Obtain an expression for the angular speed of the Earth at which Coriolis force makes objects fly from its surface.

A satellite revolves in a circular orbit around the Earth at a certain height above it. Calculate the time period of revolution of the satellite, if the radius of the Earth is significantly higher than the height at which the satellite revolves.

A man has a mass of 100\text{ kg} on earth. When he is on the space craft an observer from the earth registers his mass as 102\text{ kg}. Determine the speed of the space craft.

State Hamilton's principle for the motion of a monogenic system. Use the calculus of variation to deduce Lagrange's equation that follows from Hamilton's principle. Explain carefully all the terms used in the derivation.

Express Lagrange's equation of motion for the cyclic coordinate q_j and show that the result leads to the general conservation theorem for the generalized momentum coordinates.

Using the concept of the operator equation for a given vector quantity of a rotating body, show that \vec{v}_s = \vec{v}_r + \vec{\omega} \times \vec{r} where \vec{v}_s and \vec{v}_r are the velocities of the particle relative to space and rotating sets of axes while \vec{\omega} is the angular velocity of the earth relative to the inertial system. Obtain the expression for the Coriolis force for such a moving system of mass m.

Starting with Newton's second law of motion, establish D'Alembert's principle and discuss its significance.

Using D'Alembert's principle, show that the following relation can be obtained for a system of particles under generalized coordinates \sum_i \vec{F}_i \cdot \delta \vec{r}_i = \sum_j Q_j \delta q_j with Q_j = \sum_i \vec{F}_i \cdot \frac{\partial \vec{r}_i}{\partial q_j}. What is the significance of Q_j ?

The principal moments of inertia of a body at a point are given as 200, 300 and 450\text{ gm.cm}^2. Write down the equation of the ellipsoid of inertia at that point.

A rigid body rotates with the angular velocity '\omega' about an axis through the origin O and having direction cosines l, m, n. Show that the moment of inertia of the rigid body about the axis is I = I_{xx} l^2 + I_{yy} m^2 + I_{zz} n^2 + 2 I_{xy} l . m + 2 I_{yz} m . n + 2 I_{zx} n . l where the symbols have their usual meanings.

A particle of mass 'm' moves according to the equations x = a \cos \omega t, y = a \sin \omega t, z = c, where a, c, \omega are constants. Obtain the instantaneous velocity and linear momentum vectors in terms of the Cartesian components and hence the angular momentum. Find the force \vec{F} and the torque \vec{N} acting on the particle and verify that the angular momentum \vec{L} and the torque \vec{N} satisfy the relation \frac{d\vec{L}}{dt} = \vec{N}.

A particle of rest mass 'M' moving at a velocity 'u' collides with a stationary particle of rest mass 'm'. If the particles stick together, show that the speed of the composite ball is equal to v = \frac{u \gamma M}{(\gamma M + m)}, \quad \text{where } \gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}}

(i) An electron of rest mass m_0 moves with a velocity v such that its total energy is double of its rest mass energy. What is the electron velocity?

Establish the relation between the angular momentum and torque of a particle. Show that this relation leads one to the principle of conservation of angular momentum.

(ii) At what velocity, the electron momentum will be m_0 c?

How can one introduce the constraints of motion through the concept of generalized coordinate systems? Write down the set of transformation equations for a system of N particles relating the generalized coordinates with the real coordinates.

(i) For a particle of mass m moving with velocities \vec{v}_s and \vec{v}_r relative to the space and rotating axis, respectively, show that the equation of motion is obtained as \vec{F}-2m(\vec{\omega}\times\vec{v}_r)-m\vec{\omega}\times(\vec{\omega}\times\vec{r}) = m\vec{a}_r with \vec{\omega} and \vec{a}_r being the angular velocity and acceleration in rotating coordinates.

Show that the magnitudes of the centripetal and Coriolis accelerations of the earth while rotating counter-clockwise about the north pole are around 0\cdot 034\text{ m sec}^{-2} and (1\cdot 46 \times 10^{-4} v)\text{ m sec}^{-2}, respectively. In obtaining these results, consider the ratio of sidereal days to solar days in a year as (366\cdot 5/365\cdot 5) and the radius of the earth along the equator as 6400\text{ km}.

(iii) Discuss the important role of Coriolis forces in the circulation pattern of winds.

(ii) Which term in the above equation represents the Coriolis force?