Atwood was born in Westminster, with the date remaining unknown, but presumed to have been shortly before his baptism on 15 October 1745. He attended Westminster School and in 1765 was admitted to Trinity College, Cambridge. He graduated in 1769 with the rank of third wrangler and was awarded the inaugural first Smith's Prize. Subsequently he became a fellow and a tutor of the college and in 1776 was elected a fellow of the Royal Society of London.
In 1784 he left Cambridge and soon afterwards received from William Pitt the Younger the office of patent searcher of the customs, which required but little attendance, enabling him to devote a considerable portion of his time to mathematics and physics.
Analysis of a Course of Lectures on the Principles of Natural Philosophy (Cambridge, 1784).
Treatise on the Rectilinear Motion and Rotation of Bodies (Cambridge, 1784), which gives some interesting experiments, by means of which mechanical truths can be ocularly exhibited and demonstrated, and describes the machine, since named after Atwood, for verifying experimentally the laws of simple acceleration of motion.
Review of the Statutes and Ordinances of Assize which have been established in England from the 4th year of King John, 1202, to the 37th of his present Majesty (London, 1801), a work of some historical research.
Dissertation on the Construction and Properties of Arches (London, 1801).
Chess games recorded by Atwood were published posthumously by George Walker in London in 1835, under the name Selection of Games at Chess, actually played by Philidor and his Contemporaries. Atwood was one of a few masters that could beat Verdoni on occasion.