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Finding the Effective Mass and Spring Constant of a Force Probe from Simple Harmonic Motion
3.Michael Vollmer and Klaus-Peter Möllmann, “Bouncing poppers,” Phys. Teach. 53, 489 (Nov. 2015). High-speed videos of poppers are included.
4.If a delta-function impulse is applied to an ideal spring-mass system that is originally in equilibrium, the mass acquires an initial velocity, momentarily comes to rest after a quarter-period, and continues to oscillate thereafter. Conservation of momentum (during the impulse) and the impulse-momentum relation (subsequently) show that the initial impulse equals the integral under the F-versus-t curve during the first quarter-period, even though the time scales differ. In a representative experiment, the popper gained 0.046 kg·m/s of momentum, while the integral of force with respect to time was 0.043 N·s for the first quarter-cycle of the graph—reasonably close agreement despite damping and the impulse occurring over a finite time.
5.To determine the time when the popper left the surface, we aimed a laser beam at the impending gap between the popper and the platform. An analog light sensor intercepted this beam, and timing information was cross-referenced to the force data. The relative timing of the popper's departure and the graph peak depends on the platform mass. We used a 37.9-g platform. Repeating the experiment with a 500-g platform delays the graph from reaching its peak until several milliseconds after the popper has left the surface.
6.Similar limitations are encountered with the Vernier Force Plate and with sound level meters. The latter typically allow the user to select slow versus fast response, to discern instantaneous versus time-averaged decibel levels. No such adjustment is provided on the Vernier force sensors, but temporal smoothing could be accomplished via software filtering.
For force measurements of high-speed impacts, a quartz piezoelectric sensor is more appropriate. A typical sensor of this type has an upper frequency limit of 36 kHz and a rated stiffness of over 109
N/m. See http://www.pcb.com/TestMeasurement/Force/GenPur.aspx
8.The observed oscillations have a unitless damping ratio of less than 0.1. For such light damping, the frequency of damped harmonic motion is essentially identical to the frequency of simple harmonic motion.
9.The force sensor's response time is quoted by Vernier as 2 ms. John Gastineau, private communication (Sept. 2015).
10.Careful examination of Fig. 6 reveals a small increase in frequency as amplitude decreases. Pure DHM would not produce this effect. Its origin is presumably a deviation from Hooke's law (too minor to be obvious in Fig. 9). Discussion of nonlinear oscillators is beyond the scope of the class activity.
11.The effective mass of a uniform helical spring is equal to one-third of its mass. A good discussion can be found in A. P. French, Vibrations and Waves (Norton, New York, 1971), p. 60.
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Force probes are versatile tools in the physics lab, but their internal workings can introduce artifacts when measuring rapidly changing forces. The Dual-Range Force Sensor by Vernier (Fig. 1) uses strain gage technology to measure force, based on the bending of a beam. Strain gages along the length of the beam change resistance as the beam bends (Fig. 2). The elasticity of the beam leads to oscillations that persist after being excited by an impulsive force. How quickly the force probe freely returns to zero is thus related to the rigidity of the beam and the total mass attached to it. By varying the added mass and measuring the resulting frequency of the probe's internal free oscillations, the effective mass and spring constant of the probe's moveable parts can be found. Weighing of the probe parts and conducting a Hooke's law experiment provide static verification of these parameters. Study of the force sensor's behavior helps students to learn about damped harmonic motion, mathematical modeling, and the limitations of measuring devices.
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