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STEP Mechanics from Siklos booklet

I have been working on a mechanics problem from the Siklos core booklet. The question is as follows:

A uniform solid sphere of radius

a

and mass

m

is drawn very slowly and without slipping from horizontal ground onto a step of height

a/2

by a horizontal force of magnitude

F

which is always applied to the highest point of the sphere and is always perpendicular to the vertical plane which forms the face of the step. Find the maximum value of

F

in the motion, and prove that the coefficient of friction between the sphere and the edge of the step must exceed

\frac{1}{\sqrt 3}

.

I am not sure how to show that the coefficient of friction must exceed the value stated. I got as far as showing that

F

is equal to friction

f

by taking moments, and even that

R=mg

. However, I do not know why this leads to the conclusion that

\mu mg > F_{max}

, where

F_{max} = \frac{mg}{\sqrt 3}

(hence result). Normally

f_{max} = \mu R

so I thought that in this case

F_{max} = \mu R

also, but apparently this is not so. Can anybody explain this?

Siklos Sphere.gif1.2KB

Reply 1

atsruser

I haven't worked through the question, but in your notation, to prevent slipping as the question requires, we have to have that

f < \mu R

at all times, else the limiting value of friction is exceeded. I guess that gives the inequality.

Reply 2

ghostwalker

Aside: In the diagram your direction for

f

is incorrect. It should be in the opposite direction.

Reply 3

Brister

Original post by atsruser

I haven't worked through the question, but in your notation, to prevent slipping as the question requires, we have to have that

f < \mu R

at all times, else the limiting value of friction is exceeded. I guess that gives the inequality.

Thanks, that was the condition I needed.

Reply 4

Brister

Original post by ghostwalker

Aside: In the diagram your direction for

f

is incorrect. It should be in the opposite direction.

You are right, of course. The diagram is straight out of the Siklos booklet. That definitely contributed to my confusion.

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STEP Mechanics from Siklos booklet

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