Figure 4. A concept diagram for measuring the mass
of a spinning cylinder by equating mechanical and
another solar system, these methods could not maintain a
quantum-based mass standard along the way.
A Better Way
The gravity and gear head problems can be avoided if,
instead of lifting the mass, we measure the energy required
to spin a mass. Physics 101 tells us that the energy of a
solid spinning cylinder is:
(π2 / 4) MD2 ƒ2
where M is the mass of the cylinder in kg, D is the diameter
in meters, and ƒ is the rotational velocity in revolutions/
Now, suppose that we set the cylinder spinning with a
motor and measure the electrical energy Ee = IVT required
to spin the cylinder up to a rotational rate of ƒ in a time T.
I and V are the current and voltage applied to the motor.
Equating the electrical and mechanical energies, we have:
IVT = (π2 / 4) MD2 ƒ2 and solving for M M = 4IVT π2D2ƒ2
This equation ignores electrical resistance loss and
friction in the motor. As you will see in the circuit design
that follows, we can measure and eliminate these losses.
Figure 4 shows a motor drive and measurement
block diagram that when calibrated against standards
of time, voltage, current, and length allows an accurate
measurement of the product IVT and ƒ. Briefly, it works as
We measure D with a digital caliper (our length
standard). When the start button is pushed, a constant
current is applied to the motor. As it spins up, ƒ is
compared to a preset stop value ƒs. At the same time,
the product VT is accumulated in an integrating amplifier.
When ƒ = ƒs, the integration is shut off. I is measured with
an ammeter. VT is measured with a voltmeter, and we have
everything required to calculate M.
Figures 5 and
6 show a measurement circuit and its
PCB (printed circuit board) that is designed to power the
motor, plus make possible the accurate measurement I, VT,
and ƒs by eliminating the effects of friction and resistance
losses. While Figure 6 looks complicated, it’s just the
combination of four relatively simple circuits.
At the upper left, op-amp U3 uses positive feedback to
implement a latching comparator. The start button latches
the U3 output into positive saturation (13V), providing
the control voltage for the motor and starting the voltage
integration via relay K12. When the motor voltage rises
to a value set by trimpot VR11, U3 will flip to negative
saturation, turning off the motor and integration.
At the lower left, op-amp U1 (LM7171) provides
a motor drive with ±100 mA capability. Depending on
switch S1, the drive is constant voltage or constant current.
Constant current is achieved by returning the motor
current to ground through R3 and using the voltage across
R3 as the feedback for U1.
A small paper tab on the rotating cylinder intersects
an LED-detector pair to produce a pulse waveform at the
rotation frequency. The frequency of this pulse waveform
and thus the rotation rate can be measured with a
frequency counter or oscilloscope.
At the upper right, U7 is a programmable timer that is
used to calibrate the VT integrator U4. The LS7210 has an
internal oscillator with a frequency set by C40 and R40; in
this case, 2,795 Hz. The timer counts a specific number of
oscillations set by five digital inputs at pins 9-13. Grounding
Figure 5. The PCB for the mass
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