A rod of length L has non-uniform linear mass density (mass per unit length) \lambda, which varies as \lambda=\lambda_0\left(\dfrac{S}{L}\right); where \lambda_0 is a constant and S is the distance from the end marked `O' (as shown in the figure). Find the centre of mass of the rod.
A rod of length l_0 is kept at rest in x'\,y' plane of its rest frame making an angle \theta_0 with x' axis. What is the length and orientation of the rod in a laboratory frame (x,y) in which the rod moves to the right with velocity v?
(i) If a particle of mass m is in a central force field f(r)\hat{r}, then show that its path must be a plane curve, where \hat{r} is a unit vector in the direction of position vector \vec{r}.
(ii) A block of mass m having negligible dimension is sliding freely in x-direction with velocity \vec{v}=v\hat{\imath} as shown in the diagram.
What is its angular momentum \vec{L}_O about origin O and its angular momentum \vec{L}_A about the point A on y-axis?
A water drop of radius 0.04\ \mathrm{mm} is falling through air. If the coefficient of viscosity for air is 1.8 \times 10^{-4} poise, find its terminal velocity. If 100 such drops coalesce, what will be the new terminal velocity?
Use Gauss's theorem to calculate the gravitational potential due to a solid sphere at a point outside the sphere. Calculate the amount of work required to send a body of mass m from the Earth's surface to a height R/2, where R is the radius of the Earth.
Two capillary tubes of lengths 2l and l with internal radii r and 2r respectively are connected in series. Water flows through them in streamline. If the pressure difference across the first capillary is P, find the pressure difference across the second one.
Express angular momentum in terms of kinetic, potential and total energy of a satellite of mass m in a circular orbit of radius r.
State and explain Stokes' law. A drop of water of radius 0.01\ \mathrm{m} is falling through a medium whose density is 1.21\ \mathrm{kg/m^3} and \eta = 1.8 \times 10^{-5}\ \mathrm{N\,s/m^2}. Find the terminal velocity of the drop of water.
A ball moving with a speed of 9\ \mathrm{m/s} strikes an identical stationary ball such that after the collision the direction of each ball makes an angle 30^\circ with the original line of motion. Find the speed of the balls after the collision. Is the kinetic energy conserved in this collision?
A diatomic molecule can be considered to be made up of two masses m_1 and m_2 separated by a fixed distance r. Derive a formula for the distance of centre of mass, C, from mass m_1. Also show that the moment of inertia about an axis through C and perpendicular to r is \mu r^2, where \mu=\dfrac{m_1m_2}{m_1+m_2}.
Prove that x^2+y^2+z^2=c^2t^2 is invariant under Lorentz transformation.
Define moment of inertia and explain its physical significance. Calculate the moment of inertia of an annular ring about an axis passing through its centre and perpendicular to its plane.
Describe Michelson-Morley experiment and show how the negative results obtained from this experiment were interpreted.
A body moving in an inverse square attractive field traverses on elliptical orbit with eccentricity e and period \gamma. Find the time taken by the body to traverse the half of the orbit that is nearer the centre of force. Explain briefly why a comet spends only 18\% of its time on the half of its orbit that is nearer the sun.
Derive the expression for Coriolis force and show that this force is perpendicular to the velocity and to the axis of rotation. What is the nature of this force?
Calculate the percentage contraction in the length of a rod in a frame of reference, moving with velocity 0.8c in a direction What is the orientation of the rod in the moving frame of reference in case (ii)?
Given a proton for which \beta=0.995 measured in the laboratory. What are the corresponding relativistic energy and momentum? Take, m_p=1\cdot67\times10^{-24}\,\mathrm{g}.
(i) The distance between the centres of the carbon and oxygen atoms in the carbon monoxide (CO) gas molecule is 1.130\times10^{-10}\ \mathrm{m}. Locate the centre of mass of the molecule relative to the carbon atom.
(ii) Find the centre of mass of a homogeneous semicircular plate of radius a.
A student is working in a physics laboratory, which is at temperature 27^{\circ}\mathrm{C}, on a sonometer to study formation of stationary waves. The cross-sectional area of the sonometer wire is 0.85\times10^{-6}\ \mathrm{m^2} and a tension of 20\ \mathrm{N} is applied on it. If the rigid supports are 1.2\ \mathrm{m} apart and the temperature of the wire drops by 7^{\circ}\mathrm{C}, calculate the (i) final tension and (ii) fundamental frequency of vibration of the wire. Take, coefficient of linear expansion and isothermal Young's modulus as 1.5 \times 10^{-5}\ \mathrm{K}^{-1} and 2.0 \times 10^{11}\ \mathrm{N\,m}^{-2} respectively.
Four solid spheres A, B, C, and D each of mass m and radius a, are placed with their centres on the four corners of square of side b as shown in the figure below:
Calculate the moment of inertia of the system about one side of the square, Also, calculate the moment of inertia of the system about a diagonal of the square.