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| The Circle of Reference |
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| The figure shows a flat circular disc rotating with uniform speed in the vertical plane about a horizontal axis passing through its
center. A parallel beam of light is made to fall on the disc. A peg 'Q' on the rim of the disc executes uniform circular motion casting a shadow on the screen. This shadow 'P' of the peg is observed on a vertical plane. This shadow is found to execute SHM, with its mean position at the mid point of the line described. |
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The above figure shows a body moving with uniform speed along a circle. Let perpendiculars be drawn from different positions of the body onto any diameter, say the vertical diameter. Let T represent the time taken by the body to complete one revolution. This can be divided into
eight equal intervals namely  and so on.
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| A, B, C, D.... I represent the positions of the body at these instants. When the body is at A, the foot of the perpendicular will be at Po, the
center of the circle. When the body moves from A to B, the foot of the perpendicular moves Po to P1. |
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| When the body moves from B to C, the foot of the perpendicular moves from P1 to P2. It is seen from the figure that P2 coincides with C. When the body moves from C to D, the foot of the perpendicular moves from P2 to P3. It is seen from the figure that P3 coincides with P1. The displacements of the foot of the perpendicular are represented in the adjacent graph. The curve resembles a sine curve. |
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| From the above discussion, it follows that SHM can have an alternative definition as -“when a particle executes uniform circular motion, the foot of the perpendicular drawn from the particle on any diameter of the circle, executes SHM”. In other words, SHM is the projection of uniform circular motion on the diameter. The circle in the above case is called the circle of reference. |
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| The maximum displacement of the particle executing SHM from the equilibrium position is known as the amplitude. It will be equal to the radius of the circle of reference. Therefore, the total range of motion is 2A. |
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| It is the time taken by the particle executing SHM, to complete one oscillation i.e., a to and fro motion. It will be equal to the time taken by the body to complete one revolution or the circle of reference. |
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| The number of oscillations of a particle executing SHM in one second is called the frequency. It is the reciprocal of the period i.e.,
n = 1/T. It will be equal to the number of revolutions completed per second by the body on the circle of reference. Its unit is cycles per second or hertz. |
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