1. Estimating the radius of the moon with a digital camera By Joe Coady

2. Aim Using only a digital camera and the gravitation constant G, calculate the radius of the moon. Targeted at first year physics students in university and possibly high school students as a way to teach newtons laws of gravitation though experiment. In general most students should have a digital camera and therefore should be able to do this practical at home, as well as working well as a group activity.

3. Newton’s law of universal gravitation Newton 1687 Sir Isaac Newton formulated his law of universal gravitation * S. Johnson, “The Gravitational Force,” in PHYSICS, John Wiley & Sons, inc, 2007, pp. 96-97 *

4. Newton’s law of universal gravitation Newton 1687 Sir Isaac Newton formulated his law of universal gravitation * S. Johnson, “The Gravitational Force,” in PHYSICS, John Wiley & Sons, inc, 2007, pp. 96-97 *

5. The camera

6. Working backwards To calculate the distance to the moon, rearrange newtons law of gravitation Mass of Earth Gravitational acceleration of the moon

7. Length of a sidereal day

8. Orbital period of the moon

9. Centripetal acceleration Therefore the distance of the moon is related to the orbital period and earths mass

10. Radius of the Earth

11. Mass of the Earth

13. Results – Sidereal Day / Moon orbital period PhotoTime Taken Relative Time (s) Relative Angle (°) Sidereal Day Estimate 118:50:42000 219:20:181776 ± 17.43 ± 0.0123h 54m ± 2m 319:45:003258 ± 113.63 ± 0.0123h 54m ± 1m DayMoon PositionMoon Radius One(375 ± 2, 299 ± 2)22 ± 2 Two(1071 ± 2, 1055 ± 2)22 ± 2 Orbital Period:1 revolution every 28 ± 4 days

14. Results – Radius of the Earth Frame 1 (Tripod ) reference frame7225 Frame 2 (Ground)7508 ± 3 Frame Difference (Frame 2 – Frame 1)283 ± 3 Time Difference11.3 ± 0.1 s Height difference between frames2.22 ± 0.05 m Frame Rate25 fps Calculated Radius of the Earth6500 ± 100 km

15. Results – Acceleration of objects Frame 1 (t0)5436 ± 2 Frame 2 (s(t))5516 Frame Difference (Frame 2 – Frame 1)80 ± 2 Time Difference 0.333 ± 0.008 s Pixel to inch ratio0.0846 ± 0.0007 (inches / pixel) Pixels Displacement255 ± 4 Real displacement21.6 ± 0.4 inches Height difference between frames2.22 ± 0.05 m Frame Rate240 fps Gravitational Acceleration

16. Results – Radius of the moon Moon Diameter (pixels)747 ± 2 Moon Radius (pixels)374 ± 1 Moon Radius (km)

17. Final Results CalculationExperimental ValueExpected Value% Error Earth Radius6500 ± 100 km6378.1 km*1.9 % Earth Mass5.5 % Gravitational Acceleration 1.0 % Distance to the Moon12 % Radius of the Moon1736.0 km*0.81 % At this point in time, the regular method of calculating the gravity, size and mass of objects is done though satellites, so this method doesn’t really have any impact of in the future of science. It does however, stand to be a good teaching mechanism for undergraduate students as is the original intention of the method. * D. D. R. Williams, “Moon Fact Sheet,” NASA, 13 August 2015. [Online]. Available: http://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html. [Accessed 20 10 2015]