains constant.

Identify the Form of the Time Period Equation

For a simple harmonic oscillator, the time period (T) is given by the formula T = 2 \(pi\) \( \sqrt{\frac{m}{k}} \), where m is the mass of the oscillating object and k is the spring constant. The parameter \(\alpha\) in the options is analogous to the spring constant k.

Analyze the Given Options

Considering the correct form of the time period equation, and knowing that the time period is independent of amplitude for small oscillations, the only option that does not depend on the amplitude, A, and includes the square root of the ratio of mass to alpha is option B) \(\frac{1}{\mathrm{~A}} \sqrt{\frac{\mathrm{m}}{\alpha}}\). While this option incorrectly implies that the time period is inversely proportional to amplitude, which is not true for small amplitudes, it is still the best match we have. None of the given options correctly represent the time period's independence from amplitude.

Key concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It's characterized by its sinusoidal nature, meaning the motion can be described by sine or cosine functions.

In SHM, any object with mass (m) that’s attached to a spring with spring constant (k), and displaced from its equilibrium position will experience a restoring force (F) that tries to bring it back to equilibrium. This is where the object has no net force acting on it. Such motion is exemplified in systems like a mass on a spring, a swinging pendulum at small angles, or even the oscillation of atoms in molecules.

The time taken to complete one full cycle of motion is called the period (T), which remains constant regardless of the amplitude (A) for small oscillations. The equation for the time period in SHM for a mass-spring system is:
\[T = 2\pi\sqrt{\frac{m}{k}}.\]
Understanding SHM is fundamental for grasping various physical phenomena including sound waves, the motion of pendulums in clocks, and even the behavior of electrons in atoms.

Linear Restoring Force

A linear restoring force is pivotal in the study of oscillatory motion. This force obeys Hooke's law, which states that the force required to extend or compress a spring is proportional to the distance it's stretched or compressed. Mathematically, it is expressed as
\[F = -kx.\]
The minus sign indicates that the force acts in the direction opposite to the displacement from the spring's equilibrium position. For the simplest of harmonic oscillators—a mass attached to a spring—the magnitude of the force increases linearly with the amount of stretch or compression (x).

The 'linear' term signifies that the graph of force versus displacement is a straight line when dealing with ideal springs and small displacements. In terms of energy, the restoring force does work on the object, converting the kinetic energy to potential energy, and vice versa, throughout the oscillation. This relationship is at the heart of systems that exhibit SHM and understanding it helps to further comprehend how different mechanical systems, such as vehicle suspensions or seismographs, operate.

Hooke's Law

Hooke's law is a principle of physics that relates the force (F) needed to extend or compress a spring by some distance (x) with a constant of proportionality (k), known as the spring constant. The law is expressed as
\[F = -kx,\]
where the negative sign shows that the force exerted by the spring is in the opposite direction to the displacement.

Hooke's law is only valid for the elastic limit of the material - that means within the range where the material will return to its original shape after the force is removed. This law is widely applicable - it not only governs the behavior of springs but also is applicable in other contexts where materials exhibit elastic properties.

For students understanding periodic motion and the time period of oscillation, Hooke's law provides the foundation for why the time period of a simple harmonic oscillator does not depend on the amplitude of oscillation. In the practical sense, knowing the spring constant (k) is essential for designing systems with desired vibrational properties, such as in the construction of shock absorbers and musical instruments.