Friction in a hemispherical bowl

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In summary, the mass slides slowly down a hemispherical bowl with friction. Calculating the speed without friction and using the energy loss to calculate the new speed gives an approximate solution that is not bad for μ=0.60.
  • #1
f todd baker
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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.
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  • #2
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.
 
  • #3
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.
 
  • #4
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)).
 
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Likes mfb and berkeman
  • #5
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.
 
  • #6
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.
 

Related to Friction in a hemispherical bowl

1. What causes friction in a hemispherical bowl?

Friction in a hemispherical bowl is caused by the resistance between two surfaces when they come into contact with each other. In this case, it is the resistance between the surface of the bowl and any object placed inside it.

2. How does the shape of a hemispherical bowl affect friction?

The shape of a hemispherical bowl can affect friction in several ways. The curved surface of the bowl can create a larger contact area between the bowl and the object inside, increasing the friction. Additionally, the smoothness or roughness of the bowl's surface can also impact the amount of friction.

3. Is friction in a hemispherical bowl always constant?

No, friction in a hemispherical bowl can vary depending on several factors. The weight and material of the object placed inside the bowl, as well as the surface of the bowl itself, can all affect the amount of friction present in a given situation.

4. Can friction in a hemispherical bowl be reduced?

Yes, friction in a hemispherical bowl can be reduced by using a lubricant or by changing the surface of the bowl to one that is smoother. Additionally, reducing the weight of the object placed inside the bowl can also decrease the amount of friction.

5. How does friction in a hemispherical bowl affect the movement of objects inside it?

Friction in a hemispherical bowl can create a resistance that makes it more difficult for objects inside to move. This can affect the speed and direction of the object's movement, as well as the force required to move the object. Friction can also cause objects to come to a stop sooner than they would have without it.

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