Rolling without slipping on a curved surface

[NTesla]

Homework Statement

A solid spherical ball of mass m and radius r rolls without slipping on a rough concave surface of
large radius R. Find the magnitude and direction of friction on the ball.

Relevant Equations

$$ \tau = I\alpha $$, F = ma

Kindly see the attached pdf. My attempt to solve it, is in it.

I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction.

I'm not able to figure out, why my solution is wrong, if it is wrong .

 

[haruspex]

the ball is undergoing angular simple harmonic motion

No, that is only an approximation for small displacements. But you don’t use it in what you posted, and the question as you have posted it doesn’t ask about period. I presume there are more parts to the question.

the acceleration of point of the ball which is instantaneously in contact with the surface, must be zero.

No, its instantaneous velocity is zero, but its acceleration will be nonzero and normal to the surface, so its tangential component is zero.

Your answer looks correct.

 

[kuruman]

Your last sentence is
"The minus sign here means that the direction of friction is up the incline."
Is it always up the incline? You need to specify the direction over an entire cycle of the motion.

 

[haruspex]

Your last sentence is
"The minus sign here means that the direction of friction is up the incline."
Is it always up the incline? You need to specify the direction over an entire cycle of the motion.

The OP's analysis appears to be independent of the point in the cycle (other than assuming a nonzero magnitude for the frictional force).

 

[NTesla]

No, its instantaneous velocity is zero, but its acceleration will be nonzero and normal to the surface, so its tangential component is zero.

Yes, that's right i think. Thanks for pointing that out.

Your answer looks correct.

I dont think it is correct. The reason is that when I'm asking AI the same question, it is giving me another answer where the magnitude of $f = 2mgsinθ/3$. And it shows the calculation too. I can't find fault in the calculation shown by the AI. But I also can't find problem with my calculation. However, both can't be right. Here's the calculation shown by the AI:

 

[NTesla]

Is it always up the incline? You need to specify the direction over an entire cycle of the motion.

More than the direction, I'm worried about the magnitude of the frictional force. Kindly see post#5 above, wherein I've shown the calculation done by AI, wherein the magnitude is not $2/7mgsinθ$ but $2/3mgsinθ$. Kindly let me know which calculation is wrong and why: mine or that of AI.

About the direction of friction in a whole cycle: For all those moment when the ball is not at point C1 (the bottom most point), the friction direction will be up the incline. At the bottom most point, I suppose the friction will be zero. so the direction of friction will be indeterminate at the bottom most point.

 

[haruspex]

Yes, that's right i think. Thanks for pointing that out.

I dont think it is correct. The reason is that when I'm asking AI the same question, it is giving me another answer where the magnitude of $f = 2mgsinθ/3$. And it shows the calculation too. I can't find fault in the calculation shown by the AI. But I also can't find problem with my calculation. However, both can't be right. Here's the calculation shown by the AI:

View attachment 366163

I have confirmed your result independently. Will try to locate the error in the AI solution, but rightnowI am a passenger on a very bumpy road!

 

[NTesla]

I have confirmed your result independently. Will try to locate the error in the AI solution, but rightnowI am a passenger on a very bumpy road!

Appreciate it very much, you taking time to help out. Will definitely wait for your answer.

P.S: Hope your journey is worth the destination.