UPSC Physics Optional PYQs (CSE & IFoS)PYQs

A rocket of mass m_1 + m_2 is launched with a velocity whose horizontal and vertical components are u_x and u_y. At the highest point in its path, the rocket explodes into two parts of masses m_1 and m_2 that separate in a horizontal direction in the original plane of motion. Show that the fragments strike the ground at a distance apart given by D = \frac{u_y}{g} \sqrt{\frac{2(m_1 + m_2)K}{m_1 m_2}} where K is the kinetic energy produced by the explosion. (Neglect air resistance, the mass of explosive, any spinning motion of the fragments and assume that g is constant)

What do you understand by length contraction? Calculate the percentage length contraction of a rod moving with a velocity 0.8c in a direction at 60^\circ with respect to its own length.

Suppose the position of a particle of mass m on the xy-plane is \bar{r} = (x, y, z) = (r \cos\phi, r \sin\phi, 0) Here r and \phi are functions of time t. Find l_z, the z-component of \vec{l}, where \vec{l} is the angular momentum of the particle. Further, show that the areal velocity is equal to the angular momentum divided by 2m. Is areal velocity constant?

Derive the relativistic expression for kinetic energy by considering mass variation with velocity. Hence, establish the relation between momentum (p) and energy (E) for a relativistic particle; \frac{dE}{dp}=v.

Determine the location of the centre of mass of a uniform solid hemisphere of radius R and mass M from the centre of its base.

Obtain expressions for the moment of inertia of a solid cone about its

(i) Vertical axis and

(ii) axis passing through the vertex and parallel to its base.

A rubber cord 1\ \mathrm{mm} in diameter and 1\ \mathrm{m} long is fixed at one end and a weight of 1\ \mathrm{kg} is attached to the other end. If the Young's modulus of rubber is 0.05 \times 10^{11}\ \mathrm{dynes\,cm^{-2}}, then find the period of the vertical oscillations of the weight.

A rocket starts vertically upwards with speed v_0. Then define its speed v at a height h in terms of v_0, h, R (radius of Earth) and g (acceleration due to gravity on Earth's surface). Also calculate the maximum height attained by a rocket fired with a speed of 90% of the escape velocity.

(i) Explain Coriolis force. Compare the qualitative effects of the Coriolis forces in the two geographical hemispheres of the Earth for rivers flowing east. (ii) With schematic, explain Foucault's pendulum.

A shaft of diameter 8\,\mathrm{cm} and length 5\,\mathrm{m} is transmitting power of 8\,\mathrm{kW} at 300 revolutions per minute. If the coefficient of rigidity of the material of the shaft be 8 \times 10^{11}\,\mathrm{dynes/cm^2}, then calculate the relative shift between the ends of the shaft.

Explain the rotation of a free rigid body and show that a rigid body rotating in any manner about a fixed point has two constants of motion, L^2 and T. Here \vec{L} is angular momentum and T is kinetic energy.

A transistor having \alpha = 0\cdot 975 and a reverse saturation current I_{\text{CO}} = 10~\mu\text{A} is operated in CE configuration. What is \beta for this configuration? If the base current is 250~\mu\text{A}, calculate the emitter current and the collector current.

Explain the use of an op-amp as a differentiator and an integrator using proper circuit diagram. Write the condition to work as good differentiator and integrator.

A superconducting material has a critical temperature of 3\cdot 7~\text{K} and a magnetic field of 0\cdot 0306~\text{tesla} at 0~\text{K}. Find the critical field at 2~\text{K}.

An n-channel silicon JFET has donor concentration of 2 \times 10^{21}/\text{m}^3 and a channel width of 4~\mu\text{m}. If the dielectric constant of silicon is 12, find the pinch-off voltage. If the FET operates with a gate-source voltage of -2~\text{V}, what is the saturation voltage V_{\text{Dsat}}?

Explain the Meissner effect in superconductivity. Obtain an expression for London penetration depth of magnetic field for a superconductor.

Describe briefly the construction of an enhancement type MOSFET using a p-type silicon bar. Point out the structural difference between enhancement and depletion forms of the MOSFET.

Obtain the Boolean expression for the output Y in the given logic circuit :

Simplify the expression and show that the circuit is equivalent to an AND gate with inputs A and B.

Sketch the cross-sectional structure of an enhancement mode MOSFET and explain its principles of operation with the help of its output characteristics.

Consider a face-centred cubic lattice of side a. Deduce –

(i) The primitive translation vectors;

(ii) The volume of the primitive cell;

(iii) The reciprocal primitive translation vectors;

(iv) The volume of the reciprocal lattice