A particle is executing SHM with amplitude A and has a...

A particle is executing SHM with amplitude $A$ and has a maximum velocity $V_{o^{\circ}}$. The displacement at which its velocity will be ( $\left.V_{a} / 2\right)$ and the velocity at displacement $A / 2$ are (a) $\frac{A}{2}, \frac{V_{e}}{2}$ (b) $\frac{A}{3}, \frac{V_{n}}{3}$ (c) $\left(\frac{\sqrt{3}}{2}\right) A, \frac{\sqrt{3} V}{2}$ (d) $\frac{A}{\sqrt{2}}, \frac{V_{a}}{\sqrt{2}}$

A particle is executing SHM with amplitude $A$ and has a maximum velocity $V_{o^{\circ}}$. The displacement at which its velocity will be ( $\left.V_{a} / 2\right)$ and the velocity at displacement $A / 2$ are
(a) $\frac{A}{2}, \frac{V_{e}}{2}$
(b) $\frac{A}{3}, \frac{V_{n}}{3}$
(c) $\left(\frac{\sqrt{3}}{2}\right) A, \frac{\sqrt{3} V}{2}$
(d) $\frac{A}{\sqrt{2}}, \frac{V_{a}}{\sqrt{2}}$

Key Concepts

Velocity-Displacement Relationship

The relationship between velocity and displacement in SHM is given by the equation v = ??(A² - x²), where ? is the angular frequency, A is the amplitude, and x is the displacement. This relationship, which stems from the conservation of energy, allows one to calculate the instantaneous velocity at a given displacement and forms the basis for solving problems involving specific velocity or displacement values.

Maximum Velocity

The maximum velocity in SHM occurs as the particle passes through the equilibrium position. It is derived from the conservation of energy principle and is directly related to the amplitude and angular frequency. Understanding this concept is essential for computing velocity at any other position in the motion.

Energy Conservation in SHM

In SHM, the conservation of energy principle states that the sum of kinetic energy and potential energy remains constant throughout the motion. At the maximum displacement, the energy is entirely potential, whereas at the equilibrium position, it is entirely kinetic. This principle is fundamental for deriving relationships between displacement and velocity, such as determining the velocity at a displacement of A/2.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force acting on a particle is directly proportional to its displacement from the equilibrium position and is directed towards that equilibrium position. This leads to oscillatory motion described mathematically by sine and cosine functions, which serve as the backbone for analyzing the behavior of systems undergoing SHM.

Amplitude

The amplitude in SHM is the maximum displacement of the particle from its equilibrium position. It represents the extent of the oscillation and is a crucial parameter in determining other attributes such as the maximum velocity and energy distribution of the oscillating system.

Transcript

Please give Ace some feedback