Question

Consider a particle of mass m that moves frictionlessly on the inner surface of a paraboloid defined by x^2 + y^2 = az as shown in the figure. a) Choose a coordinate system suitable for the symmetry of the problem and write the Lagrangian function of the system. (U_g(z = 0) = 0) b) By writing the bond condition, using the Lagrangian Factors method, Get the equations of motion. c) At constant z=h height, the object is in a horizontal circular orbit. Determine the Lagrangian multiplier based on the angular velocity ??, taking into account that the olarak belirleyiniz.

          Consider a particle of mass m that moves frictionlessly on the inner surface of a paraboloid defined by x^2 + y^2 = az as shown in the figure.

a) Choose a coordinate system suitable for the symmetry of the problem and write the Lagrangian function of the system.

(U_g(z = 0) = 0)

b) By writing the bond condition, using the Lagrangian Factors method, Get the equations of motion.

c) At constant z=h height, the object is in a horizontal circular orbit.

Determine the Lagrangian multiplier based on the angular velocity ??, taking into account that the olarak belirleyiniz.

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Consider a particle of mass m that moves frictionlessly on the inner surface of a paraboloid defined by x^2 + y^2 = az as shown in the figure.

a) Choose a coordinate system suitable for the symmetry of the problem and write the Lagrangian function of the system.

(Ug(z = 0) = 0)

b) By writing the bond condition, using the Lagrangian Factors method, Get the equations of motion.

c) At constant z=h height, the object is in a horizontal circular orbit.

Determine the Lagrangian multiplier based on the angular velocity ??, taking into account that the olarak belirleyiniz.

Added by Amber D.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sam Stansfield

09/19/2024

Step 1

Since the problem has cylindrical symmetry, it is convenient to use cylindrical coordinates (r, θ, z). The position vector of the particle can be written as: r = r(cosθ, sinθ, z) Now, we need to find the velocity vector of the particle. We differentiate the Show more…

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Consider a particle of mass m that moves frictionlessly on the inner surface of a paraboloid defined by x^2 + y^2 = az as shown in the figure. a) Choose a coordinate system suitable for the symmetry of the problem and write the Lagrangian function of the system. (U_g(z=0) = 0) b) By writing the bond condition, using the Lagrangian Factors method, Get the equations of motion. c) At constant z=h height, the object is in a horizontal circular orbit. Determine the Lagrangian multiplier based on the angular velocity ̙ϕ, taking into account that the olarak belirleyiniz.

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Transcript

00:01 So i think even though it doesn't say it explicitly that we're supposed to take gravity into account.

00:06 We have a particle of mass m that is moving on the surface of a paraboloid that's defined like this.

00:20 We're going to treat that as a constraint.

00:24 So our lagrangian is t -mi -sv.

00:29 I'm going to use cylindrical coordinates.

00:40 Okay.

00:40 So our constraint then is that row squared is a times z and our potential energy function, our v actually, our gravitational potential is mgz.

01:01 So that it's z equals zero, the potential energy is zero.

01:08 So this is half m, row dot squared plus roe squared, phi dot squared minus, or plus z dot squared minus mgs.

01:24 That's our lagrangian.

01:28 All right.

01:44 So we go and find our lagrangian equations, partial l with respect to row dot.

02:12 So we get the lagrangian equations.

02:34 And then that has to equal the derivative of this with respect to row, it's called this f with our negrange multiplier.

02:54 Actually, i'm going to write an f not like that, like this.

03:11 And then the phi part, row squared phi dot, and there's no phi dependence.

03:28 So you've got d by dt or, there's our second equation, and our constraint doesn't depend on phi...

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