- #1

abercrombiems02

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Perhaps this occurs when the horizontal component of the velocity is greater than the rate of change of the sphere's curvature in the x direction?

Any hints, ideas? Thanks

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- Thread starter abercrombiems02
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- #1

abercrombiems02

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Perhaps this occurs when the horizontal component of the velocity is greater than the rate of change of the sphere's curvature in the x direction?

Any hints, ideas? Thanks

- #2

Doc Al

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- #3

abercrombiems02

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Hmmm, I seem to be having some trouble finding a soltuion. If i define u_r to be radially outward, and u_theta, to be tangential to the circle I obtain the follwoing EOMS.

N - mgcos(theta) = -m*omega^2*r (u_r equation)

mgsin(theta) = m*omega_dot*r (u_theta equation)

The condition that when the two separate is when N = 0, so I have the set of equations

g/r*cos(theta)=(d_theta/dt)^2

g/r*sin(theta) = d2_theta/dt^2

I tried solving the bottom one in simulink. Its actually quite a simple model. I just let r = 9.8 m so that the ratio g/r = 1 s^-2. Then I plotted

g/r*cos(theta)-(d_theta/dt)^2 as a function theta. (This is the equation for the normal force), however it never is zero.

Are my EOM's incorrect? I believe I did the dynamics of this problem correctly...

N - mgcos(theta) = -m*omega^2*r (u_r equation)

mgsin(theta) = m*omega_dot*r (u_theta equation)

The condition that when the two separate is when N = 0, so I have the set of equations

g/r*cos(theta)=(d_theta/dt)^2

g/r*sin(theta) = d2_theta/dt^2

I tried solving the bottom one in simulink. Its actually quite a simple model. I just let r = 9.8 m so that the ratio g/r = 1 s^-2. Then I plotted

g/r*cos(theta)-(d_theta/dt)^2 as a function theta. (This is the equation for the normal force), however it never is zero.

Are my EOM's incorrect? I believe I did the dynamics of this problem correctly...

Last edited:

- #4

abercrombiems02

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BTW, is there an analytical solution to the set of equations, or is the only way to solve them using numerical methods such as all the built in junk in MATLAB?

- #5

Doc Al

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That's one equation that you need, but I'll rewrite it in terms of tangential speed instead of omega (taking N = 0 as the condition for breaking contact):abercrombiems02 said:N - mgcos(theta) = -m*omega^2*r (u_r equation)

[tex]mg \cos \theta = m v^2 / r [/tex]

Combine this with an expression for speed (or kinetic energy) as a function of theta (use conservation of energy) and you'll get a simple analytical solution.

- #6

abercrombiems02

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