The kinetic and potential energies for a mechanical system with two generalized coordinates q1 and q2 are
i) Write down the Lagrangian for the system.
ii) Obtain the expressions for the generalized momenta.
iii) Derive the Euler Lagrange equations of motion for the system.
iv) Obtain the energy function for the system.
i) Lagrangian
The Lagrangian \( L \) is given by the difference between the kinetic energy \( T \) and the potential energy \( V \). If the system has two generalized coordinates \( q_1 \) and \( q_2 \), ______ ____ _________ __________ _________ __________ ____ ____ __________.
________ __________ ________ ________ _______ _______ ___ _________ __________ ___.
__________ ________ _________ _______ _________.
____ ____ ____ ___ _________ ______ ______ __________ __________ _________ _____.
______ ______ _________ _______ ____ ________ _____ ___ _______ __________.
______ _________ ______ _________ ________ __________ ________ ________ ______ ______ _________ ______.
__________ _____ __________ ______ ____.
_________ ________ ___ _________ __________ _____ _________ _______ ________ _____ ___ ___.
_____ ________ _____ _______ ______ _____.
________ _____ __________ ____ _____ _____ ______ ________.
_________ ____ ______ _____ ____.
_____ ______ _______ ____ _________ __________ ____ _______ _____.
___ ________ _________ __________ _________ ____ _______ _____ _______ ______.
_____ ____ ____ ____ _________ _________.
________ _________ ____ _____ _______.
____ ______ _____ _______ ________ ___.
_____ _______ ________ __________ ____ ____ __________ __________ ________ _________ _____ _____.
____ __________ ________ __________ _________ ___ _______.
____ __________ _____ ___ _____ __________ _______.
____ __________ __________ ____ _________ _____ ____.
________ ____ _______ ______ ______ _______ _____ ____.
Get Full Answer on WhatsApp
A particle of mass m moving to the right with an initial velocity u collides elastically with a particle of unknown mass M coming from the opposite direction. After the collision m has a velocity u/2 at right angles to the incident direction, and M is deflected back making an angle of 45 degrees to its incident direction as shown below. Calculate the ratio M/m.
Write down the equations of constraint/s for the following systems and classify them as holonomic/nonholonomic and scleronomic/rheonomic:
i) a particle of mass m sliding down the surface of a sphere of radius R without friction, under the action of gravity.
ii) a simple pendulum of length L for which the point of support is oscillating vertically with an angular speed and amplitude A.
Using the D' Alembert’s Principle determine the equation of motion for an Atwood’s machine with masses M and 3M
A particle is scattered by a rigid sphere of radius R , as given in the figure. Determine the differential scattering cross-section and the total scattering cross-section.
A particle moves under a central force, . Is this an attractive or a repulsive force? Find the radius and energy of the circular orbits.
The kinetic and potential energies for a mechanical system with two generalized coordinates q1 and q2 are
i) Write down the Lagrangian for the system.
ii) Obtain the expressions for the generalized momenta.
iii) Derive the Euler Lagrange equations of motion for the system.
iv) Obtain the energy function for the system.
Consider three identical springs and two masses m1 = m and m2 = 2m , as shown in the figure. The motion of the two masses are constrained along the line joining them. Find the normal mode frequencies and the normal coordinates.
