He hung his head in shame when he learned about his son's failure.

Using your words, you can say: ‘He hung his head in shame when he came to know of his son’s failure.’ The only change needed is for ‘hanged.’

To hang is an unusual verb because it has two past participles, the irregular 'hung' and the regular 'hanged', 'hung' can always be used but 'hanged' is an alternative if referring to an execution. So in your sentence you should choose 'hung'.