The Cavendish Experiment by Matthew Chen COSMOS Cluster 2 Physics in Electro-optics and Nuclear Technology Chen 1 Abstract Scientists during the 18th century pondered over the concept of gravity. By this time Isaac Newton had already established his Law of Universal Gravitation. Interested in finding the density of the Earth and ultimately the mass of the Earth, Henry Cavendish, an English experimental physicist performed what is known today as ‘The Cavendish Experiment’ by observing an oscillating torsion bar system to mathematically calculate the value of the Gravitational Constant in Newton’s Law of Universal Gravitation. By calculating the Gravitational Constant, Cavendish could derive the mass of the Earth as well. These findings in addition to his numerous other findings throughout his life in the field of science made Cavendish well regarded among the scientific community as an influential experimentalist. Introduction In the 18th century, scientists have already made substantial contributions to the fields of physics and chemistry. Mechanics was a fundamental part of physics and the invention of calculus by Isaac Newton greatly bolstered the mathematical treatment of physics. Instruments of the era have also been greatly improved including the telescope, barometer, and mercury thermometer. In addition, Kepler’s Laws had already been established and Galileo had already made his controversial findings. The latest well regarded scientist of the time was Isaac Newton. In his book Principia published in 1687, Newton established that the gravitational force between two objects was proportional to the product of the two masses divided by the distance between them squared. This is known as Newton’s Law of Universal Gravitation and the sheer idea that the same type of force is applied to all objects in the universe was astounding at the time. During his research into gravity, Newton already knew that the oscillation of a pendulum varied according to the latitude of the pendulum on the Earth. Newton, in fact, did not calculate the gravitational constant, G, and this gravitational constant was not of interest to many scientists for years afterwards. This concept of gravity was debated on among the scientific community. In earlier times it was believed that “Any action of one body A on another body B at a distance from it must be capable of being traced all the way across from A to B.” Another interesting question at the time was whether or not the gravitational force depended on the substance as well as the mass. Thus, Roland von Eotvos, a Hungarian scientist, investigated this question and determined that the gravitational force did not depend on the substance. From a mathematical standpoint it didn’t matter if gravitational attraction required a medium or not. This concept of gravity interested an aristocrat named Henry Cavendish. Henry Cavendish, born on October 10, 1731 (just four years after Newton’s death), was an English Physicist well known for his contributions to the field of science. As a social recluse, he took Chen 2 great care into the measurements performed in his scientific experiments. He began working with scientific experiments while helping his father, also an experimental physicist. Cavendish knew about the uncertainties present in the study of gravity. Before his well known experiment known as the “Cavendish Experiment,” Cavendish stated: ‘I know of but 2 practicable ways of finding the density of the earth first by the going of a pendulum in the foregoing manner and secondly by finding the deviation of a plumb line at the bottom of a mountain by taking the meridian altitudes of stars.’ At the time, a main interest among scientists was to find the density of the Earth and ultimately “weigh” the Earth. There have been several attempts to calculate the density of the Earth including experiments from Bouguer who measured the length in seconds of a pendulum at 3 elevations. Bouguer was able to determine that the period of oscillation for a pendulum varied decreased with height and that the nearby mountains played a role in this value. By the time Cavendish began what is known as ‘The Cavendish Experiment’, numerous experiments had found the density of the Earth to be anywhere from 4.48 to 5.3 times denser than water, too wide of a range. Thus, Cavendish, at the old age of 67, began his experiment to “weigh” the Earth. Method In 1798, Cavendish sought to find the density of the Earth by utilizing the torsion balance system devised by John Michell. The torsion balance system consisted of two small lead balls suspended by a lightweight, strong bar suspended at the middle by a thin, twisted wire. Two larger balls were held in place nearby, allowing the two smaller balls to rotate about the horizontal plane towards the larger balls. The attraction of gravity and the resisting force of the string ultimately cause the bar to oscillate back and forth about the equilibrium point. By measuring the oscillation period and by knowing the masses and distances apart the balls are, Chen 3 Cavendish could calculate the torsion coefficient and ultimately the Gravitational constant to find the density of the Earth. The gravitational force exerted on a ball is extremely small in comparison to its mass. Cavendish naturally was extremely careful about making his measurements as accurately as possible. To do this, he placed the torsion balance system inside a separate mahogany casing to prevent any outside wind from affecting the experiment. He also set it up so he could move the weights without entering the room with a rope attachment. In addition, Cavendish determined the position of the arm by placing a small slip of ivory at the end of each arm and as close to it as possible, allowing him to make measurements to the hundredths of an inch. The only light that was allowed in the room was light that reflected from a mirror onto the scales. In his original experiment, Cavendish used these values: Diameter of large ball M = 12 inches Diameter of small ball m = 2 inches Mass of large ball M = 158 kg (348 lbs) Radius of large ball rM = 30.5 cm (12 in)? Mass of small ball m = 0.73 kg (1.6 lbs) Radius of small ball rm = 5 cm (2 in)? Separation of large balls L = 1.86 m (73.3 in) Separation of small balls L = 1.86 m (73.3 in) Distance between large and small balls R = 0.225 m (8.85 in) Length of suspending wire = 0.997m (39.25 in) Cavendish set out to observe the two extreme points of the oscillation, calculate the middle point, and used the elapsed time divided by the number of vibrations to calculate the time Chen 4 of one oscillation. This value could be further used to ultimately calculate the weight of the Earth. Results Cavendish was able to derive the gravitational constant using the following derivation. 1)The torque of one arm equals Tarm (tau) = F * (L (length of bar)/2) Therefore the net gravitational torque is Tarm = F * L 2)The wire resists the twisting motion, similar to Hooke’s Law for springs. Thus resistance is proportional to the angle of displacement. Twire = k(torsion coefficient) * Ɵ (angle displaced) 3)At the equilibrium point the torques are equivalent: kƟ = FL F = (kƟ)/L 4)Newton’s Universal Gravitation Equation at the equilibrium point is Fgravity = (G * M * m)/Re2 5)Substitute Fgravity with Fgravity = (k * θ)/L (kθ)/L = (GMm)/Re2 G = (kθRe2)/(LMm) All of the values presented here are known except for k, the torsion coefficient. 6)Utilizing the Oscillation Period T = 2* pi * (I (moment of inertia)/k)(1/2) Since the balls are heavy and the bar is considerably lightweight in comparison, the mass of the bar is considered negligible and not a major component in calculating inertia. 7)Solving for k, the torsion coefficient in the equation: T2 = (4π2I)/k k = (4π2I)/T2 8)The moment of inertia of smaller ball is I = (mL2)/2 because of the Parallel Axis Theorem which states that the moment of inertia of a rotating body is I = I(center of mass) + md2. The I(center of mass) = 0 since the bar is negligible I = m(L/2)2 + m(L/2)2 I = (mL2)/2 9)Substituting the moment of inertia into the equation leads to this equation: k = (2π2mL2)/T2 10)Therefore by substituting k (torsion coefficient), G equals: Chen 5 G = (2π2LθRe2)/(T2M) G - Universal Gravitation Constant L - length of torsion bar θ - angle bar turns Re - equilibrium point distance between M and m T - oscillation frequency M - mass of larger object Cavendish calculated the value of G to be 6.71 * 10-11cm3g-1s-2, a difference of less than 1% of most accurate modern value obtained. The modern actual measurement of the gravitational constant G is 6.672 * 10-11 cm3g-1s-2. 11)After he calculated the value of the gravitational constant, Cavendish utilized Newton’s universal law of gravitation and set up the equation: mg = GmM/Re2 where m is the mass of the object, M is the mass of the Earth and r is the distance between the two centers of mass. 12) Solving for M yields: M = gRe2/G Since the radius of the Earth was already known at the time, the mass of the Earth could finally be calculated. 13) Finally, the density of the Earth could be calculated from utilizing the equation: Density = Mass/Volume ⍴ = M/((4/3) * π Re3) 14) Substituting M into the equation yields: ⍴ = (3g)/(4 π ReG) Seventeen experiments (first three with a normal wire, the remainder with a stiffer wire) were performed. He ultimately concluded that the density of the Earth is 5.48 times that of water. This is also relatively close to the more precise density found years later with more accurate instruments: 5.51 times the density of water. These experiments took him a very long time to complete, mainly because of the slow oscillation rate of the torsion bar system. It took him about 2.5 hours to complete just one experiment. Discussion Cavendish finally reported his findings in a 57 page paper titled Experiments to Determine the Density of the Earth in June 1798. Initially Cavendish received criticism from his Chen 6 work which was to be expected with any major scientific breakthrough. The well regarded “Cavendish Experiment” did allow Henry Cavendish calculate the value of interest: the density of the Earth. Today this experiment is well known for his accurate calculation of the gravitational constant G as well as the Earth’s density. The Cavendish experiment had a profound impact on the future of science. Not only were astronomers able to calculate the mass of the Sun and other planets by knowing the Gravitational constant and Newtonian mechanics, they made great progress by discovering new planets as well. Cavendish’s experiment has been repeated numerous times in modern day science. some scientists have even varied the experiments by placing plates instead of spheres on the ends of the arms to examine gravity at very small distances. At such a small scale, no deviation from Newton’s inverse square law has been found. Summary/Conclusion Henry Cavendish, a well known experimental physicist during the 18th century sought out to calculate the mass of the Earth. Through the torsion bar experiment, Cavendish was able to not only calculate the mass of the Earth but also find the Gravitational Constant used in Newton’s gravitational force equation. His work has led to great leaps in the study of physics and, more specifically, in the study of gravity and astronomy. Today it is still recognized historically as an elegant physics experiment. Chen 7 Citations Works Cited Berry, Arthur John. Henry Cavendish: His Life and Scientific Work. London: Hutchinson, 1960. Print. Jungnickel, Christa, and Russell McCormmach. Cavendish. Philadelphia, PA: American Philosophical Society, 1996. Print. Jungnickel, Christa, and Russell McCormmach. Cavendish: The Experimental Life. Lewisburg, PA: Bucknell, 1999. Print. Kurtus, Ron. "Cavendish Experiment to Measure Gravitational Constant." School for Champions. N.p., 20 Feb. 2015. Web. 24 July 2015. Kurtus, Ron. "Derivation of Gravitational Constant from Cavendish Experiment." School for Champions. N.p., 20 Feb. 2015. Web. 23 July 2015. Urone, Paul Peter. "Newton's Universal Law of Gravitation." College Physics. Australia: Brooks/Cole, 2001. N. pag. Print. Photography Citations Cavendish Experiment. N.d. Inside Science. Web. 26 July 2015. <http://www.insidescience.org/sites/default/files/Cavendish_Experiment.jpg>. Cavendish Experiment. N.d. Openstax CNX. Web. 26 July 2015. <http://cnx.org/resources/69df41f15dea06389647e33c45c17901a0c03fff/Figure_07_05_08 a.jpg>. Henry Cavendish. N.d. Web. <http://ffden2.phys.uaf.edu/211_fall2010.web.dir/Smith_Elliot/Cav_Photo_Face.jpg>. Chen 8 Acknowledgements Thank you to my Professors Niels and Diego, my Teacher Fellow Creed, my Teacher Assistant Vache, and to my Residential Advisors Alex and Monica. Appendices Cavendish was very thorough with his error analysis. He noticed how difficult it was to place the point at the end of the arm midway between two scratches that are so near together without having it slip towards one of the scratches. He considered a solution using a beam compass with only one point, a possible area for improvement. He also noticed errors in “manner of observing” and air resistance (though very minimal disturbances). There was also a very small excess attraction of iron rods that he could not figure out. He suspected magnetism but he could not figure out the issue even when he switched the 2 inch balls with 10 inch magnets. However this error was deemed negligible. He demonstrates one of grand principles of experimental physics: if you suspect a problem, make it bigger to see how bad it really is. Ultimately, Henry Cavendish’s main goals in his scientific studies was efficiency; he lacked any appreciation of beauty or sense.