Turning over our papers, the chemistry examination began. (the chemistry exam turned over the papers?) Turning over our papers, we began the chemistry exam. (this one is correct)
King Tut's tomb was unearthed while digging for artifacts. (King Tut's tomb was digging for artifacts?) Archaeologists unearthed King Tut's tomb while they were digging for artifacts. (Correct version) Sitting serenely in the laboratory, the apple tasted great to her. (The apple was sitting?) Sitting serenely in the laboratory, she enjoyed the great taste of the apple. (Correct version)