An Interesting Physics Problem
 

What is the maximum slope that a solid sphere of uniform density can roll down without slipping, if the coefficient of static friction between the sphere and the incline is 1/2?

Re: An Interesting Physics Problem
 

26.6 degrees (from the horizontal)?

That can't be it or else this wouldn't be interesting...I give up.

Edit - Can I have multiple answers? I want to add an answer of 63.4 degrees from the horizontal.

Re: An Interesting Physics Problem
 

Neither of those two answers appear to be correct.

Perhaps if you provided a worked out solution.

Edit: Ah, never mind. I see how you got that.

Definitely not.

Re: An Interesting Physics Problem
 

Vertical.











(you didn't say I couldnt add some, uuummmm.... prespin [img]/images/graemlins/smile.gif[/img] )

Re: An Interesting Physics Problem
 

[ QUOTE ]

Vertical.


[/ QUOTE ]

Wrong.









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(you didn't say I couldnt add some, uuummmm.... prespin [img]/images/graemlins/smile.gif[/img] )

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Prespin all you want. It will still slip. Do you see why?

Re: An Interesting Physics Problem
 

I think it's atan(2*mu), which here is 45deg. Post a solution in a bit.

Re: An Interesting Physics Problem
 

[ QUOTE ]
Neither of those two answers appear to be correct.

Perhaps if you provided a worked out solution.

Edit: Ah, never mind. I see how you got that.

Definitely not.


[/ QUOTE ]

Ya, the first one I was sure was wrong and I just realized I did the second one wrong, so here is a reworked solution.

x direction parallel to slope, y direction perpendicular, angle is theta

Sum moments bout point of contact, P:

mgrsin(theta)=Ip*alpha

Ip=2/5mr²+mr²

d²x/dt² = a_x = r/2*alpha

solveing for acceleartion in x direction a_x

a_x = (5/14)*g*sin(theta)

Now summing forces in x direction:

-Ff + mgsin(theta) = m*a_x = m*(5/14)*g*sin(theta)

where Ff = mu*N = 0.5*N = 0.5mg*cos(theta)

-0.5mgcos(theta)+mgsin(theta)=5/14*mg*sin(theta)

solving for theta:

theta = 37.9 degrees from the horizontal.

Re: An Interesting Physics Problem
 

Solid attempt, but no.

Re: An Interesting Physics Problem
 

[ QUOTE ]
I think it's atan(2*mu), which here is 45deg. Post a solution in a bit.

[/ QUOTE ]

Nope.

Let me just give the numerical answer so you'll know when you've gotten it.

The slope is 7/4.

Re: An Interesting Physics Problem
 

[ QUOTE ]
[ QUOTE ]
I think it's atan(2*mu), which here is 45deg. Post a solution in a bit.

[/ QUOTE ]

Nope.

Let me just give the numerical answer so you'll know when you've gotten it.

The slope is 7/4.

[/ QUOTE ]

Oops I think I made one other silly mistake. I get this if I modify one line in my solution:

Instead of:
d²x/dt² = a_x = r/2*alpha

If I use:
d²x/dt² = a_x = r*alpha

I get a slope of 7/4 or angle of 60.3 deg.

Re: An Interesting Physics Problem
 

Not at all how I solved it, but just effective.

I will say the choice of placing your axis of rotation through the point of contact seems conceptually strange to me, since the sphere is simply rotating about an axis through it's center of mass. To each his own though!

Re: An Interesting Physics Problem
 

I started with energy conservation, since the negative work done by friction is equal to the positive work done by the torque due to the friction. So after a displacement x:

mgx sinq = (7/10) mv^2

Differentiating with respect to time:

mgv sinq = (7/5) mva

a = (5/7)g sinq

Newton's 2nd Law:

Fnet_x = mg sinq - F_f = ma = (5/7) mg sinq

(2/7) mg sinq = F_f = mu mg cosq

tanq = (7/2)mu = 7/4

In general, for a round object of moment of inertia I = cmr^2,

slope = tanq = (1 + 1/c) mu

Which is a neat formula.

Re: An Interesting Physics Problem
 

[ QUOTE ]
Not at all how I solved it, but just effective.

I will say the choice of placing your axis of rotation through the point of contact seems conceptually strange to me, since the sphere is simply rotating about an axis through it's center of mass. To each his own though!

[/ QUOTE ]

This approach is typically taught in undergrad engineering courses as one alternative method of solving these types rigid body dynamics problems. It has the advantage that the point of contact is an instantaneous inertially fixed axis of rotation, which can simplify some problems that have a center of mass which is both translating and rotating, like the one you posted.

Re: An Interesting Physics Problem
 

[ QUOTE ]
[ QUOTE ]
Not at all how I solved it, but just effective.

I will say the choice of placing your axis of rotation through the point of contact seems conceptually strange to me, since the sphere is simply rotating about an axis through it's center of mass. To each his own though!

[/ QUOTE ]

This approach is typically taught in undergrad engineering courses as one alternative method of solving these types rigid body dynamics problems. It has the advantage that the point of contact is an instantaneous inertially fixed axis of rotation, which can simplify some problems that have a center of mass which is both translating and rotating, like the one you posted.

[/ QUOTE ]

Yeah, that makes sense.

In fact I probably learned that 15 years ago. Bit decay. [img]/images/graemlins/frown.gif[/img]