Experiment 12: The Gravitation Constant
You will measure the universal gravitation constant,
repeating the work of Henry Cavendish (1798). A torsion
balance, shown here, measures the force between known
masses which are a known distance apart. Thus, you know
everything in the inverse square law except G, which you
solve for. The balance contains a pair of small masses on a
rod, hung from a very thin metal strand. They are
gravitationally attracted to a pair of larger masses, which
twists the strand. The amount the rod turns is measured by
bouncing a laser off a mirror on it, and seeing how far the
resulting spot of light moves across the wall. To calibrate the
balance (determine how much torque on it corresponds to
how much movement), its period of oscillation is timed as it
swings back and forth: Like any oscillating system, the
stiffer it is, the faster it vibrates.
PROCEDURE:
Please avoid disturbing the balance. Do not touch its table and walk lightly if you must pass nearby.
Steps 2 and 3 take at least half an hour, most of which is just waiting around. To make the best use
of this time, start working on the calculations. The period is a little over ten minutes, s is roughly 35
cm. You can pencil these numbers in, and then replace them when the better values are available.
1. Measure r, the distance from the apparatus to the
spot of light on the wall.
2. Determine xi and T. A tape measure stuck to the
blackboard acts as an x axis which the spot moves
along. The instructor will move the large masses
to the "before" position as shown. You need to
know where the spot will settle down. The trouble
is, it takes about two hours for it to stop moving.
The spot’s position will oscillate as the metal strand twists back and
forth. If this was simple harmonic motion, you could watch it until you
knew xr (the farthest it goes to the right) and xl (the farthest it goes to the
left). xeq (equilibrium) would be
halfway between.
But friction is present, so we have damped harmonic motion
instead. The amplitude keeps decreasing. Let’s find xeq by
considering time t2. At that moment, xr is about halfway between
x1 and x3: xr = ½(x1 + x3). xl is the same as x2. xeq is in the middle
at xeq = ½(xr + xl ).
Use the position of the spot’s right side for x. (An arbitrary choice. Left would be just as good.) At
three successive times, record x when the spot stops going one way and starts going the other way.
You also need the period, T. This can be measured most accurately between equilibrium crossings
because the spot is moving relatively fast, minimizing doubt about when to start or stop the clock.
From earlier observations, the instructor will know approximately where equilibrium is. A couple
of people should start stopwatches when the spot first crosses this point. They should stop them the
next time it is there moving in the same direction.
So, record x1, x2, x3 and T, and calculate the initial xeq. Call its uncertainty + 1 cm, mostly due to
random influences such vibration the balance picks up from the counter.
3. Determine xf and recheck T. With the large masses to the “after” position, repeat step 2.
4. Average your two values for T. Estimate an uncertainty based on how much the two values
differed.
Calculations: HINT: Watch your units! Forgetting to convert minutes to seconds, centimeters to
meters, etc., is a frequent source of problems.
1. What angle is the balance turned by the lead ball's gravity?
a. Find how far equilibrium moved due to reversing the forces on the balance by subtracting xf xi. Also, find the uncertainty in this distance. (After this, you can leave the uncertainties until
the end.)
b. From s and r, as shown, find the angle which
gravity shifts the beam's direction. (This is
probably easiest if you say that such a thin triangle
is hardly any different from a small sector of a
circle, and use the methods of sec. 7. Do not
convert radians into degrees: You'd just have to
convert back later.)
c. Divide this angle by 4 for the following
reasons:
- You want the angle from equilibrium:
The position where there is no net
gravitational torque is in the middle,
halfway between xi and xf. That's a
factor of 2.
- You want the angle turned by the
balance, not the angle swept through by
the beam. They aren't the same; as
shown by the example at right, that's another factor of 2.
2. How much torque does it take to twist the balance that far?
a. Find the stiffness of the metal strand: Somewhat like a harmonic oscillator (sec. 10), the
period of a torsional vibration is T = 2π I / k , where I is the moment of inertia, and k is the
stiffness constant. (The stiffer something is the faster it vibrates; more inertia slows it down.)
From the balance's mass and dimensions, I = 7.50 x 10-5 kg.m2. Find k.
b. The angle through which gravity turns the balance (step 1c) depends on how much torque is
trying to twist it and how much stiffness that torque must overcome: τ = kθ, where θ is in
radians (rotational version of Hooke's Law). Find τ, the torque due to the lead balls' gravity.
3. How large are the forces which produce this torque?
a. There are two torques on the balance; one due to the force on each mass. Divide by 2 to get
one individual torque.
b. From this and the fact that the masses are 5 cm from the axis, find the force on one of the
masses.
c. This force is actually the resultant of the attractions of both
large masses, + as shown. From the distance and direction
to the more distant ball, it can be shown that the component of
pulling opposite to
is about 7.5% of F1. So, to get the
attraction of just the closer mass, multiply your force by 1.075.
4. According to the manufacturer, the large mass is 1.5 kg, the small one is 15 grams, and their
centers are 4.65 cm apart. From this and your measured force, calculate G.
5. Double the percent uncertainty in the period (because T was squared), and add to this the percent
uncertainty in the deflection (step 1a) to get the percent uncertainty in G.
6. In your discussion, the procedure section should, as usual, explain how the experiment works,
rather than just reciting the steps of what to do. In your conclusion, compare your value to the
accepted one.
(A footnote: Once G is known, simple calculations give the masses of bodies such as the earth and
sun, so this experiment is sometimes called "weighing the earth.")
PHY 131
Experiment 12: The Gravitation Constant
r = _____________ ± 0
Finding xi:
Finding xf:
x1 =
x1 =
x2 =
x2 =
x3 =
x3 =
T=
T=
Average T = _____________ ± _____________
Calculate xi and xf:
s = xf - xi = _____________ ± _____________
Show step by step calculation of G. (Continue on back or another sheet.)