Define displacement. Establish the relation D = √(A² + B²), and write the importance of periodic sine and cosine functions.

**OR**

Derive the expression for the total energy for a particle executing simple harmonic motion.

Question

Define displacement. Establish the relation D = √(A² + B²), and write the importance of periodic sine and cosine functions.

**OR**

Derive the expression for the total energy for a particle executing simple harmonic motion.

Accepted Answer

>

> [!TLDR]
> (a) Combining two same-frequency sine/cosine displacements as $A\sin\omega t+B\cos\omega t$ can be rewritten as a single SHM of amplitude $D=\sqrt{A^2+B^2}$, which is why sine and cosine — the two functions satisfying the SHM equation — are the fundamental building blocks of periodic motion. (b) The total energy of an SHM oscillator stays constant at $	frac12m\omega^2A^2$, as kinetic and potential energy continuously trade off.
>
> **(a) Displacement and $D=\sqrt{A^2+B^2}$:** Displacement, in oscillatory motion, is the distance of the oscillating particle from its mean (equilibrium) position at a given instant, taken with sign/direction. Consider two SHM-type displacements of the same angular frequency $\omega$ added together:
> $$x = A\sin\omega t + B\cos\omega t.$$
> We can write this as a single sinusoid $x = D\sin(\omega t+\phi)$ by expanding: $D\sin(\omega t+\phi) = D\cos\phi\sin\omega t + D\sin\phi\cos\omega t$. Comparing coefficients with the original expression,
> $$D\cos\phi = A, \qquad D\sin\phi = B.$$
> Squaring and adding these two equations:
> $$D^2\cos^2\phi + D^2\sin^2\phi = A^2+B^2 \;\Rightarrow\; D^2(\cos^2\phi+\sin^2\phi) = A^2+B^2 \;\Rightarrow\; D = \sqrt{A^2+B^2}.$$
> So the sum of two perpendicular-phase SHM terms is itself a pure SHM of amplitude $D=\sqrt{A^2+B^2}$ (this is exactly why $y=a\sin\omega t+b\cos\omega t$, as in Q5, is genuine SHM). **Importance of sine and cosine functions:** they are the unique periodic, bounded functions that satisfy the SHM differential equation $\dfrac{d^2y}{dt^2}=-\omega^2y$; because any linear combination of them is again such a function (as shown above), *every* possible simple harmonic oscillation — regardless of its starting phase — can be written using sine and cosine. This is also why they are the essential building blocks (via Fourier analysis) for describing more complex periodic motions.
> …

Q.Define displacement. Establish the relation D = √(A² + B²), and write the importance of periodic sine and cosine functions. OR Derive the expression for the total energy for a particle executing simple harmonic motion.

Concept understanding — Simple Harmonic Motion

Simple Harmonic Motion: The Natural Rhythm of Things

The Intuition: A Restoring Force That Fights Displacement

The Precise Statement

What Does This Motion Look Like?

The Mathematical Description (Derived from $F = -kx$ )

Q.Define displacement. Establish the relation D = √(A² + B²), and write the importance of periodic sine and cosine functions. OR Derive the expression for the total energy for a particle executing simple harmonic motion.

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