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.