May I do laundry on Friday afternoon:

It is permitted in a time of need to do laundry Friday afternoon, as turning on the machine is considered a minor labor. However, initially, this should be done on Thursday, and by the very latest, on Friday before Mincha Ketana. Past Mincha Ketana, one may only turn on the machine, and should not hang or fold the laundry being that it is laborious activity.

Background:

It is forbidden to do Melacha on Friday afternoon, starting from Mincha Ketana, which is 2.5 Halachic fluctuating hours prior to sunset on Erev Shabbos unless the work being done is not real labor. Turning on the machine is not real labor, and is hence permitted even past Mincha Ketana.

Sources: See Admur 251:1 and 3; Ateres Paz 6; Kol Menachem 251

Doing laundry on Erev Shabbos-Part 1 – Shulchanaruchharav.com