A $60^\circ$ rotation around the origin in the counter-clockwise direction is applied to $3 \sqrt{3} - 5i.$  What is the resulting complex number?
A $60^\circ$ rotation around the origin in the counter-clockwise direction corresponds to multiplication by $\operatorname{cis} 60^\circ = \frac{1}{2} + \frac{\sqrt{3}}{2} i.$

[asy]
unitsize(0.5 cm);

pair A = (3*sqrt(3),-5), B = rotate(60)*(A);

draw((-2,0)--(8,0));
draw((0,-6)--(0,3));
draw((0,0)--A,dashed);
draw((0,0)--B,dashed);

dot("$3 \sqrt{3} - 5i$", A, S);
dot("$4 \sqrt{3} + 2i$", (4*sqrt(3),2), NE);
[/asy]

Thus, the image of $3 \sqrt{3} - 5i$ is
\[(3 \sqrt{3} - 5i) \left( \frac{1}{2} + \frac{\sqrt{3}}{2} i \right) = \boxed{4 \sqrt{3} + 2i}.\]