CBSE (Science) Class 11CBSE
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Figures Correspond to Two Circular Motions. the Radius of the Circle, the Period of Revolution, the Initial Position, and the Sense of Revolution (I.E. Clockwise Or Anti-clockwise) Are Indicated on Each Figure Obtain the Corresponding Simple Harmonic Motions of The X-projection of the Radius Vector of the Revolving Particle P, in Each Case - CBSE (Science) Class 11 - Physics

ConceptSimple Harmonic Motion and Uniform Circular Motion

Question

Figures correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i.e. clockwise or anti-clockwise) are indicated on each figure

Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P, in each case.

Solution 1

1) Let A be any point on the circle of reference of the figure (a) From A, draw BN perpendicular on x axis

if  angle POA = theta, then

angleOAM =  theta = omegat

:. In triangle OAM,

(OM)/(OA) = sintheta

:. (-x)/3 = omegat  = sin (2pi)/T t

:. x = -3 sin (2pi)/2 t or x = -3sin pit which is the equation of SHM.

2) Let B be any point on the circle of reference of figure (b). From B draw BN perpendicular on x-axis

Then triangleBON =  theta =  omegat

:. In triangleONB, cos theta = (ON)/(OB)

or ON = OB cos theta

:. - x = 2 cos omega t

=> x =- 2 cos (2pi)/T t = -2 cos  (2pi)/4 t

:. x = - 2 cos  pi/4 t which is equation of SHM

Solution 2

a) Time period, = 2 s

Amplitude, A = 3 cm

At time, = 0, the radius vector OP makes an angle pi/2 with the positive x-axis, i.e.,  phase angle phi = + pi/2

Therefore, the equation of simple harmonic motion for the x-projection of OP, at time t, is given by the displacement equation:

x = A cos[(2pit)/T + phi]

= 3 cos ((2pit)/2 + pi/2) = -3sin ((2pit)/2)

:. x = - 3 sinpit " cm"

(b) Time period, = 4 s

Amplitude, a = 2 m

At time t = 0, OP makes an angle π with the x-axis, in the anticlockwise direction. Hence, phase angle, Φ = + π

Therefore, the equation of simple harmonic motion for the x-projection of OP, at time t, is given as:

x = acos ((2pit)/T + phi ) = 2 cos ((2pit)/4 + pi)

:. x = - 2 cos (pi/2 t) m

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Solution Figures Correspond to Two Circular Motions. the Radius of the Circle, the Period of Revolution, the Initial Position, and the Sense of Revolution (I.E. Clockwise Or Anti-clockwise) Are Indicated on Each Figure Obtain the Corresponding Simple Harmonic Motions of The X-projection of the Radius Vector of the Revolving Particle P, in Each Case Concept: Simple Harmonic Motion and Uniform Circular Motion.
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