gravitational constant - gravitational attraction - torsion balance
What it shows:
The gravitational attraction between lead spheres. The data from the demonstration can also be used to calculate the universal gravitational constant G.
How it works:
The Cavendish apparatus basically consists of two pairs of spheres, each pair forming dumbbells that have a common swivel axis (figure 1). One dumbbell is suspended from a quartz fiber and is free to rotate by twisting the fiber; the amount of twist measured by the position of a reflected light spot from a mirror attached to the fiber. The second dumbbell can be swiveled so that each of its spheres is in close proximity to one of the spheres of the other dumbbell; the gravitational attraction between two sets of spheres twists the fiber, and it is the measure of this twist that allows the magnitude of the gravitational force to be calculated.
figure 1. the twin dumbbells of the Cavendish experiment
The Cavendish apparatus we employ is built by Leybold Scientific. 1 The quartz fiber and smaller dumbbell are enclosed in a metal case with glass window for protection. A plan view of the spheres and dimensions are given in figure 2. A HeNe laser is used to provide the spot reflection. When the apparatus is used quantitatively, the swing-time method is usually employed to calculate G.
figure 2. Plan view of double dumbbell layout
The large dumbbell is rotated on its axis so that the spheres press up against the glass shield next to the smaller spheres (see figure 2). The gravitational attraction between the spheres exerts a torque on the quartz fiber which twists through a small angle. The position of the reflected spot is noted and the large dumbbell is moved to its second position on the other side of the glass; gravitational attraction twists the fiber in the opposite direction. The response time of the spot to move to the second position and the final spot position are noted. The speed with which the fiber can respond to the move depends upon its torsional constant κ, which can be calculated by measuring the period of oscillation of the fiber,
The applied torque due to the gravitational attraction τ=κθ where θ is the maximum angle of deflection of the light spot. At this maximum deflection, the force between a large sphere and a small sphere is
where r is the distance between sphere centers. It is related to the torque by τ=F(L/2) where L is the length of the small dumbbell. So the gravitational constant can be calculated by
Note that, as the mirror turns through an angle θ, the reflected light moves through 2θ. So by reversing the dumbbell an angle of 4θ is measured.
Data for this particular apparatus are given in table 1.
table 1. Cavendish apparatus data
|torsion constant κ|
|Period T||10 Min (approx.)|
|max. deflection θ|
|sphere separation r|
|dumbbell length L||25 cm|
|large sphere mass M||1500g|
|small sphere mass m||15g|
Setting it up:
Because the apparatus needs to be moved between lecture halls, it is mounted on a sturdy mobile platform 1m high with leveling and locking screws. It is positioned on one side of the hall such that the laser spot can be reflected onto the far wall. The laser, a 3mW HeNe is mounted on an optics bench with a tilting post; the position of the optics bench - on the lecture bench or on a cart - depends upon the dimensions of the hall and is at the discretion of the demonstrator. There will be three reflections off the apparatus; a front surface reflection off the protective glass screen (this is static and so is useful for laser positioning), the actual mirror reflection and a mirror-glass-mirror reflection whose spot will move at twice the speed of the true reflection. Tape should be provided to mark the maximum deflection points of the spot on the wall.
The apparatus once in place takes about 45 minutes to settle down, although this can be speeded up by force damping - moving the large dumbbell back and forth to damp the oscillations. NOTE: securing screws are provided in the apparatus to secure the inner dumbbell and fiber during transit; these should be in place before the apparatus is moved.
The apparatus was originally invented by the Rev. John Michell in 1795 to measure the density of the Earth. It was modified by Henry Cavendish in 1798 to measure G and subsequently by Coulomb to measure electrical and magnetic attraction and repulsion. Apart from the historical significance of the experiment, it's really neat to see that you can measure such an incredibly weak force using such a simple device.
In a lecture hall setting the Cavendish apparatus is too small for the audience to see its workings. A large scale model of the dumbbell and fiber components are a good idea to help explain what's going on. We have built such a model from wood and brass, with dumbbell arm lengths of 50cm and the small dumbbell hanging from a copper wire. The larger spheres, made of wood, have magnets enclosed and the smaller spheres, of Styrofoam, have steel ball bearings at their centers. Rating ****
1. M.H.Shamos, Great Experiments in Physics, (Henry Holt & Co. New York 1959) p.75, contains Cavendish's original paper
2. R.E. Crandall, Am J Phys 54, 367, 1983.
3. J.Cl. Dousse and C. Rheme, Am J Phys 55, 706, 1987.
4. Y.T. Chen and A. Cook, Gravitational Experiments in the Laboratory, (Cambridge University Press, 1993). The most up-to-date and complete reference.
1 available from CENCO 33210C, and PASCO SE-9633