# Simple Harmonic Motion And Circular Motion

There is a relation between **Simple Harmonic Motion** And circular motion. A body performing the simple harmonic motion will be in circular motion also.

**Simple Harmonic Motion**

An indirect geometrical method can be used to find mathematical expressions for describing the simple harmonic motion. Let a mass m attached to the end of a vertical spring vibrates simply harmonically with period T, frequency f, and amplitude X0.

The motion of the mass is displayed by the pointer P1 on the line BC with A as the mean position and B and C as the extreme position.

At t= 0, when no force is acting on, the pointer P_{1 }is at means position A.

The pointer moves between extreme positions B and C in a way that it moves from A to B to A, A to C, and then back to A at time instants T/4, T/2, 3T/4, and T respectively. This will complete one cycle of vibration. During this time, the amplitude of vibration will be x_{0.}

**Simple Harmonic Motion And circular motion**

The concept of Simple Harmonic Motion And circular motion is introduced by considering that a point P moves on a circle of radius X_{0}, with uniform angular frequency Ï‰= 2 Ï€ /T. The radius of the circle is the same as the amplitude of vibration of the pointer’s motion. Draw a perpendicular PN on the vertical diameter of the circle and point N is called the projection of the point P on the diameter DE which is parallel to the line of vibration of the pointer.

The points D and E are at the same level as the points B and C. As point P describes the uniform circular motion with a constant **angular velocity **Ï‰, N oscillates to and fro on the diameter DE with the time period T. The position of P at t= 0 is 0_{1} on the circle and N is at the mean position O.

After time T/4 T/2 3T/4 and T the point N will be at D, O, E, and O respectively.

Comparing the motions of N and pointer P_{1}, it is clear that both are identical.

**Instantaneous Displacement**

It is the distance of the projection of point N from the mean position 0 at any instant. As it is clear from the figure that point P makes angle <O_{1}OP= Î¸ then angle NPO is also Î¸. ON/OP=sin Î¸

X/X0=sin Î¸

x=x_{0 }sin Î¸

x=x_{0}sin Ï‰ t

This will also be the displacement of the pointer P_{1}at the instant t.

** The phase of Vibration:**

The value of x is the function of Î¸. it draws a similar waveform that was traced experimentally but here it has been traced theoretically by linking SHM with circular motion.

When Î¸ =0 radian then x= 0

When Î¸ =90^{0}= Ï€ /2radianthen x=x_{0}

When Î¸ =180Â° = Ï€ radian then x = 0

When Î¸ =270^{0}=3 Ï€/2 radian then x=-x_{0}

When Î¸= 360Â°= 2 Ï€ radian then x = 0

Here Î¸ is called the phase of vibration. This phase is also related to the circular motion aspect of SHM.

**Instantaneous velocity**

The linear **velocity** of point p at any instant of time t will be directed along the tangent to the circle and its magnitude will be

v_{p} =r Ï‰

r=x_{0}

v_{p}=x_{0} Ï‰

At the mean position where x=0, velocity is maximum

V= Ï‰ âˆšx_{o}^{2}-x^{2}= Ï‰ âˆšx_{o}^{2}-0= Ï‰ âˆšx_{o}^{2}= Ï‰ x_{o}

At the extreme position where x=x_{o} , velocity is minimum

V= Ï‰ âˆšx_{o}^{2}-x^{2}= Ï‰ âˆšx_{o}^{2}-x_{o}^{2}= Ï‰ âˆšx_{o}^{2}=0

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