Friction in a hemispherical bowl

[f todd baker]

I have been trying to find the speed of a point mass sliding with friction (f=μN) in a hemispherical bowl. Start at the top at rest. So far I have -μN+mgcosθ=ma and N=mgsinθ+mv²/R where θ is the angle below horizontal from the center of the sphere and R is the radius of the sphere. I know it likely does not have a simple solution but would be happy with an approximate solution for small μ.

 

[mfb]

For an exact solution you'll have to solve a differential equation.

For an approximation for small μ, you can calculate the speed without friction, then determine friction and energy loss based on that speed, then calculate a new speed based on the initial value and the energy loss. That gives an easier integral instead of a differential equation.

 

[f todd baker]

Right you are, and as I see it there are two coupled second-order, nonlinear equations. Good idea to use v(θ,μ)≈v((θ,0) to calculate energy lost. I'll try it.

 

[f todd baker]

I think the suggestion by mfb was a good one. Using the frictionless v(θ)=√(2gRsinθ) I find v≈√(2gR(1-3μ)) at the bottom. For comparison, a path on a 45⁰ incline with the same drop has v=√(2gR(1-μ)).

 

[mfb]

Out of curiosity, I simulated the setup. With μ=0.001 and m=R=g=1 I got E=0.99701, in agreement with your result. μ=0.01 leads to E=0.9704. Even with μ=0.1, the approximation is not bad: E=0.732.

The mass stops at the center for μ>0.60. Tested with 100, 200 and 500 steps: The critical value is somewhere between 0.603 and 0.605. There is no obvious mathematical constant in that range.

 

[f todd baker]

This is beautiful. I will post your results on my web site. (I have had my wrist slapped here before by mentioning it explicitly, so will not give you the exact link!) I will link back here so that you will be properly acknowledged. Thanks.