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 nature of the central force depends on its direction relative to the particle. A central force is generally written as F⃗=f(r)r^\vec{F} = f(r) \hat{r}, where f(r)f(r) is a scalar function of the distance ______ ___ ____ ____ _________ _____ _________ _________ ___ _________ __________ _____.
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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.
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.
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.
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.
A particle moves under a central force, . Is this an attractive or a repulsive force? Find the radius and energy of the circular orbits.
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.
