PYQ Bank

Prove that the expression x^2+y^2+z^2-c^2t^2 is invariant under Lorentz transformation.

How do we infer the law of conservation of linear momentum from Newton's laws of motion? A stationary bomb explodes and on explosion, it fragments into three parts. Two of these part which are of equal masses fly apart perpendicular to each other with a velocity of 60 m/s each. The third part has a mass four times the other two. Find the magnitude and the direction of the velocity of the third part.

Write a short note on Geostationary satellites.

Distinguish between elastic and inelastic collision. Find the minimum distance an \alpha-particle with kinetic energy of 0.4 Mev can approach a stationary but free Lithium nucleus in a head-on collision.

How is an electron volt conned to other units of energy like the Joule or the org ? Determine the speed of an election of energy 1.3\text{ Mev} assuming its rest energy to be 0.5\text{ Mev}.

Consider a spiral spring of length L and mass M suspended vertically from a rigid support. A mass m is attached to the lower end of the spring. The mass m is now pulled down through a small distance and is then released. If M and m are comparable describe the motion. What elastic constant of the material of the spiral spring provides the restoring force?

A steel ball 1.00\text{mm} in diameter at constant speed of 0.176\text{ cm s}^{-1} in a large vessel filled with oil. Calculate the dynamic viscosity of oil. Given density of steel = 7700\text{ kg m}^{-3} and that of oil = 900\text{ kg m}^{-3}. Deduce the mathematical relation needed for the calculation.

Prove that the path of particle moving in a central force field is a plane curve and that the angular momentum of the particle remains constant in time.

A smooth sphere A of mass m and speed u impinges obliquely on a sphere B of mass (M>m) at rest. The velocity of A before collision makes an angle \theta with the line of centres at the time of impact. Show that A is deflected through a right angle if \tan^2\theta=e\frac{M-m}{M+m} where e is the coefficient of restitution.

Write a short note on Gyroscope.

A block A of mass 6\text{ kg} is placed on a plane inclined to the horizontal at an angle 30^\circ. It is joined to another block B of mass 18\text{ kg} by means of a massless string going round a pulley. B hangs vertically. The pulley can be approximated as a uniform circular disc of mass 2\text{kg} and radius 10\text{cm}. The string from A to the pulley is parallel to the plane. The kinetic friction coefficient is \frac{1}{3\sqrt{3}}. Calculate the acceleration of B and the tensions in the two segments of the string.

Show that the angular momentum of a particle located at position \vec{r} relative to the origin of co-ordinates is given by \vec{l} = \vec{r} \times \vec{p} where \vec{p} is the linear momentum of the particle. Using this result prove that the angular momentum of a system of particles can be expressed as the sum of their angular momentum around the centre of mass and the angular momentum around the origin of a single particle of mass equal to the total mass of the system located at the centre of mass.

A \mu meson travels towards the earth's surface from high up in atmosphere with a speed of 0.99\text{ C}. It decays after travelling a distance of 6\text{ km}. In what time does the \mu meson decay as measured by observers in reference frames (i) bound to the earth and (ii) bound to the meson itself.

Explain processional motion of a top. A solid sphere of radius 2\text{ cm} and mass 50\text{ gm} has a thin nail of length 5\text{ mm} fixed perpendicular to its surface. When this sphere spins like a top with a speed of 20 rev./second what will be its processional speed?

Distinguish between streamline and turbulent motion. Define Reynolds number. When a sphere of radius 1.2\text{ mm} moves in glycerine, the laminar flow observed if the velocity of sphere does not exceed 0.23\text{ ms}^{-1}. At that minimum velocity of sphere of radius 5.5\text{ cm} will the flow in water become turbulent? The viscosities of glycerine and water are 11.9\text{ P} and 0.011\text{ P} respectively. The density of glycerine is 1.2 \times 10^{-3}\text{ Kg/m}^3.

Write a short note on Motion of rocket.

Moving in a closed elliptical orbit, Halley's comet goes round the Sun once every 75 years. At its point of closest approach its distance from the Sun is 8.9 \times 10^7\text{ km}. If mass of the Sun is 2 \times 10^{30}\text{ kg}, what will be the greatest distance of the comet from the Sun ?

A particle is moving under a central force field. Obtain the equations of motion and prove that in such a field the areal velocity of the particle remains constant.

The tides in oceans on Earth are produced by gravitational pull of the Sun and the Moon. The pull due to Sun is much greater than the pull due to Moon. Explain, why tides due to Sun are only about 0.4 times as high as that due to Moon. The mass of the Sun is 2 \times 10^7 times the mass of the Moon and the distance between Sun and Earth is 400 times the distance between Moon and Earth. The radius of the Earth is 6.4 \times 10^6\text{ m}.

Write a short note on Coriolis force and its effects.