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Friday, 18 November 2016
Problem set 27
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)?
\alpha+i\beta k+i\gamma k^2
\alpha+\beta k-\gamma k^2
\alpha-i\beta k-\gamma k^2
i\alpha+\beta k-i\gamma k^2
For Dirac delta function, we have,
\int\limits f(x)\delta(x-y)\:dx=f(y)\int\limits f(x)\delta'(x-y)\:dx=-f'(y)\int\limits f(x)\delta''(x-y)\:dx=(-1)^2f''(y)\int\limits f(x)\delta^n(x-y)\:dx=(-1)^nf^n(y)\begin{align*}
\tilde{f}(k)&={\scriptstyle\int\limits_{-\infty}^{\infty}dx\:e^{ikx}\left(\alpha\delta(x)+\beta\delta'(x)+\gamma\delta''(x)\right)}\\
&=\alpha f(0)+\beta(-f'(0))+\gamma f''(0)\\
&=\alpha-i\beta k-\gamma k^2
\end{align*}
Hence, answer is (C)
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
\omega\propto R
\omega\propto R^{-1}
\omega\propto R^{1/4}
\omega\propto R^{-2/3}
Force acting on particle is given by
\begin{eqnarray}
\begin{array}{c}\vec F(r)=-\nabla V(r)\\
{\scriptstyle=-\left(\hat r\frac{\partial}{\partial r}+\hat\theta\frac{1}{r}\frac{\partial}{\partial\theta}+\hat\phi\frac{1}{r\sin\theta}\frac{\partial}{\partial\phi}\right)\left(kr^4\right)}\\
=-4kr^3\hat r\end{array}
\end{eqnarray}
For circular motion in radius R, we must have
\frac{mv^2}{R}=F(R)=-4kR^3
Using V=R\omega\frac{mR^2\omega^2}{R}=-4kR^3\omega^2=\frac{-4kR^2}{m}\omega\propto R
Hence, answer is (A)
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
\sqrt{\frac{k}{2m}}\sqrt{\frac{3k}{m}}
\sqrt{\frac{k}{2m}}(13\pm\sqrt{73})
\sqrt{\frac{5k}{2m}}\sqrt{\frac{k}{m}}
\sqrt{\frac{k}{2m}}\sqrt{\frac{6k}{m}}
Lagrange equation is
\frac{d}{dt}\left(\frac{\partial L}{\partial\dot q_j}\right)-\frac{\partial L}{\partial q_j}
Substituting value of L for generalized coordinates q_1 and q_2 we get
m\ddot q_1+k\left(\frac{5}{2}q_1-2q_2\right)=0\quad (1)4m\ddot q_2+k\left(4q_2-2q_1\right)=0\quad (2)
Form normal modes, both particles moves simple harmonically with same frequency \omega, hence, we have
\ddot q_1=-\omega^2q_1\ddot q_2=-\omega^2q_2
Substituting values of \ddot q_1 and \ddot q_2 in equations (1) and (2), we get
\left(\frac{5}{2}k-m\omega^2\right)q_1=2kq_2\quad (3)\left(4k-4m\omega^2\right)q_2=2kq_1\quad (4)
From equation (3) and (4) we get
\frac{q_2}{q_1}=\frac{\left(\frac{5}{2}k-m\omega^2\right)}{2k}\frac{q_2}{q_1}=\frac{2k}{\left(4k-4m\omega^2\right)}
Hence, we have
\frac{\left(\frac{5}{2}k-m\omega^2\right)}{2k}=\frac{2k}{\left(4k-4m\omega^2\right)}
Solving this equation, we get
2m^2\omega^4-7km\omega^2+3k^2=0
This is a quadratic equation in \omega^2. Solving this equation we get
\omega^2=\frac{k}{2m}\quad\text{or }\frac{3k}{m}
or we get
\omega=\sqrt{\frac{k}{2m}}\quad\text{or }\sqrt{\frac{3k}{m}}
Hence, answer is (A)
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
1.25\times10^{-5}
5.0\times10^{-5}
7.5\times10^{-5}
1.0\times10^{-4}
T_R=(\gamma-1)mc^2
where, \gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}=1.00005000375031T_R=5.000375031\times10^{-5}mc^2T_N=0.5\times mv^2=5\times10^{-5}mc^2\frac{(T_R-T_N)}{T_R}=7.5\times10^{-5}
Hence, answer is (C).
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
2ma^2, 4ma^2
4ma^2, 8ma^2
4ma^2, 4ma^2
8ma^2, 8ma^2
Inertia tensor is given by
{\scriptscriptstyle{I}=\begin{pmatrix}
\sum\limits_\alpha m_\alpha\left(x_{\alpha,2}^2+x_{\alpha,3}^2\right)&-\sum\limits_\alpha m_\alpha x_{\alpha,1}x_{\alpha,2}&-\sum\limits_\alpha m_\alpha x_{\alpha,1}x_{\alpha,3}\\
-\sum\limits_\alpha m_\alpha x_{\alpha,2}x_{\alpha,1}&\sum\limits_\alpha m_\alpha\left(x_{\alpha,1}^2+x_{\alpha,3}^2\right)&-\sum\limits_\alpha m_\alpha x_{\alpha,2}x_{\alpha,3}\\
-\sum\limits_\alpha m_\alpha x_{\alpha,3}x_{\alpha,1}&-\sum\limits_\alpha m_\alpha x_{\alpha,3}x_{\alpha,2}&\sum\limits_\alpha m_\alpha\left(x_{\alpha,1}^2+x_{\alpha,2}^2\right)
\end{pmatrix}
}
where, \alpha runs over number of particles. Here, x_{\alpha,1} means x-coordinate of particle number \alpha, x_{\alpha,2} means y-coordinate of particle number \alpha and x_{\alpha,3} means z-coordinate of particle number \alpha
For a two-dimensional case, we have
{\scriptstyle{I}=\begin{pmatrix}
\sum\limits_\alpha m_\alpha\left(x_{\alpha,2}^2\right)&-\sum\limits_\alpha m_\alpha x_{\alpha,1}x_{\alpha,2}\\
-\sum\limits_\alpha m_\alpha x_{\alpha,2}x_{\alpha,1}&\sum\limits_\alpha m_\alpha\left(x_{\alpha,1}^2\right)
\end{pmatrix}
}\begin{align*}
I_{11}&= \sum\limits_\alpha m_\alpha\left(x_{\alpha,2}^2\right)\\
&=ma^2+ma^2+2ma^2+2ma^2\\
&=6ma^2
\end{align*}\begin{align*}
I_{12}&= -\sum\limits_\alpha m_\alpha x_{\alpha,1}x_{\alpha,2}\\
&=-(ma^2+ma^2-2ma^2-2ma^2)\\
&=2ma^2
\end{align*}\begin{align*}
I_{21}&= -\sum\limits_\alpha m_\alpha x_{\alpha,2}x_{\alpha,1}\\
&=-(ma^2+ma^2-2ma^2-2ma^2)\\
&=2ma^2
\end{align*}\begin{align*}
I_{22}&= \sum\limits_\alpha m_\alpha\left(x_{\alpha,1}^2\right)\\
&=ma^2+ma^2+2ma^2+2ma^2\\
&=6ma^2
\end{align*}{I}=\begin{pmatrix}
6ma^2&2ma^2\\
2ma^2&6ma^2
\end{pmatrix}
The principal moments of inertia are obtained by solving
{I}=\begin{vmatrix}
6ma^2-\lambda&2ma^2\\
2ma^2&6ma^2-\lambda
\end{vmatrix}=0
Solving this determinant we get
\lambda^2-12ma^2\lambda+32m^2a^4=0
Solving this quadratic equation in \lambda we get,
\lambda_1=8ma^2 and \lambda_2=4ma^2
Hence, answer is (B)
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