Problem 3 A hoop of mass m and radius R, rolls without slipping down an inclined plane of angle a. The inclined plane of mass M moves horizontally. Let X be the abscissa of point O' with respect to 0, and s be the abscissa of the center of mass of the rolling hoop with respect to O'. Given: the moment of inertia of the hoop is I mR, and the condition of rolling without slipping is expressed ass = RO: 1. Write the Lagrangian of the system. 2. Assuming that the Lagrangian of the sy stem (inclined plane and hoop) is given by: L = ms" +(m + M)X" + ms'X'cos(a) + mg(s)sin(a) 3. Write the expression of the Lagrangian function

Problem 3 A hoop of mass m and radius R, rolls without slipping down an inclined plane of angle a. The inclined plane of mass M moves horizontally. Let X be the abscissa of point O' with respect to 0, and s be the abscissa of the center of mass of the rolling hoop with respect to O'. Given: the moment of inertia of the hoop is I mR, and the condition of rolling without slipping is expressed ass = RO: 1. Write the Lagrangian of the system. 2. Assuming that the Lagrangian of the sy stem (inclined plane and hoop) is given by: L = ms" +(m + M)X" + ms'X'cos(a) + mg(s)sin(a) 3. Write the expression of the Lagrangian function

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Forming and Shaping

Metal forming is a category of metalworking process through which the material can be given the desired shape and size by subjecting it to plastic deformation, where the component undergoes bulk deformation process upon application of loads. Unlike the shearing and machining process, the material does not undergo up to a failure point as indicated in the stress-strain curve. Hence, the stresses induced during the forming process are greater than yield strength but less than the failure strength of the material. The process does not lead to the loss of material, so it can be said to be an economical process. During forming, the material is subjected to tensile forces, compressive forces, bending, and shearing loading conditions. The material should possess the property of ductility to undergo the forming process.

Question

Problem 3
A hoop of mass m and radius R, rolls without slipping down an inclined
plane of angle a. The inclined plane of mass M moves horizontally. Let X
be the abscissa of point O' with respect to 0, and s be the abscissa of the
center of mass of the rolling hoop with respect to O'. Given: the moment of
inertia of the hoop is I = mR, and the condition of rolling without slipping
is expressed as s = RO:
1. Write the Lagrangian of the system.
2. Assuming that the Lagrangian of the system (inclined plane and hoop)
is given by:
L = ms" +(m + M)X" + ms'X'cos(a) + mg(s)sin(a)
3. Write the expression of the Lagrangian function
4. Derive the Euler Lagrange equations
5. Find s" and X" in terms of the masses (m,M), angle a and g
Reference level
m,R
M
a
ta"

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