The equation of a particle executing SHM is given by x=3 cos((π)/(2)) t cm, where t is in secon

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The equation of a particle executing SHM is given by x=3...

The equation of a particle executing SHM is given by $x=3 \cos \left(\frac{\pi}{2}\right) t \mathrm{~cm}$, where $t$ is in second. The distance travelled by the particle in the first $8.5 \mathrm{~s}$ is $\left(24+\frac{n}{\sqrt{2}}\right)$ then the value of $n$ is.

Key Concepts

Total Distance Traveled in Oscillatory Motion

The total distance traveled in oscillatory motion in SHM is not simply the net displacement but the sum of the absolute displacements over the duration of the motion. This involves accounting for the directional changes at each turning point, making it necessary to consider the path taken during full and partial oscillation cycles.

Period of Oscillation

The period of oscillation is the time taken to complete one full cycle of motion in SHM. It is inversely related to the angular frequency and is an important parameter for understanding the timing of events within the oscillation, such as maxima, minima, and zero crossings of the displacement function.

Amplitude

Amplitude is the maximum displacement of the oscillating particle from its equilibrium position. In SHM, it represents the peak value of the oscillation, and it is a key parameter because it determines the maximum extent of the motion, though it does not affect the period or frequency of the oscillations.

Simple Harmonic Motion (SHM)

SHM is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It is characterized by sinusoidal oscillations with constant amplitude and period, making it a fundamental concept in physics and engineering that models many natural and mechanical systems.

Angular Frequency

Angular frequency is the rate at which the oscillation cycle is completed, measured in radians per second. It is derived from the argument of the sinusoidal function describing the motion and is critical in determining the period of the oscillation as well as the dynamic behavior of the system.

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