Classical Mechanics - Statics - Mass and overhang

[Lito]

Homework Statement

A block of mass M is positioned underneath an overhang that makes an angle θ > 0 with the vertical. You apply a horizontal force of Mg on the block, as shown in the figure. Assume that the friction force between the block and the overhang is large enough to keep the block at rest.

a. Make a free-body diagram of the block, indicating all external forces acting on it.

b. What are the normal force N and the friction forces F that the overhang exerts on the block?

c. Show that the overhang θ can be at most 45◦ if there is any chance that the setup is static.

d. Suppose the coefficient of friction is µ. For what range of angles θ does the block in fact remain at rest?

Homework Equations

(F⃗ net)x = ΣFx = 0

(F⃗ net)y = ΣFy = 0

fs≤μsN

The Attempt at a Solution

[/B]
a.

b.
$$ \Sigma F_x=0 => N= Mgcos\theta-Mgsin\theta$$
$$ \Sigma F_y=0 => F_f= Mgcos\theta+Mgsin\theta$$
and also $$ F_f= \mu*N= \mu*Mg(cos\theta-sin\theta) $$

c.
I'm not sure what am I supposed to do…
Is it enough to state that in order that the friction will be positive the term (cos θ -sin θ) has to be positive.
Therefore for 0 < θ < 45 => 0 < (cos θ -sin θ) < 1 ?

d.
$$ F_{f(max)}= \mu*N \geq Mg(cos\theta+sin\theta) $$
$$\mu*Mg(cos\theta-sin\theta) \geq Mg(cos\theta+sin\theta) $$
$$\mu*cos\theta-cos\theta) \geq sin\theta+\mu*sin\theta $$
$$\mu-1 \geq tan\theta-\mu* tan\theta $$
$$\frac{\mu-1}{1-\mu} \geq tan\theta $$
$$-1 \geq tan\theta $$
then i get
$$90 \geq \theta \geq 135 $$
but it make no sense according to the previous section...

Thanks a lot :)

 

[haruspex]

In part b), reconsider your "and also". What are you assuming there that is beyond the information given?
Your answer to c) is on the right lines, but the logic sequence doesn't quite work. Start with "if theta > 45^o then..."
Your algebra for d) has a sign error in the fourth line. Correct that and get a new expression for bounds on tan theta as a function of mu.