Lagrangian problems, inclined planes

Problem:

A wedge of mass M rests on a horizontal frictionless surface.  A point mass m is placed on the wedge, whose surface is also frictionless.  Find the horizontal acceleration a of the wedge.

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Solution:

Problem:

A wedge of mass M = 4.5 kg sits on a horizontal surface.  Another mass m = 2.3 kg sits on the sloping side of the wedge.  The incline is at an angle θ = 31.7 degrees with respect to the horizontal, and g = 9.8 m/s.  All surfaces are frictionless.  The mass m is released from rest on mass M, which is also initially at rest.  Use Lagrangian mechanics to determine the vector values of the accelerations of both M and m once the mass is released.

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Solution:

Problem:

The system, shown in the diagram, consists of a wedge placed on a horizontal surface and a block connected to a string that passes over a massless pulley attached to the wedge.  The other end of the string is attached to the wall so that the string is horizontal between the wall and the pulley, and parallel to the inclined surface between the pulley and the block. The wedge and the block have equal masses m.  The angle θ is given.  When the system is released from rest the block begins to slide along the inclined surface of the wedge.  Neglecting all friction, find the acceleration of the wedge.

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Solution:

Problem:

Consider a hoop of mass m and radius r rolling without slipping down an incline.
(a)  Determine the Lagrangian L(x, dx/dt) of this one-degree-of-freedom system.  Derive from it the Lagrange equation and its solution for initial condition x0 = 0, dx/dt|0 = 0.
(b)  Determine the alternative Lagrangian L(x, dx/dt, θ, dθ/dt)  and the holonomic constraint f(x, θ) = 0 that must accompany it.  Derive the associated three equations of motion for the two unknown dynamical variables x and θ, and the undetermined Lagrange multiplier λ.  Solve these equations for the same initial conditions as in (a) and determine the static frictional force of constraint between the hoop and the incline.

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Solution:

Problem:

Two spheres are of the same radius R and mass M, but one is solid and the other is a hollow shell (of negligible thickness).  Both spheres roll (without sliding) down a ramp of incline θ.
(a)  Which sphere will have the greater acceleration down the ramp?
(b)  Determine the Lagrangian for the motion of the sphere and derive the equation of motion for both cases.

Solution: