Alarge crate with mass m rests on a horizontal floor. the static and kinetic coefficients of friction between the crate and the floor are μs and μk, respectively. a woman pushes downward on the crate at an angle θ below the horizontal with a force f⃗ .
a)what is the magnitude of the force vector f⃗ required to keep the crate moving at constant velocity?
express your answer in terms of m, g, θ, and μk.
b)if μs is greater than some critical value, the woman cannot start the crate moving no matter how hard she pushes. calculate this critical value of μs.
express your answer in terms of θ.

a)

Here the crate is moving at constant velocity, so no acceleration:

a = 0

Let's analyze the forces acting along the horizontal and vertical direction.

- Vertical direction: the equation of the forces is

(1)

where

R is the normal reaction of the floor (upward)

is the component of the force F in the vertical direction (downward)

mg is the weight of the crate (downward)

- Horizontal direction: the equation of the forces is

(2)

where

is the horizontal component of the force F (forward)

is the force of friction (backward)

From (1) we get

And substituting into (2)

b)

In this second case, the crate is still at rest, so we have to consider the static force of friction, not the kinetic one.

The equations of the forces will be:

(1)

(2)

In this second case, we want to find the critical value of such that the woman cannot start the crate: this means that the force of friction must be at least equal to the component of the force pushing on the horizontal direction, .

Therefore, using the same procedure as before,

And solving for ,

Now we analyze the expression that we found. We notice that if the force applied F is very large, , therefore we can rewrite the expression as

So, this is the critical value of the coefficient of static friction.

a charge 3q is at the origin, and a charge −2q is on the positive x axis at x=a. part a where on the x-axis would you place a third charge so it would experience no net electric force? enter an expression for the exact answer, which will involve a square-root sign. (do not enter an approximate decimal answer.)