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Physics Resonance: Problem set 27 -->

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Friday, 18 November 2016

Problem set 27

  1. The Fourier transform of f(x) is \tilde{f}(k)=\int_{-\infty}^{\infty}dx\:e^{ikx}f(x). If f(x)=\alpha\delta(x)+\beta\delta'(x)+\gamma\delta''(x), where \delta(x) is the Dirac delta-function (and prime denotes derivative), what is \tilde{f}(k)?
    1. \alpha+i\beta k+i\gamma k^2
    2. \alpha+\beta k-\gamma k^2
    3. \alpha-i\beta k-\gamma k^2
    4. i\alpha+\beta k-i\gamma k^2
  2. A particle moves in three-dimensional space in a central potential V(r)=kr^4, where k is a constant. The angular frequency \omega for a circular orbit depends on its radius R as
    1. \omega\propto R
    2. \omega\propto R^{-1}
    3. \omega\propto R^{1/4}
    4. \omega\propto R^{-2/3}
  3. The Lagrangian of a system is given by {\scriptstyle L=\frac{1}{2}m\dot q_1^2+2m\dot q_2^2-k\left(\frac{5}{4}q_1^2+2q_2^2-2q_1q_2\right)} where m and k are positive constants. The frequencies of its normal modes are
    1. \sqrt{\frac{k}{2m}}\sqrt{\frac{3k}{m}}
    2. \sqrt{\frac{k}{2m}}(13\pm\sqrt{73})
    3. \sqrt{\frac{5k}{2m}}\sqrt{\frac{k}{m}}
    4. \sqrt{\frac{k}{2m}}\sqrt{\frac{6k}{m}}
  4. Consider a particle of mass m moving with a speed v. If T_R denotes the relativistic kinetic energy and T_N its non-relativistic approximation, then the value of (T_R-T_N)/T_R for v=0.01\:c, is
    1. 1.25\times10^{-5}
    2. 5.0\times10^{-5}
    3. 7.5\times10^{-5}
    4. 1.0\times10^{-4}
  5. Two masses, m each, are placed at the points (x,y)=(a,a) and (-a,-a). Two masses, 2m each, are placed at the points (a,-a) and (-a,a). The principal moments of inertia of the system are
    1. 2ma^2, 4ma^2
    2. 4ma^2, 8ma^2
    3. 4ma^2, 4ma^2
    4. 8ma^2, 8ma^2

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