If force (F) versus displacement (x) and displacement (x)...

If force $(F)$ versus displacement $(x)$ and displacement $(x)$ versus time graph of a particle performing SHM is shown in Fig. 9.33. Then choose correct statement. (A) Mass of the particle $\frac{160}{\pi^{2}} \mathrm{~kg}$. (B) Mass of the particle $160 \pi^{2} \mathrm{~kg}$. (C) Maximum kinetic energy of particle is $80 \mathrm{~J}$. (D) Maximum kinetic energy of particle is $40 \mathrm{~J}$.

If force $(F)$ versus displacement $(x)$ and displacement $(x)$ versus time graph of a particle performing SHM is shown in Fig. 9.33. Then choose correct statement.
(A) Mass of the particle $\frac{160}{\pi^{2}} \mathrm{~kg}$.
(B) Mass of the particle $160 \pi^{2} \mathrm{~kg}$.
(C) Maximum kinetic energy of particle is $80 \mathrm{~J}$.
(D) Maximum kinetic energy of particle is $40 \mathrm{~J}$.

Key Concepts

Simple Harmonic Motion (SHM)

SHM is a type of oscillatory motion in which the restoring force is directly proportional to the displacement from an equilibrium position and acts in the opposite direction. This creates a sinusoidal motion characteristic, where parameters like amplitude, period, and frequency play key roles in describing the motion.

Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to its displacement from the equilibrium position, with the constant of proportionality termed the spring constant. In the context of SHM, this relationship is the underlying principle governing the restoring force, linking mechanics with oscillatory behavior.

Force-Displacement Graph

The force-displacement graph is a graphical representation that illustrates the relationship between the force acting on a particle and its displacement. In SHM, this graph is typically linear, where the slope is directly related to the spring constant or system stiffness, enabling analysis of the dynamic characteristics of the motion.

Displacement-Time Graph

The displacement-time graph of a particle in SHM shows the sinusoidal variation of displacement over time. This graph provides key information about the amplitude, period, and phase of the motion, and is instrumental in determining the oscillatory parameters and the temporal behavior of the system.

Angular Frequency

Angular frequency is a measure of how quickly an oscillatory system moves through its cycle, defined as 2? divided by the period of oscillation. In SHM, it relates directly to both the system's mass and the stiffness (spring constant) via the relation ?² = k/m, thereby serving as a crucial parameter for analyzing the motion's dynamics.

Kinetic Energy in SHM

In SHM, the kinetic energy of the moving particle is a function of both its mass and velocity. The maximum kinetic energy occurs when the particle passes through the equilibrium point, where its speed is greatest. Understanding this energy exchange between kinetic and potential forms is essential for describing the conservation of energy within oscillatory systems.

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