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Adam Johnson

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Missing homework template due to originally being posted in other forum.

I am trying to solve a problem using Newtonian Mechanics. I was able to solve it using Lagrangian Mechanics, and got the correct answer from the textbook, but when I use Newtonian I get the wrong answer. I want to be able to do it both ways in order to understand/appreciate the difference between Newtonian and Lagrangian.

A particle of mass m rests on a smooth plane. The plane is raised to an inclination angle θ at a constant rate α (θ=0 at t=0), causing the particle to move down the plane. Determine the motion of the particle.

If we designate the starting position of the particle to be r0 (distance from the axis around which the plane rises), and r(t) to be the distance traveled by the particle down the ramp at a given time t, then r(t)=∫r'(t)dt from 0→t, where r'(t)=∫r''(t)dt from 0→t.

Now r''(t) is just the component of gravity directed down the ramp, so r''(t)=g*sin(αt).

∴ r'(t)=∫g*sin(αt)dt=g/α (1-cos(αt)). Then r(t)=∫r'(t)dt=g/α∫(1-cos(αt))dt=g/α (t - 1/α sin(αt))

If you do it using Lagrangian mechanics, you get r0(1-cosh(αt))-(g/2α^2)*(sin(αt)-sinh(αt)).

Any insight on logical errors I have made with my Newtonian approach would be greatly appreciated. Thanks!

__The problem__A particle of mass m rests on a smooth plane. The plane is raised to an inclination angle θ at a constant rate α (θ=0 at t=0), causing the particle to move down the plane. Determine the motion of the particle.

__My attempt at a solution__If we designate the starting position of the particle to be r0 (distance from the axis around which the plane rises), and r(t) to be the distance traveled by the particle down the ramp at a given time t, then r(t)=∫r'(t)dt from 0→t, where r'(t)=∫r''(t)dt from 0→t.

Now r''(t) is just the component of gravity directed down the ramp, so r''(t)=g*sin(αt).

∴ r'(t)=∫g*sin(αt)dt=g/α (1-cos(αt)). Then r(t)=∫r'(t)dt=g/α∫(1-cos(αt))dt=g/α (t - 1/α sin(αt))

If you do it using Lagrangian mechanics, you get r0(1-cosh(αt))-(g/2α^2)*(sin(αt)-sinh(αt)).

Any insight on logical errors I have made with my Newtonian approach would be greatly appreciated. Thanks!