Min/Max angle of a force applied to stationary mass.

[pirland]

Ok, I've been having some real trouble with this problem. Either I’m missing some key piece of reasoning or it’s harder then it seems.

”If a block of mass m is pressed against a vertical surface by a constant force Fh, what are the maximum and minimum angels that that force can be applied and still have the block remain stationary? Express in terms of m, g, theta, and mu.”

Now I assume that what is needed is to equate the maximum static friction force to the total force applied parallel to the surface, but I’m having trouble finding an equation that will give me my answer. Any tips or pointers would be greatly appreciated.

 

[Justin Lazear]

Draw a free-body diagram for the block.

You have two components of force going down, gravity and the y component of the applied force. Gravity is given by mg and, supposing theta is the angle of elevation, the y-component of the applied force is $F\sin{\theta}$. The normal force is the only force going up, and is opposite and equal to the sum of the downward forces, since there cannot be any net y-component of force.

You have one force in each horizontal direction. On one side you have the x-component of applied force, $F\cos{\theta}$. On the other side you have the force of static friction, $F_n \mu$, where $F_n$ is the normal force. Again, these two forces must balance.

Can you take it from there?

--J

 

[pirland]

Clarification

Sorry, I should have specified more, the problem that I'm having is with the resolving of the actual equation. Since the block has not passed through the wall the horizontal force is being countered by the normal force, causing balance in the x axis. What I believe I'm looking for is a way to solve for theta in a situation where the static force is exactly equal to the sum of all forces in the y axis. When I put that together mathmatically I get something along the lines of:

-cos(theta)Fh*mu=sin(theta)Fh+Fg

But I can't seem to resolve it into anything useful, mainly the maximum and minimum values of theta. If the above equation is in fact correct could you point me in the righ direction for solving it?

 

[Justin Lazear]

Oh, I totally misread the question. Sorry about that.

In the x-direction, there's the applied force and the normal force.

$$0 = F_h \cos{\theta} + F_N = 0$$

Note that the normal force will be negative, as it's pointing in the opposite direction.

In the y-direction, there's the applied force, gravity, and the static friction force.

$$0 = F_h \sin{\theta} + F_g + F_s$$

$F_g$, as I'm sure you know, is mg. $F_s = F_N \mu$, so you must solve the first equation for $F_N$ and substitute it in there. This gives you an equation you can work with, as it has only one nonconstant, i.e. $\theta$.

In order to find the maximum and minimum values, you must use calculus.

--J

 

[pirland]

Sorry to be so long in responding, I got a nasty flu bug. The above information was just what I needed to break through my mental block, thanks for your prompt and informative response.