If the displacement of a body of mass m is given by x=a sinωt, then its kinetic energy is (a) (1

step 1 we have problem number 86 which is quite a fundamental problem simple problem we have given disp

If the displacement of a body of mass m is given by x=a...

If the displacement of a body of mass $m$ is given by $x=a \sin \omega t$, then its kinetic energy is (a) $\frac{1}{2} \operatorname{ma}^{2} \omega^{2}[1+\cos \omega \mathrm{t}]$ (b) $\frac{1}{4} \operatorname{ma}^{2} \omega^{2}[1+\sin 2 \omega \mathrm{t}]$ (c) $\frac{1}{4} \operatorname{ma}^{2} \omega^{2}[1+\cos 2 \omega \mathrm{t}]$ (d) $\frac{1}{4} \operatorname{ma}^{2} \omega^{2} \sin ^{2} \omega \mathrm{t}$

If the displacement of a body of mass $m$ is given by $x=a \sin \omega t$, then its kinetic energy is
(a) $\frac{1}{2} \operatorname{ma}^{2} \omega^{2}[1+\cos \omega \mathrm{t}]$
(b) $\frac{1}{4} \operatorname{ma}^{2} \omega^{2}[1+\sin 2 \omega \mathrm{t}]$
(c) $\frac{1}{4} \operatorname{ma}^{2} \omega^{2}[1+\cos 2 \omega \mathrm{t}]$
(d) $\frac{1}{4} \operatorname{ma}^{2} \omega^{2} \sin ^{2} \omega \mathrm{t}$

Key Concepts

Simple Harmonic Motion (SHM)

Simple harmonic motion describes the oscillatory motion of systems where the restoring force is proportional to the displacement from equilibrium. The motion is typically represented by sinusoidal functions, characterized by parameters like amplitude and angular frequency, which define the periodic nature of the system.

Velocity Derivation from Displacement

When a body undergoes SHM, its velocity is obtained by differentiating its displacement function with respect to time. This process introduces the angular frequency and results in a trigonometric function that describes the instantaneous rate of change of displacement, which is essential for kinetic energy calculations.

Kinetic Energy Calculation

The kinetic energy of an object is determined by the expression (1/2)mv², where m is the mass and v is the instantaneous velocity. In contexts like SHM, substituting the velocity function derived from the displacement leads to an expression involving squared sine or cosine functions, reflecting the periodic variations in energy.

Trigonometric Identities

Trigonometric identities, especially the double-angle formulas, simplify the expressions that arise from squaring sinusoidal functions. These identities enable the conversion of squared sine or cosine terms into forms involving cosine of double the angle, which can make the resulting kinetic energy expressions more compact and easier to interpret.

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