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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2010

  • question_answer
        The displacement of a particle executing periodic motion is given by \[y=4{{\cos }^{2}}\frac{t}{2}\sin 1000t\] The expression may be considered to be a result of superposition of ......... independent harmonic motions.

    A)  1                                            

    B)  3

    C)  4                                            

    D)  5

    Correct Answer: B

    Solution :

                    \[y=4{{\cos }^{2}}\frac{t}{2}.\sin 1000t\] \[=2\left( 2{{\cos }^{2}}\frac{t}{2} \right).\sin 1000t\] \[=2(1+\operatorname{cost}).\sin 1000t\] \[=2\sin 1000t+2\cos t.\sin 1000t\] \[=2\sin 1000t+\sin (1000+1)t\]                                                 \[\sin (1000t-t)\] \[=2\sin 1000t+\sin 1001t+\sin 999t\] \[[\because 2\sin A\cos B=\sin (A+B)+\sin (A-B)]\]                 \[={{y}_{1}}+{{y}_{2}}+{{y}_{3}}\] So, the given expression may be considered as the result of superposition of 3 independent harmonic motions.


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