Why is using Newton's First Law failing me? A factory worker pushes a 29.7kg crate a distance of 5.0m along a level floor at constant velocity by pushing downward at an angle of 32$^\circ$ below the horizontal. The coefficient of kinetic friction between the crate and floor is 0.24.
What magnitude of force must the worker apply to move the crate at constant velocity?
So because constant velocity 
\begin{align}\sum F&=0\\
F_x\times \cos{(360^\circ-32^\circ)} - f_k &= 0\\
F_x\times \cos{(360^\circ-32^\circ)} - (mg)\times 0.24 &= 0\\
F_x\times \cos{(328^\circ)} - 69.8544\,\mathrm N &= 0\\
F_x  &= 69.8544\,\mathrm N/\cos{(328^\circ)}\\
F_x&=82.3708\,\mathrm N
\end{align}
That turns out to be wrong, the expected answer is $f_k$ the force of kinetic friction. Why? Shouldn't I be able to use $\sum F=0$ in this problem to find the answer? 
I also tried to find $\sqrt{F_x^2+F_y^2} -f_k$ but that didn't help either.
 A: Horizontal component of force
$$F_x=F\cos32^\circ-f$$
Vertical component of force
It must be zero as block will remain intact with ground.
$$F_y=N-mg-F\sin32^\circ = 0$$
Normal force (due to ground)
What you are missing is that you haven't added the increase in $N$ due to $F\sin32^\circ$
$$N = mg + F\sin32^\circ$$
Frictional force
$$f=\mu N=\mu (mg + F\sin32^\circ)$$

The horizontal component must be infinitesimally small positive force. In other words, you can equate it to zero to get $F$.
$$F_x=F\cos32^\circ-\mu (mg + F\sin32^\circ)\to0^+$$
$$\large F=\frac{\mu mg}{\cos32^\circ-\mu \sin32^\circ}\approx100\,\text{N}\left(98.88\right)$$

A: The normal force is not only provided by the weight of the body but also by the force you have applied. Take it into consideration and you will get the answer
A: I'm going to just use $32^∘$ below; it doesn't make a difference. Your equation isn't correct. You should have
$F_x−f_k = 0$
or
$F\cos(32) −f_k = 0$.
The x-component of $F$ is $F_x = F\cos(32)$. Writing $F_x \cos(32)$ doesn't make logical sense.
Why? Shouldn't I be able to use $\sum F=0$ in this problem to find the answer?
Yes, you can.
