Question

A particle of mass M is constrained to move in a vertical plane along a trajectory given by x = Acos(?) , y = Asin(?) , where A is constant. The Lagrangian function can be given as : (Hint take V=0 at Y=0) take two generalized coordinates (r, ?) y r sin ? r ? r cos ? x 1/2 mA²??² + mgcos(?) 1/2 mr??² - mgr 1/2 mA²??² - mgsin(?) 1/2 mA?? - mgcos(?) 1/2 mA??² + mgsin(?)

          A particle of mass M is constrained to move in a vertical plane along a trajectory given by x = Acos(?) , y = Asin(?) , where A is constant. The Lagrangian function can be given as : (Hint take V=0 at Y=0) take two generalized coordinates (r, ?)

y
r sin ?
r
?
r cos ?
x

1/2 mA²??² + mgcos(?)
1/2 mr??² - mgr
1/2 mA²??² - mgsin(?)
1/2 mA?? - mgcos(?)
1/2 mA??² + mgsin(?)

A particle of mass M is constrained to move in a vertical plane along a trajectory given by x = Acos(?) , y = Asin(?) , where A is constant. The Lagrangian function can be given as : (Hint take V=0 at Y=0) take two generalized coordinates (r, ?)

y
r sin ?
r
?
r cos ?
x

1/2 mA²??² + mgcos(?)
1/2 mr??² - mgr
1/2 mA²??² - mgsin(?)
1/2 mA?? - mgcos(?)
1/2 mA??² + mgsin(?)

Added by Andrea L.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Manish Jain

01/12/2022

Step 1

- The Lagrangian function \(L\) is given by \(L = T - V\), where \(T\) is the kinetic energy and \(V\) is the potential energy. - The kinetic energy \(T\) is expressed as \(T = \frac{1}{2}M(\dot{r}^2 + (r^2)(\dot{\theta}^2))\), where \(\dot{r}\) and Show more…

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A particle of mass M is constrained to move in a vertical plane along a trajectory given by X = Acos(θ) and Y = Asin(θ), where A is constant. The Lagrangian function can be given as (Hint: take V=0 at Y=0). Take two generalized coordinates (r, θ): r = A*cos(θ) z = A*sin(θ) The Lagrangian function is: L = T - V where T is the kinetic energy and V is the potential energy. The kinetic energy T can be expressed as: T = (1/2)*M*(ṙ^2 + (r^2)*(θ̇^2)) where ṙ and θ̇ are the time derivatives of r and θ, respectively. The potential energy V can be expressed as: V = -M*g*z where g is the acceleration due to gravity. Therefore, the Lagrangian function L can be written as: L = (1/2)*M*(ṙ^2 + (r^2)*(θ̇^2)) + M*g*A*sin(θ)