Conservation Laws
With an appropriate diagram, show that in the Rutherford scattering, the orbit of the particle is a hyperbola. Obtain an expression for the impact parameter. [10 Marks][2011]
Define a conservative field. Determine if the field given below is conservative in nature: $$ \vec{E} = c [y^2 \hat{i} + (2xy+z^2) \hat{j} + 2yz \hat{k}] V/m $$ where c is a constant. [12 Marks][2012]
Consider a uniform half-sphere of radius $R$ and mass $M$. The half-sphere is supported by a frictionless horizontal plane as shown in the figure. The half-sphere lies in the region $z<0$. Find the centre of mass of the half-sphere. [15 Marks][2012]
If the forces acting on a particle are conservative, show that the total energy of the particle which is the sum of the kinetic and potential energies is conserved. [20 Marks][2013]
Prove that as a result of an elastic collision of two particles under non – relativistic regime with equal masses, the scattering angle will be $90 \degree$. Illustrate your answer with a vector diagram. [5 Marks][2013]
Discuss the problem of scattering of charged particle by a coulomb field. Hence, obtain an expression for Rutherford scattering cross-section. What is the importance of the above expression? [25 Marks][2014]
Write down precisely the conservation theorems for energy, linear momentum, and angular momentum of a particle with their mathematical forms. [10 Marks][2015]
Draw a neat diagram to explain the scattering of an incident beam of particles by a centre of force. [10 Marks][2015]
Show that the differential scattering cross-section can be expressed as $$ \sigma(\theta) = \frac{s}{\sin \theta} |\frac{ds}{d \theta}| $$ where $s$ is the impact parameter and $\theta$ is the scattering angle. [15 Marks][2015]
i) The distance between the centres of the carbon and oxygen atoms in the carbon monoxide (CO) gas molecule is $1.130 \times 10^{-10} m$. 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 $a$. [10 Marks][2016]
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 $\mu = \frac{m_1 m_2}{m_1 + m_2}$. [15 Marks][2017]
A ball moving with a speed of $9 \space m/s$ strikes an identical stationary ball such that after the collision the direction of each ball makes an angle $30 \degree$ with the original line of motion. Find the speed of the balls after the collision. Is the kinetic energy conserved in this collision? [15 Marks][2017]
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 $\vec{r}$. [10 Marks][2018]
ii) A block of mass $m$ having negligible dimension is sliding freely in $x$-direction with velocity $\vec{v}=v \hat{i}$ as shown in the diagram. What is its angular momentum $\vec{L_0}$ about origin $O$ and its angular momentum $\vec{L_A}$ about the point $A$ on $y$-axis? [10 Marks][2018]
A rod of length $L$ has non-uniform linear mass density (mass per unit length) $\lambda$, which varies as $\lambda = \lambda_0 (\frac{S}{L})$; 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. [15 Marks][2018]
i) What is central force? Give two examples of the central force.
ii) Show that the angular momentum $(\vec{L})$ of the particle in a central force field is a constant of motion. [10 Marks][2019]
Show that the cross-section for elastic scattering of a point particle from an infinitely massive sphere of radius $R$ is $\frac{R^2}{4}$. What is the inference of this result? [10 Marks][2019]
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. [10 Marks][2020]
A particle moving in a central force field describes the path $r=ke^{\alpha \theta}$, where $k$ and $\alpha$ are constants. If the mass of the particle is $m$, find the law of force. [10 Marks][2021]
i) Calculate the mass and momentum of a proton of rest mass $1.67 \times 10^{-27} kg$ moving with a velocity of $0.8c$, where $c$ is the velocity of light. If it collides and sticks to a stationary nucleus of mass $5.0 \times 10^{-26} kg$, find the velocity of the resultant particle. [8 Marks][2021]
ii) Calculate the mass of the particle whose kinetic energy is half of its total energy. Find the velocity with which the particle is travelling. [7 Marks][2021]
Show that the mean kinetic and potential energies of non-dissipative simple harmonic vibrating systems are equal. [10 Marks][2022]
Show that for very small velocity, the equation for kinetic energy, $K=\Delta m c^2$ becomes $K=\frac{1}{2} m_0 v^2$, where notations have their usual meanings. [10 Marks][2022]
A particle of mass $m_1$ collides with another particle $Q$ of mass $m_2$ at rest. The particles $P$ and $Q$ travel at angles $\theta$ and $\phi$, respectively, with respect to the initial direction of $P$. Derive the expression for the maximum value of $\theta$. [15 Marks][2022]
A force $\vec{F}$ is given by $\vec{F} = x^2 y \hat{x} + zy^2 \hat{y} + xz^2 \hat{z}$. Determine whether or not the force is conservative. [10 Marks][2023]
i) Prove that the separation of two colliding particles is same, when observed in centre of mass and laboratory system. [10 Marks][2023]
ii) Determine the kinetic energy of a thin disc of mass $0.5 kg$ and radius $0.2 m$ rotating with $100$ rotations per second around the axis passing through its centre and perpendicular to its plane. [5 Marks][2023]
Central Force Motion and Gravitation
A planet revolves around the Sun in an elliptic orbit of eccentricity e. If T is the time period of the planet, find the time spent by the planet between the ends of the minor axis close to the Sun. [10 Marks][2012]
A particle is moving in a central force field on an orbit given by $ r=ke^{\alpha\theta} $, where $k$ and $\alpha$ are positive constants, $r$ is the radial distance and $\theta$ is the polar angle. $ \\ $ (a) Find the force law for the central force field. $ \\ $ (b) Find $\theta(t)$. $ \\ $ (c) Find the total energy. $ \\ $ [20 + 20 + 20 Marks][2012]
A particle describes a circular orbit under the influence of an attractive central force directed towards a point on the circle. Show that the force varies as the inverse fifth power of distance. [15 Marks][2013]
A charged particle is moving under the influence of a point nucleus. Show that the orbit of the particle is an ellipse. Find out the time period of the motion. [15 Marks][2014]
The density inside a solid sphere of radius a is given by $ \rho=\frac{\rho_0a}{r} $, where $\rho_0$ is the density at the surface and $r$ denotes the distance from the centre. Find the gravitational field due to this sphere at a distance $2 a$ from its centre. [10 Marks][2014]
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. [10 Marks][2016]
Express angular momentum in terms of kinetic, potential and total energy of a satellite of mass m in a circular orbit of radius $r$. [10 Marks][2017]
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. [15 Marks][2018]
The radius of the Earth is $ 6.4\times{10}^6\mathrm{\ }m $, its mean density is $ 5.5\times{10}^3\mathrm{\ }kg/m^3 $ and the universal gravitational constant is $ 6.66\times{10}^{-11}{\rm Nm}^2/{\rm kg}^2 $. Calculate the gravitational potential on the surface of the Earth. [10 Marks][2021]
Rotating Frames of Reference
With an appropriate diagram, show that in the Rutherford scattering, the orbit of the particle is a hyperbola. Obtain an expression for impact parameter. [10 Marks][2011]
Prove that the time taken by the earth to travel over half of its orbit separated by the minor axis remote from the sun is two days more than half a year. Given, the period of the earth is 365 days and eccentricity of the orbit =1/60. [10 Marks][2011]
A rigid body is spinning with an angular velocity of $ 4\frac{rad}{s} $ about an axis parallel to the direction $ 4 \hat{j} - 3 \hat{k}$ passing through the point A with $ \vec{OA}=2 \hat{i} +3 \hat{j} - \hat{k} $, where O is the origin of the coordinate system. Find the magnitude and direction of the linear velocity of the body at point P with $ \vec{OP}=4 \hat{i} - 2 \hat{j} + \hat{k} $ [12 Marks][2012]
Suppose that an $S^\prime$ -frame is rotating with respect to a fixed frame having the same origin. Assume that the angular velocity $ \vec{\omega} $ of the $ S^\prime$-frame is given by $ \vec{\omega}=2t \hat{i} - t^2 \hat{j}+(2t+4) \hat{k} $ where $t$ is time and the position vector $\vec{r}$ of a typical particle at time $t$ as assumed in $ S^\prime$-frame is given by $\vec{r}=\left(t^2+1\right) \hat{i} - 6t \hat{j} + 4t^3 \hat{k} $. Calculate the Coriolis acceleration at $t=1$ second. [10 Marks][2013]
Calculate the horizontal component of the Coriolis force acting on a body of mass $ 0.1\mathrm{\ }kg $ moving northward with a horizontal velocity of $ 100{\mathrm{\ }ms}^{-1} $ at $ {30}^\degree N $ latitude on the Earth. [15 Marks][2013]
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? [10 Marks][2016]
Consider two frames of reference $S$ and $S^\prime$ having a common origin $O$. The frame $S^\prime$ is rotating with respect to the fixed frame $S$ with a uniform $\vec{\omega}=3a_x rads^{-1}$. A projectile of unit mass at position vector $\vec{r}=7a_x+4a_y m $ is moving with $ \vec{v}=14a_y ms^{-1}$. Calculate in the rotating frame $S^\prime$ the following forces on the projectile: $\\$ (i) Euler's force $\\$ (ii) Coriolis force $\\$ (iii) Centrifugal force $\\$ [15 Marks][2022]