In order for the crate to move at constant velocity, all forces
must add up to zero.
In the horizontal direction (x), the horizontal component of the
push force (Px) must fully oppose the kinetic friction force
(F).
In the vertical direction, the normal force (N) will be as large as
necessary, to prevent the crate from sinking through the floor.
Structural capabilities of floor and crate, of course are required.
The normal force must oppose both the weight of the crate (m*g),
and the woman's downward component of the push force (Py).
It is of interest for us to develop an equation for the normal
force, because the horizontal force of friction is proportional to
it.
Component-ize the woman's push force:
Px = P*cos(theta)
Py = P*sin(theta)
Vertical force balance:
N = m*g + Py
Horizontal force balance:
Px = F
Apply trigonometry for Px and Py:
N = m*g + P*sin(theta)
P*cos(theta) = F
Use F = muk*N to find the friction force in terms of normal
force:
F = muk*N
Thus:
F = muk*(m*g + P*sin(theta))
Substitute:
P*cos(theta) = muk*(m*g + P*sin(theta))
Multiply through by muk:
P*cos(theta) = muk*m*g + muk*P*sin(theta)
Gather P-terms to one side:
P*cos(theta) - muk*P*sin(theta) = muk*m*g
Factor:
P*(cos(theta) - muk*sin(theta)) = muk*m*g
Solve for P, and our result becomes:
P = muk*m*g/(cos(theta) - muk*sin(theta))
--------------Answer--------------
Notice that for this equation to have physical meaning, the
denominator must be a positive number. It cannot be zero, and it
cannot be negative.
Solve for where it equals zero:
cos(theta) = muk*sin(theta)
muk = 1/tan(theta)
tan(theta) = 1/muk
It is required that tan(theta) < 1/muk
If tan(theta) were to equal 1/muk, then infinite push force would
be required to produce constant speed. In otherwords, the angle
would be too steep, that the push force is only increasing the
total contact force (friction + normal force as vectors), which is
guaranteed to constrain against it.
If tan(theta) were to exceed 1/muk, then the push force would be
self-locking, as more push force would be causing it to slow down,
rather than to speed up.
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