CMI CM-I Problem Sheet EndSem 2012

1. Consider a particle thrown from a height of H from the ground vertically up with a speed $U$ at a latitude $\lambda$. Taking the earth's rotation up to the first order in it's angular velocity $\vec{\omega}$ find the position where it will fall on the ground.  Further find the height H for which the particle will fall vertically down on the ground.( let east be along $\hat{\epsilon}_1$, and vertical along $\hat{\epsilon}_3$.)

2. a rod is moving along with velocity $v\hat{\epsilon}_1$ and makes an angle $\theta$ with $x_1$ - axis as observed from frame S. A frame S' is moving with a velocity $U\hat{\epsilon}_2$. What will be angle made by the rod is S' with respect to the  $x'_1$ axis?
3. Consider two particles having masses $m_1,m_2$ and described by  coordinates $x_1\,,\,x_2$ moving under the  potential $$ V(x_1,x_2)\ =\ \frac{\Omega^2}{M}(m_1\,A\,x_1^2\ +\ 2\,(m_1\,m_2)^{1/2}\,B\,x_1\,x_2\ +\ m_2\,A\, x_2^2) $$ where $A\, ,\, B$ are dimensionless constants. The normal modes have frequencies $ \sqrt{3}\Omega$ and $\sqrt{2}\Omega$. Find the normal modes in terms of $x_1\, , \, x_2$ and the constants $A\, ,\,   B$.
3. A comet moves around the sun starting from far away with an initial velocity $\vec{v}_0$. The sun's mass is M and its radius is R. Using conservation laws find the cross section of the comet striking the sun in terms of $G\, , M\, , R\, $ and $v_0$.

4. (a) A particle of mass m moves on the surface $x_3^2\ =\ x_1^2\,+\,x_2^2$ with( $x_3\,>\, 0)$ is under the influence of the gravitational potential $V\, =\, \,g\,x_3$. Choosing . Choosing appropriate coordinates write down the Lagrangian and obtain the Euler-Lagrange equations. Obtain the constants of motion.
   (b) Obtain the Hamiltonian for the above problem. Show that the Hamiltonian equations lead to the same equation as the Euler-Lagrange equations.

5. (a) An inextensible massless string passes over a massless pulley attached to a ceiling. The two  masses $m_1\, , \, m_2$ are attached to the two ends of the string. Using the concept of virtual work show that the  work done by constraint forces is zero. (5 marks)
   (b) Consider the ( two- dimensional)  motion of a particle under the action of a force
$$ \vec{F}(x_1,x_2)\, = \, (A\, x_2^2\, +\ B\, x_1\,x_2)\hat{\epsilon}_1\, +\, (C\, x_1^2\,+\,D\,x_1\,x_2)\hat{\epsilon}_2 $$
where $A,B,C$ and $D$ are constants.  If the force can be expressed in terms of a potential find relations between the constants. Obtain the potential. (5 marks)  

6. Consider two particles having masses $m_1,m_2$ and described by  coordinates $x_1\,,\,x_2$ moving under the potential
$$ V(x_1,x_2)\ =\ \frac{\Omega^2}{M}(m_1\,A\,x_1^2\ +\ 2\,(m_1\,m_2)^{1/2}\,B\,x_1\,x_2\ +\ m_2\,A\, x_2^2) $$
where $A\, ,\, B$ are dimensionless constants. The normal modes have frequencies $ \sqrt{3}\Omega$ and $\sqrt{2}\Omega$.

Find the constants $A\, ,\, B$ .