The mass-spring system. The kinetic friction force and velocity are sketched to show the dependence of the direction of the friction force on the direction of the velocity.
The final block position for different initial positions of a damped harmonic oscillator in the presence of both static and sliding friction forces. Regions corresponding to different integer numbers of half oscillations are highlighted by vertical dashed lines. The gap between the black lines together with the value of allows the static and the dynamic friction coefficients to be estimated. The slope of the lines is 1.
(a) Experimental approach to estimate the dynamic friction coefficient . (b) Measurement of the threshold value of the force needed to produce the motion of the block. Measurements are performed by placing the wooden plane horizontally. Both a force sensor attached to the block and a motion sensor are used.
Measurements of the (a) position and (b) velocity of the block during upward and downward motion. The block is pushed from the bottom to the top of the inclined wooden plane, and a motion sensor records its position and velocity. The sensor is placed at the top. The linear interpolating curves (continuous lines) in the velocity versus time plot allow us to obtain the mean acceleration in each part of motion from the fit parameters. This method reduces the large experimental errors present in the acceleration versus time graphs obtained by the software.
Static friction coefficient estimated from the linear relation between and defined as the threshold force value before the motion starts. The values of and are obtained by the statistical interpolation of the experimental data.
(a) The apparatus for obtaining periodic damped motion consists of a block-spring system placed on a inclined wooden plane and a motion sensor at the bottom. Four small plastic supports are placed beneath the block. (b) Position versus time for a damped harmonic oscillator in the presence of a static and a kinetic friction force. Continuous tilted lines show the linear amplitude decay. Continuous horizontal lines highlight the presence of two centers at symmetrical with respect to the position for each half period depending on the direction of the motion. Oscillatory motion ends after six half oscillations at .
The final position versus the initial position in the presence of static and kinetic friction forces. Both positions are recorded by the motion sensor at the bottom of the inclined wooden plane. Regions corresponding to different integer number of half oscillations are separated by vertical lines. Experimental points are placed on two discontinuous lines as highlighted by a linear fit in the region of ; the slope is close to 1, as expected. The values obtained for the width of the first two regions are and .
(a) Phase space representation (the plane, where and are the oscillator position and velocity at the step, respectively) generated by the code for . The continuous vertical lines represent the two oscillation centers , and the static region is delimited by vertical dashed lines at . (b) Experimental phase space derived from the data shown in Fig. 6(b) (filled points) and phase space computed with the parameters , , , , and (dashed black line). Symmetrical centers are at (continuous vertical lines), and the static region is delimited at (dashed vertical lines). All six half oscillations are observed and good agreement is found between experimental and numerical data. [(c) and (d)] The ratio as a function of for (see Ref. 4) and , respectively. The straight lines represent the final positions after an increasing integer number of half oscillations . The slanted lines are shifted upward or downward because .
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