14.11 Figures 14.25 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.
At time, t = 0, the radius vector makes an angle with the positive x-axis, i.e. phase angle = +
Therefore, the equation of simple harmonic motion for the x-projection of the radius vector, at time t is given by the displacement equation:
x = Acos = 3cos = -3sin ( ) = -3sin cm
(b) Time period, T = 4 s, Amplitude A = 2 m
At time, t = 0, the radius vector makes an angle with the positive x-axis, i.e. phase angle = +
Therefore, the equation of simple harmonic motion for the x-projection of the r
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(a) Time period, T = 2 s, Amplitude A = 3 cm
At time, t = 0, the radius vector makes an angle with the positive x-axis, i.e. phase angle = +
Therefore, the equation of simple harmonic motion for the x-projection of the radius vector, at time t is given by the displacement equation:
x = Acos = 3cos = -3sin ( ) = -3sin cm
(b) Time period, T = 4 s, Amplitude A = 2 m
At time, t = 0, the radius vector makes an angle with the positive x-axis, i.e. phase angle = +
Therefore, the equation of simple harmonic motion for the x-projection of the radius vector, at time t is given by the displacement equation:
x = Acos = 2cos = -2cos ( m
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<p><strong> (a)</strong> Time period, T = 2 s, Amplitude A = 3 cm</p><p>At time, t = 0, the radius vector makes an angle <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>π</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac></math></span> with the positive x-axis, i.e. phase angle <span title="Click to copy mathml"><math><mi>φ</mi></math></span> = + <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>π</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac></math></span></p><p>Therefore, the equation of simple harmonic motion for the x-projection of the radius vector, at time t is given by the displacement equation:</p><p>x = Acos <span title="Click to copy mathml"><math><mfenced open=" [" close="]" separators="|"><mrow><mrow><mfrac><mrow><mrow><mn>2</mn><mi>π</mi><mi>t</mi></mrow></mrow><mrow><mrow><mi>T</mi></mrow></mrow></mfrac><mo>+</mo><mi></mi><mi>φ</mi></mrow></mrow></mfenced></math></span> = 3cos <span title="Click to copy mathml"><math><mfenced open=" [" close="]" separators="|"><mrow><mrow><mfrac><mrow><mrow><mn>2</mn><mi>π</mi><mi>t</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac><mo>+</mo><mi></mi><mfrac><mrow><mrow><mi>π</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac></mrow></mrow></mfenced></math></span> = -3sin ( <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>2</mn><mi>π</mi><mi>t</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac></math></span> ) = -3sin <span title="Click to copy mathml"><math><mi>π</mi><mi>t</mi></math></span> cm</p><p> </p><p><strong> (b)</strong> Time period, T = 4 s, Amplitude A = 2 m</p><p>At time, t = 0, the radius vector makes an angle <span title="Click to copy mathml"><math><mi>π</mi></math></span> with the positive x-axis, i.e. phase angle <span title="Click to copy mathml"><math><mi>φ</mi></math></span> = + <span title="Click to copy mathml"><math><mi>π</mi></math></span></p><p>Therefore, the equation of simple harmonic motion for the x-projection of the radius vector, at time t is given by the displacement equation:</p><p>x = Acos <span title="Click to copy mathml"><math><mfenced open=" [" close="]" separators="|"><mrow><mrow><mfrac><mrow><mrow><mn>2</mn><mi>π</mi><mi>t</mi></mrow></mrow><mrow><mrow><mi>T</mi></mrow></mrow></mfrac><mo>+</mo><mi></mi><mi>φ</mi></mrow></mrow></mfenced></math></span> = 2cos <span title="Click to copy mathml"><math><mfenced open=" [" close="]" separators="|"><mrow><mrow><mfrac><mrow><mrow><mn>2</mn><mi>π</mi><mi>t</mi></mrow></mrow><mrow><mrow><mn>4</mn></mrow></mrow></mfrac><mo>+</mo><mi></mi><mi>π</mi></mrow></mrow></mfenced></math></span> = -2cos ( <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>π</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac><mi>t</mi><mo>)</mo></math></span> m</p>
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