Maximum and minimum values of a force to keep a block pinned.

[DocZaius]

Homework Statement

Your hand presses a block of mass m against a wall with a force F acting at an angle . Find the minimum and maximum possible values of abs(F) that can keep the block stationary, in terms of m, g, theta, and us (the coefficient of static friction between the block and the wall.)

Homework Equations

abs(normal force) = F*cos(theta)
Static Friction = Vertical component of net forces unless they are > abs(us*abs(normal force))

The Attempt at a Solution

The following must be true for the block not to move:
abs(F)*sin(theta) - mg >= -(us*abs(F)*cos(theta))
abs(F)*sin(theta) - mg <= us*abs(F)*cos(theta)

The vertical component (always positive) of the force, minus the force of gravity, needs to be less than the coefficient of static friction times the normal force (which is equal to the horizontal component of the applied force).

It also needs to be greater than the negative equivalent of the max static force.

My problem is that the applied force cancels out of the equations, and I can't give an answer that shows the upper and lower limits of the force since they cancel out. Any hints?

Thanks.

 

[tiny-tim]

My problem is that the applied force cancels out of the equations, and I can't give an answer that shows the upper and lower limits of the force since they cancel out.

Hi DocZaius!

I don't understand …

your equations show F = g times a function of theta …

just minimise/maximise that function to find min/max for F.

 

[DocZaius]

Hi DocZaius!

I don't understand …

your equations show F = g times a function of theta …

just minimise/maximise that function to find min/max for F.

Could you get the process started a bit or elaborate a bit more? I am not familiar with minimization/maximization.

 

[DocZaius]

And here is a cleaner version of my findings from the first post. Is the following at least correct?

$$\mu_{s}|F|cos(\vartheta)\geq|F|sin(\vartheta)-mg\geq-\mu_{s}|F|cos(\vartheta)$$

 

[tiny-tim]

And here is a cleaner version of my findings from the first post. Is the following at least correct?

$$\mu_{s}|F|cos(\vartheta)\geq|F|sin(\vartheta)-mg\geq-\mu_{s}|F|cos(\vartheta)$$

… now put it in the form F ≥ or ≤ g times a function of theta.

 

[DocZaius]

This is the closest I have been able to come to fulfilling your request. Unfortunately, the |F| is still on both sides of the inequality. Could you provide some additional help that explains a litte more how to accomplish your instructions?

Thank you.

$$|F|\geq\left|\frac{|F|tan(\vartheta)}{\mu_{s}}-\frac{mg}{\mu_{s}cos(\vartheta)}\right|$$

 

[tiny-tim]

This is the closest I have been able to come to fulfilling your request. Unfortunately, the |F| is still on both sides of the inequality. Could you provide some additional help that explains a litte more how to accomplish your instructions?

Thank you.

$$|F|\geq\left|\frac{|F|tan(\vartheta)}{\mu_{s}}-\frac{mg}{\mu_{s}cos(\vartheta)}\right|$$

Hi DocZaius!

ah … now I see what the difficulty is …

you're trying ot keep it as one (in)equation.

Split it up:

F(sinθ - µcosθ) ≤ mg and F(sinθ + µcosθ) ≥ mg

Then you need to minimise/maximise sinθ + µcosθ and sinθ - µcosθ …

which you can do either by diferentiating, or by finding a φ and a ψ to put them into the form sin(θ + ψ) and sin(θ + ψ).