Find the minimum height of the obstacle so that the sphere remains in equilibrium.

- ParthKohli

Find the minimum height of the obstacle so that the sphere remains in equilibrium.

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- Astrophysics

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- ParthKohli

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- ganeshie8

obstacle is fixed to the ramp is it ?

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## More answers

- ParthKohli

yes

- ganeshie8

Okay, then I think we want the torque about the point \(P\) to be in counter clockwise direction for equilibrium :
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- ganeshie8

then that counterclockwise torque would be balanced by the normal force from the ramp

- ganeshie8

so i think we simply need to see when the torque becomes 0

- ganeshie8

having dealt with the physics part, rest is just geometry... let me see if i can set up the equations correctly..

- lochana

is the surface rough?

- ganeshie8

friction shouldn't matter for this problem..

- lochana

okay:)

- ganeshie8

we need to check when that gravity force aligns over the pivot point \(P\)

- ParthKohli

you did it, haha

- ganeshie8

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- ganeshie8

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- ganeshie8

@lochana
friction dosn't matter here because, the sphere is not sliding or accelerating along the surface of the ramp.
for breaking equilibrimum, we are lifting the sphere about the point P.
while doing that, the sphere gets rotated "perpendicular" to the ramp, about the point P.
notice that the sphere is not accelerating along the surface of ramp... so there won't be any frictional force on the sphere...
thats my understanding.. i could be wrong though...

- ganeshie8

i think below statement is true, but its not so obvious w/o messing with the math :
```
friction is 0 in rolling motion when the velocity is constant
friction acts only when the velocity changes
```

- ParthKohli

yes, friction is zero because it acts only when relative motion occurs between the bottommost point and the surface. in pure rolling no friction acts if we leave a sphere undisturbed.

- ganeshie8

Ahh right, when the sphere is in pure rolling motion, the velocity of bottommost point, relative to the ramp is 0

- ganeshie8

how do you explain why the friction exists when the sphere accelerates ?

- ParthKohli

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- ParthKohli

When \(v_{CM}\) increases or decreases (acceleration), friction acts to oppose relative motion between the bottommost point. It acts in a way that \(R\omega\) becomes equal to \(v_{CM}\) again.

- ganeshie8

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- ganeshie8

is that right ?

- ParthKohli

that direction is seen when the wheel is rotating way too fast and is translating slowly. so static friction acts forward in order that the translational velocity increases, rotational velocity decreases, and rolling is attained again.

- ganeshie8

what would be the direction of static friction in our case, if the obstacle is removed and sphere starts to accelerate along the ramp under the influence of gravity ?

- ganeshie8

would it be forward or backward ?

- ParthKohli

backwards

- ganeshie8

why

- ParthKohli

because see, it tends to translate downwards due to gravity, but gravity does not provide any torque to the wheel so as to make it rotate. so it needs something to rotate clockwise, which is provided by the static friction of course.

- ganeshie8

gravity produces no torque about the center of mass of sphere

- ParthKohli

exactly, so friction has to do that job

- ganeshie8

isn't it conflicting with the previous explanation of why frction must be in the same direction as acceleration of the CM ?
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- ParthKohli

i never said that static friction is in the same direction as v_cm.

- ganeshie8

on a horizontal ramp, if the sphere is accelerating, the static friction is in the same direction is v_cm right ?

- ParthKohli

not at all... it depends.
think about it this way:
if a wheel is being rotated but not translated by any force, then it needs friction to translate it.
if a wheel is being translated but not rotated by any force, then it needs friction to rotate it.
in absence of friction, this is the second case, right? because gravity is translating it and no other force is available to rotate it other than friction.

- ParthKohli

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- ParthKohli

this is the first case ^
think about the direction of friction here

- ganeshie8

that is true only when "forces" act on the wheel
friction will be 0 when the center-of-mass of the wheel moves with constant velocity

- ganeshie8

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- lochana

So how we get inequalities for the minimum height?
what is the argue?

- ganeshie8

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- lochana

@ganeshie8 agreed!

- ganeshie8

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- lochana

are sure about gravity and reaction at p on the same line and opposite?

- ganeshie8

are you asking why must the force of gravity pass through the point P for minimum height ?

- ganeshie8

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- lochana

yes. ah okay. It is possible though. After that that equilibrium, it is definitely going to rotate.

- ganeshie8

exactly! check this
http://physics.stackexchange.com/questions/258/what-determines-the-minimum-angle-at-which-a-domino-falls-over

- ganeshie8

https://youtu.be/yNQi5g8JfPg
A bob is suspended from a string connected to the block at it's center of mass. Raise the inclined plane until the block tips over. Observe that the block does not tip over until the center of mass exceeds the base of the block, illustrated when the string and mass clear the bottom edge of the block.

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