For the harmonic travelling wave y = 2 cos 2π (10t–0.0080x + 3.5) where x and y are in cm and t is second. What is the phase difference between the oscillatory motion at two points separated by a distance of

(a) 4 m

(b) 0.5 m

(c) 2 λ

(d) 3 4 λ (at a given instant of time)

(e) What is the phase difference between the oscillation of a particle located at x = 100cm, at t = T s and t = 5 s

<p><span data-teams="true">This is a long answer type question as classified in NCERT Exemplar</span></p><p>Wave functions y= 2cos2<span contenteditable="false"> <math> <mi>π</mi> </math> </span>(10t-0.0080x + 3.5)</p><p>= 2cos(20<span contenteditable="false"> <math> <mi>π</mi> <mi>t</mi> <mo>-</mo> <mn>0.016</mn> <mi>π</mi> <mi>x</mi> <mo>+</mo> <mn>7</mn> <mi>π</mi> </math> </span>)</p><p>Now standard equation of travelling wave can be written as</p><p>Y= a cos (wt-kx+<span contenteditable="false"> <math> <mi>∅</mi> </math> </span>)</p><p>So by comparing with above equation</p><p>a= 2cm</p><p>w=20<span contenteditable="false"> <math> <mi>π</mi> <mi>r</mi> <mi>a</mi> <mi>d</mi> <mo>/</mo> <mi>s</mi> </math> </span></p><p>k=0.016<span contenteditable="false"> <math> <mi>π</mi> </math> </span></p><p>path difference =4cm</p><p>(a) phase difference <math><mo>?</mo><mi>∅</mi><mo>=</mo><mfrac><mrow><mrow><mn>2</mn><mi>π</mi></mrow></mrow><mrow><mrow><mi>λ</mi></mrow></mrow></mfrac><mo>×</mo></math>&nbsp;path difference</p><p><math><mo>?</mo><mi>∅</mi><mo>=</mo><mn>0.016</mn><mi>π</mi><mo>×</mo><mn>4</mn><mo>×</mo><mn>100</mn></math></p><p><math><mo>=</mo><mn>6.4</mn><mi>π</mi><mi>r</mi><mi>a</mi><mi>d</mi></math></p><p><math><mo>(b) ?</mo><mi>∅</mi><mo>=</mo><mfrac><mrow><mrow><mn>2</mn><mi>π</mi></mrow></mrow><mrow><mrow><mi>λ</mi></mrow></mrow></mfrac><mo>×</mo><mfenced separators="|"><mrow><mrow><mn>0.5</mn><mo>×</mo><mn>100</mn></mrow></mrow></mfenced><mo>=</mo><mn>0.8</mn><mi>π</mi><mi>r</mi><mi>a</mi><mi>d</mi></math></p><p><math><mo>(c) ?</mo><mi>∅</mi><mo>=</mo><mfrac><mrow><mrow><mn>2</mn><mi>π</mi></mrow></mrow><mrow><mrow><mi>λ</mi></mrow></mrow></mfrac><mo>×</mo><mfrac><mrow><mrow><mi>λ</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac><mo>=</mo><mi>π</mi><mi>r</mi><mi>a</mi><mi>d</mi></math></p><p><math><mo>(d) ?</mo><mi>∅</mi><mo>=</mo><mfrac><mrow><mrow><mn>2</mn><mi>π</mi></mrow></mrow><mrow><mrow><mi>λ</mi></mrow></mrow></mfrac><mo>×</mo><mfrac><mrow><mrow><mn>3</mn><mi>λ</mi></mrow></mrow><mrow><mrow><mn>4</mn></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><mrow><mn>3</mn><mi>π</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac><mi>r</mi><mi>a</mi><mi>d</mi></math></p><p>(e) T= <math><mn>2</mn><mi>π</mi></math>&nbsp;/w=2&nbsp;<math><mi>π</mi></math>&nbsp;/20&nbsp;<math><mi>π</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>10</mn><mi>s</mi></math></p><p>At x=100cm</p><p>t=T</p><p><math><msub><mrow><mrow><mi>∅</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math>&nbsp;=20&nbsp;<math><mi>π</mi><mi>T</mi><mo>-</mo><mn>0.016</mn><mi>π</mi><mfenced separators="|"><mrow><mrow><mn>100</mn></mrow></mrow></mfenced><mo>+</mo><mn>7</mn><mi>π</mi></math></p><p>= 20&nbsp;<math><mi>π</mi><mfenced separators="|"><mrow><mrow><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><mn>10</mn></mrow></mrow></mfrac></mrow></mrow></mfenced><mo>-</mo><mn>1.6</mn><mi>π</mi><mo>+</mo><mn>7</mn><mi>π</mi><mo>=</mo><mn>2</mn><mi>π</mi><mo>-</mo><mn>1.6</mn><mi>π</mi><mo>+</mo><mn>7</mn><mi>π</mi></math></p><p>Again at x=100cm t=5s</p><p><math><msub><mrow><mrow><mi>∅</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math>&nbsp;=20&nbsp;<math><mi>π</mi><mi>T</mi><mo>-</mo><mn>0.016</mn><mi>π</mi><mfenced separators="|"><mrow><mrow><mn>100</mn></mrow></mrow></mfenced><mo>+</mo><mn>7</mn><mi>π</mi></math></p><p>=100&nbsp;<math><mi>π</mi><mo>-</mo><mn>1.6</mn><mi>π</mi><mo>+</mo><mn>7</mn><mi>π</mi></math></p><p>From above two equation phase difference&nbsp;<math><msub><mrow><mrow><mi>∅</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub><mo>-</mo><msub><mrow><mrow><mi>∅</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math></p><p>=(100&nbsp;<math><mi>π</mi><mo>-</mo><mn>1.6</mn><mi>π</mi><mo>+</mo><mn>7</mn><mi>π</mi></math>&nbsp;)-&nbsp;<math><mn>2</mn><mi>π</mi><mo>-</mo><mn>1.6</mn><mi>π</mi><mo>+</mo><mn>7</mn><mi>π</mi></math></p><p>= 100&nbsp;</p>

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