JEE (Advanced) 2011 Paper-1
For the following questions answer them individually
A spherical metal shall A of radius $$R_A$$ and a solid metal sphere B of radius $$R_B(< R_A)$$ are kept far apart and each is given charge '+Q'. Now they are connected by a thin metal wire. Then-
A metal rod of length 'L' and mass 'm' is pivoted at one end. A thin disk of mass 'M' and radius 'R' (< L) is attached at its centre to the free end of the rod. Consider two ways the disc is attached :(case A) The disc is not free to rotate about its center and (case B) the disc is free to rotate about its center. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is (are) true ?
An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is/are true ?
A composite book is made of slabs A, B, C, D and E of different thermal conductivities (given in terms of a constant K) and sizes (given in terms of length, L) as shown in the figure. All slabs are of same width. Heat 'Q' flow only from left to right through the blocks. Then in steady state-
Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.
The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $$E_1$$ and $$E_2$$ are the total mechanical energies respectively. Then-
Consider the spring-mass system, with the mass submerged in water, as shown in figure. The phase space diagram for one cycle of this system is-
A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let 'N' be the number density of free electrons, each of mass 'm'. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency 'wp', which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency w, where a part of the energy is absorbed and a part of it is reflected. As w approaches wp, all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
Taking the electronic charge as 'e' and the permittivity as $$'ε_0'$$, use dimensional analysis to determine the correct expression for $$\omega_p$$.
Estimate the wavelength at which plasma reflection will occur for metal having the density of electrons $$N \approx 4 \times 10^{27} m^{-3}$$. Take $$ε_0 = 10^{-11}$$ and $$m \approx 10^{-30}$$, where these quantities are in proper SI unit-
For the following questions answer them individually
A block is moving on an inclined plane making an angle $$45^\circ$$ with the horizontal and the coefficient of friction is $$\mu$$. The force required to just push it up the inclined plane is 3 times the force required to just prevent it from sliding down. If we define N = 10 $$\mu$$, then N is.
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