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.
For each of the given systems, let's write the equations of constraint and classify them as holonomic/nonholonomic _________ __________ _____ ___ _________ _______.
__________ __________ ___ ________ ______ ___ ________ _____ _________ ___.
____ _____ ________ ________ _________ _______ ____ ________ _____ _______.
_______ _____ ____ _____ _______ _____ ____.
_________ __________ _________ ______ ____ _________ ________ ________ _____.
__________ ______ _____ _______ _____ __________.
_______ ____ ______ __________ __________ _______ _______ ____.
______ ____ __________ ______ ______ _____ ______ __________ _______ _______.
________ ___ _____ _______ ____ ___ ____ ________ ______ ________.
___ __________ __________ _____ _______ ____ ______.
_____ ___ _________ ________ _________.
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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.
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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.
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