C.1.2 Time period of oscillatory systems

Period of Oscillations in a Mass-Spring System

1. In a mass-spring system, a mass $m$ is attached to a spring with a spring constant $k$.
2. When the mass is displaced from its equilibrium position and released, it oscillates back and forth.
3. The period of these oscillations is given by:

$$T=2\pi \sqrt{\frac{m}{k}}$$

Derivation of the Formula

• Restoring Force: The spring exerts a force proportional to the displacement $x$, described by Hooke’s Law:

$$F=-kx$$

• Newton’s Second Law: The force causes an acceleration $a$, so $$ma=-kx$$
• Acceleration in SHM: In simple harmonic motion, $a=-{\omega }^{2}x$, where $\omega$ is the angular frequency.
• Equating the Two Expressions:

$$ma=-kx⟹a=-\frac{k}{m}x$$

• Angular Frequency: Comparing with $a=-{\omega }^{2}x$, we find

$${\omega }^2=\frac{k}{m}$$

• Period: The period $T$ is related to angular frequency by $T=\frac{2\pi }{\omega }$, so:

$$T=2\pi \sqrt{\frac{m}{k}}$$

Key Insights

• The period increases with mass $m$. A heavier mass oscillates more slowly.
• The period decreases with a stiffer spring $k$. A stiffer spring pulls the mass back more quickly.

Period of Oscillations in a Simple Pendulum

1. A simple pendulum consists of a mass $m$ (the bob) attached to a string of length $l$.
2. When displaced and released, it swings back and forth under the influence of gravity.
3. The period of a simple pendulum is given by:

$$T=2\pi \sqrt{\frac{l}{g}}$$

Derivation of the Formula

• Restoring Force: When the pendulum is displaced by a small angle $\theta$, the component of gravitational force acting along the arc is: $$F=-mg\sin \theta$$
• Small Angle Approximation: For small angles, $\sin \theta \approx \theta$ (in radians), so: $$F\approx -mg\theta$$
• Angular Displacement: The arc length $x$ is related to the angle by $x=l\theta$, so: $$\theta =\frac{x}{l}$$
• Substitute: $$F\approx -mg\frac{x}{l}=-\frac{mg}{l}x$$
• Newton’s Second Law: For rotational motion, $F=ma$ becomes: $$m\frac{d^{2}x}{d{t}^2}=-\frac{mg}{l}x$$
• Angular Frequency: Comparing with $a=-{\omega }^{2}x$, we find $${\omega }^2=\frac{g}{l}$$
• Period: The period $T$ is related to angular frequency by $T=\frac{2\pi }{\omega }$, so:

$$T=2\pi \sqrt{\frac{l}{g}}$$

Key Insights

• The period depends only on the length of the pendulum and the gravitational acceleration$g$.
• It is independent of the mass of the bob and the amplitude (as long as the amplitude is small).

Comparing the Two Systems

Both the mass-spring system and the simple pendulum exhibit simple harmonic motion, but their periods depend on different factors:

• Mass-Spring System: The period depends on the mass $m$ and the spring constant $k$.
• Simple Pendulum: The period depends on the length $l$ and gravitational acceleration $g$.

Reflection and Applications

Understanding these systems helps in designing clocks, measuring gravitational acceleration, and analyzing vibrations in engineering structures.