2 a bead of mass m is threaded on a frictionless wire that...

2. A bead of mass $m$ is threaded on a frictionless wire that is bent into a helix with cylindrical polar coordinates $(\rho, \phi, z)$ satisfying $z = c\phi$ and $\rho = R$, where $c$ and $R$ are positive constants. The $z$ axis is points vertically up and gravity acts vertically down. (a) Using $\phi$ as the generalized coordinate, write down the Lagrangian $L$ of the bead. (b) write down the equation of the motion of the bead.

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00:01 Hi students in this question.

00:02 A beat in cylindrical polar coordinates is being given whose hamiltonian equation has to be determined.

00:07 Now the mass of this bead is taken a small letter m and it is given that the value of z ist taken to be c -5.

00:15 On the other hand, value of r is taken to be a value of p which is a constant value.

00:20 Now by taking the first derivatives, we get z dot to be equal to c into 5 dot.

00:27 On the other hand value of r dot will be equal to p dot which will be equal to a value of zero.

00:33 Let us assume these two equations as equation number a.

00:36 Now the kinetic analogy of this particular beat can be taken as t which is equal to 1 by 2 into m into p dot square plus p square into 5 dot square plus z dot square.

00:51 Which on solving we get the value of kinetic energy t to be equal to 1 by 2 into m into 5 .2.

01:00 2 .2 .m.

01:00 Into 5 .2.

01:00 Into r square into r square plus c square.

01:03 Let this be equation number 1.

01:05 Similarly, the potential energy of the bead is taken as v which is equal to m g into z which is equal to m g into c5.

01:15 Let this be equation number 2.

01:17 Now the lgarine of this bead is taken as capillator l which is equal to the kinetic energy t minus potential energy b...