Show that the motion of a particle represented by y = sinω t – cos ω t is simple harmonic with a period of 2π/ω.

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Payal Gupta | Contributor-Level 10

4 months ago

This is a short answer type question as classified in NCERT Exemplar

As we know displacement y=sinwt-coswt

= $\sqrt[]{2} (\frac{1}{\sqrt[]{2}} s i n w t - \frac{1}{\sqrt[]{2}} c o s w t)$

= $\sqrt[]{2} (c o s \frac{π}{4} s i n w t - s i n \frac{π}{4} c o s w t)$

= $\sqrt[]{2} [s i n (w t - \frac{π}{4})]$

To comparing with standard equation

Y= asin (wt+ $\emptyset$ )

So T=2 $π$ /w

This is a short answer type question as classified in NCERT ExemplarAs we know displacement y=sinwt-coswt= <math> <mroot> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> <mrow></mrow> </mroot> <mfenced separators="|"> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mroot> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> <mrow></mrow> </mroot> </mrow> </mrow> </mfrac> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>w</mi> <mi>t</mi> <mo>-</mo> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mroot> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> <mrow></mrow> </mroot> </mrow> </mrow> </mfrac> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>w</mi> <mi>t</mi> </mrow> </mrow> </mfenced> </math> = <math> <mi> </mi> <mroot> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> <mrow></mrow> </mroot> <mo> (</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> </mfrac> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>w</mi> <mi>t</mi> <mo>-</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> </mfrac> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>w</mi> <mi>t</mi> <mo>)</mo> </math> = <math> <mroot> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> <mrow></mrow> </mroot> <mfenced open=" [" close="]" separators="|"> <mrow> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mfenced separators="|"> <mrow> <mrow> <mi>w</mi> <mi>t</mi> <mo>-</mo> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfenced> </mrow> </mrow> </mfenced> </math> To comparing with standard equationY= asin (wt+ <math> <mi>∅</mi> </math> )So T=2 <math> <mi>π</mi> </math> /w

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Show that the motion of a particle represented by y = sinω t – cos ω t is simple harmonic with a period of 2π/ω.

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