How Do You Calculate the Starting Force for a Mop on a Frictional Surface?

[Pythagorean]

I got a) and b) here. I can't find the mathematical expression of c).

THE SETUP:

A floor mop of mass m is pushed with a force F directed along the handle, which makes an angle THETA with the vertical. The coefficient of friciton with the floor is MU.

a) (i got this) draw the diagram
b) (i got this) for given THETA, MU, find the force F required to slide the mop with uniform velocity across the floor.

MY ANSWER: in the x direction, F = MUmg/sin(THETA)
(all of which are constants, so the velocity will be constant, i assume)

c) show that if THETA is less than the angle of repose, the mop cannot be started across the floor by pushign along the handle.

My issue is that I can't find a way to mathematically express an angle more than the angle of repose without using numbers. (This is a theoretical problem, no numbers given).

I tried it with THETA equaling the angle of repose, which resulted in

F = mg/cosTHETA which seems to imply that the force has changed it's direction to down/up, but that's a loose association.

 

[Pythagorean]

perhaps I have to setup equations for both the x AND y directions?

 

[Pythagorean]

hrm... I must have gotten b) wrong too, because it asks for the Force applied to the handle, not the force in the x direction.

I must be going senile. I can break it down intlo component vectors fine, but I can't bring them back together.

I guess I could do pythagoras c^2 = a^2 + b^2 but then I have to multiply it by the unit vector? But that would give me vector form again, and I want sum...

I'm going to have to sleep on it, I guess. It's getting late.

 

[radou]

The b) answer seems correct, since the only component of the force that is of interest is the horizontal. Considering c), what exactly do you mean by the angle of 'repose'?

 

[Pythagorean]

actually, the answer in the back of the book is different from mine, there's a -MUcosTHETA term in the denominator

The book, for b), says:

F = MUmg/(sinTHETA - MUcosTHETA)

because it's asking how much force to apply tot he handle, where I've only given it the Fx force it gets in the x direction from that force applied to the handle (the rest goes into the floor)

the angle of repose is also known as the critical angle or the angle of friction. For instance, if you have a brick sitting on a slope, and you vary the slope, there will be an angle (when MU = tanTHETA specifically) where the brick won't slide unless it's that angle or greater.

 

[Doc Al]

actually, the answer in the back of the book is different from mine, there's a -MUcosTHETA term in the denominator

The book, for b), says:

F = MUmg/(sinTHETA - MUcosTHETA)

The book's answer is correct. Hint: What's the normal force between mop and floor?

 

[Pythagorean]

The book's answer is correct. Hint: What's the normal force between mop and floor?

yeah, I was totally assuming the Normal force was just gravity in the opposite direction, neglecting the additinal down force of the janitor pushing on the handle.

Thanx for the input