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### Repository of WebQuests (All Areas)

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BOOK:  TABLE OF CONTENT

kapoor 6 days 19 hours ago

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kapoor 6 days 19 hours ago

### Problem/EM-05004

A point charge $$q$$ is embedded at the center of a sphere with dielectric constant $$\kappa_1$$. The sphere is itself embedded in an infinite volume with dielectric constant $$κ_2$$. Find the electric field out side the sphere and surface charge density on  the surface of the sphere.

kapoor 6 days 19 hours ago

### Problem/CM-01001; Periodic Motion

Question : {A particle moves in a force field given by the potential $V(x) = A |x|^n$  Find the time period as function of energy.

Answer: { $\displaystyle {2\over n}\sqrt{2m\pi\over E} {\left(E\over A\right)^{1/n}{\Gamma(1/n)\over \Gamma({n+2\over 2n})}}$}

kapoor 6 days 21 hours ago

### Problem/CM-01002 Periodic Motion

Question : For a particle moving in a potential well $$\displaystyle V(x) = -V_0\,\text{sech}^2(\alpha x),\qquad -V_0 < E < 0$$  find the time period as a function of energy of the particle.

Answer : $\displaystyle ({\pi\over \alpha})\sqrt{(2m/|E|)}$

kapoor 6 days 21 hours ago

### Problem/CM-01003 Periodic Motion

Question : For a particle in one dimensional potential well  $\displaystyle V(x) = V_0 \tan^2(\alpha x)$ find the time period as function of the energy of the particle.
Answer : $\displaystyle ({\pi\over\alpha})\sqrt{2m\over (E+V_0)}$

kapoor 6 days 21 hours ago

### Problem/CM-01004 Periodic Motion

Question : A particle in one dimension is subject to a force
$$F = -k x +{ \lambda\over x^3}$$
Find the potential energy and the exact period of oscillations for
$\lambda = 6ka^4, E= 5k a^2$.

kapoor 6 days 21 hours ago

### Problem/CM-01005 Periodic Motion

Question Plot the Morse potential
$$V(x) = V_0 \left( \exp(-2x/\alpha) - 2 \exp(-x/\alpha)\right)$$
and find the period of small oscillations for a particle having energy
$E$ and moving in fore field described by the Morse potential.}

kapoor 6 days 21 hours ago

### Problem/CM-01006 Periodic Motion

Question: Show that the time period of a relativistic oscillator as  function of amplitude $a$ is given by $$T\approx \frac{2\pi}{\omega}\left[ 1 + \frac{3\omega^2a^2}{16 c^2} + \cdots \right]$$ where the restoring force is $- kx$ and $k=m \omega2$.

kapoor 6 days 21 hours ago

### Problem/CM-01007 Periodic Motion

Question : Determine the period of oscillations of a simple pendulum ( a particle of mass $m$ suspended by a string of length $L$ in a gravitational field) as a function of the amplitude of the oscillations.}

Answer : {$T= 2\pi \sqrt{(L/g)}(1 + \theta_0 ^2/16 + \cdots )$ }

kapoor 6 days 21 hours ago