A metal ring of mass \(M\) and radius \(R\) is set rotating about an axis going through the diameter. The experiment is conducted in space at an infinite distance. Its angualar velocity is \(\omega\). Find tension as a function on angle \(\theta\) from horizontal. Solution
A table with a smooth horizontal surface is rotating with constant angular velocity \(\omega\) about the vertical axis. A groove is made radially, and a particle is placed at a distance \(a\) from the centre. Find the trajectory of the particle as a function of the angular displacement \(\theta\). Solution
Consider a 3D shape whose left end has radius \(1\)m. The shape is circular, with its radius increasing continually as \(r = a^x\), \(a \in \mathbb{R^+}\). Assume forces to be acting in the centre of the side (circular) faces on both sides. Consider it to have Young’s modulus \(Y\). Find elongation \(\Delta L\). Solution
Consider the previous 3D shape. Assume the function to be \(r = e^x\). This shape is rotated with constant angular velocity \(\omega\) about the axis touching the smaller end. Find elongation \(\Delta L\). Solution
A frustum is made with inner radius \(a_1\) and outer radius \(a_2\) on the short end, and radii \(b_1\) and \(b_2\) on the larger side. Find the net thermal resistance offered by the hollow frustum if its total length is \(L\) and thermal conductivity is \(K\).Solution
Consider a hollow frustum, closed on both ends. An ideal gas of molar mass M and temperature T is inside it. It is rotated about its short end with constant angular velocity \(\omega\). The radius on its left end is \(a\), and its right end is \(b\). Pressure on its left end is \(P_0\). The total length is \(L\). Find pressure profile. (\(P(x)\)) Solution
Find the moment of inertia of an ellipse (lamina) about an axis passing through the centre, perpendicular to the lamina. Consider the ellipse to have semi minor axis of length \(b\), and semi major axis of length \(a\). Solution
Find the moment of inertia of a torus about its axis of symmetry. Assume mass \(m\), major radius \(R\), minor radius \(a\). Solution
Find the time of collision of two (point sized) bodies kept at a distance \(R\) from each other, initially at rest, having equal masses \(M\). Answer
Consider a charged non conducting hollow cylinder of surface change density \(\sigma\), radius \(R\), and length \(L\). A particle of mass \(m\), change \(q\) is dropped at one end of the tube. Find velocity at a general position \(x\) from the top of cylinder. Given: \(q\sigma < 0\). Neglect gravity. Answer
Consider an isolated system of length \(L\) and cross section \(A\) as shown. There is a moveable, adiabatic wall placed at \({L \over 2}\). Find net displacement of the wall, final temperatures and pressure. Consider adiabatic gas constant to be \(\gamma\). Also find work done by the gases on the left and right portions. Answer
Find the force of water on a conical dam as shown below. Take the density to be \(\rho\). Half angle of the cone is \(\phi\). Answer
In a hollow cylinder, filled with water till a height \(h\), being rotated at angular velocity \(\omega\). Its radius is \(R\). Consider the density of water to be \(\rho\). Find the moment of inertia of the rotating liquid as function of angular velocity \(\omega\). Answer
In Q 10, assuming the cylinder to be of sufficient length, we can say that the particle’s velocity will eventually reach the speed of light, \(c\). This will clearly happen at the midpoint of the cylinder. Find the necessary length of the cylinder assuming the same assumptions as made before. Neglect relativistic effects, and use newtonian mechanics. Answer
Solve Q 10, but consider a conducting neutral cylinder instead. Answer
Find the equation of the curve that light traces as it passes from vacuum to a medium of variable refractive index as given in each sub-part. Answer
Consider a ‘liquid giant’. A planet made completely of a liquid of density \(\rho\), and of radius \(R\). Consider the density of planet to remain constant with depth. A small spherical object if density \(\rho_s\) is gently dropped from the surface of the planet. Find its position as a function of time. Answer
Find the variation in radius and density of a small sphere of initial radius \(r_0\), and its bulk modulus being \(\beta\), with depth in a liquid giant of radius \(R\) and constant density \(\rho\). Answer
Find the resistance of an ellipse with major axis \(a\), and minor axis \(b\), rotated about its major axis, with the parts beyond \(\pm c\) being cut off as shown in diagram. Its resistivity is \(\rho\). Answer
Find the temperature of a resistor due to self heating as a function of time. Consider the resistor to be of resistance \(\rho_0\), length \(l_0\) and cross sectional area \(A_0\) at temperature \(T_0\). Assume the resistor to be connected to an ideal cell of constant electromotive force, \(\mathcal{E}\). Consider its mass to be \(m\) and its specific heat capacity to be \(s\). Assume \(\eta\) to be the fraction of heat the is used by the resistor to heat up, and the rest(\(1-\eta\)) is lost to surroundings. Its coefficients of linear thermal expansion is \(K\), and its thermal coefficient of resistivity to be \(\alpha\). Assume it to be an isotropic material. Answer
Consider a ring of radius \(R\), and charged with a linear charge density \(\lambda\), placed on a smooth frictionless surface, in a region of varying magnetic field that varies with time \(t\) as \(B = B_0e^{-\alpha t}\). Let its mass be \(m\). Find its angular velocity \(\omega\) as a function of time. Also find the tension \(T\) in the ring as a function of time. Neglect the interaction of the charges of the ring with itself. Answer
Consider a hollow cylinder of mass \(m\), radius \(R\) and length \(L\) placed on a smooth frictionless surface as shown in figure. Let its curved surface area be charged with a uniform surface charge density, \(\sigma\). It is placed in a region of time varying magnetic field, \(B = B_0\ln(1+\Omega t)\). At time, \(t=0\), sand starts to pour into the cylinder at a rate of \(\lambda\) (kg \(\text{s}^{-1}\)). The density of the sand is \(\rho\). Assume that when the cylinder is completely filled, the magnetic field becomes constant at whatever value it is then. Further assume that as the sand falls, it always forms a perfect cylinder. Answer
Consider the following cyclic decay of nuclei, with the given rate constants. Find the quantity of each type of nucleus as a function of time. Assume initially they all start with \(A_0\), \(B_0\) and \(C_0\) of nuclei. Answer
An infinitely long \(\Pi\) shaped conductor as shown is fixed horizontally on a flat surface. A capacitor of capacitance \(C\) is connected as shown. A straight conducting rod of length \(l\), mass \(m\) and resistance \(R\) is placed parallel to the line joining the capacitor as shown. Assume frictionless contact between the rod and the \(\Pi\) shaped conductor. A constant magnetic field of magnitude \(B\) is set up vertically, into the plane. The rod is initially given a velocity \(v_0\) towards the right by an impulsive force. Find the velocity of the rod and the charge on the capacitor as a function of time. Neglect the magnetic fields produced by the wires themselves. Answer
Consider a similar setup as the last question. Instead of just a capacitor, we now have a capacitor of capacitance \(C\), and an inductor of inductance \(L\). Other conditions remain the same. Find the charge on capacitor, the current through the circuit, and the velocity of the rod all as functions of time. Neglect the magnetic fields produced by the wires themselves. Answer
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