# : Lagrange Multipliers E,=0 A disc of mass m and radius R has a string wrapped around it with the end attached to a fixed

: Lagrange Multipliers

E,=0

A disc of mass m and radius R has

a string wrapped around it with the

end attached to a fixed support. The

string is unwinding as the disk falls.

Take I =

mR. We will denote by r

the length of the string and by 0′, the

angular speed of the disc.

1. Express the constraint cquation, of the rolling without slipping dise, in

polar coordinate (r.0)

2. Write the modified Lagrangian equation

3. Derive the differential equations of motion for each coordinate

4. Calculate the Lagrange multiplier A. in terms of m, g

5. Calculate the equation of constraints Q, and Qe : Lagrangian Dynamics

Ep = 0

A pendulum of length / and mass m is

mounted on a block of mass M. The

block can move freely without friction

on a horizontal surface as shown in the

adjacent figure

H.

1. Find the velocity of mass m, w.r.t the origin O

2. Write the Lagrangian of the system

3. Derive the Euler Lagrange equations