Modeling a falling slinky
Tension versus length diagrams for a compression spring (left) and a tension spring (right). The tension in each spring is zero for spring length assuming Hooke's law applies (this length is not achieved for the tension spring). The turns of the spring touch for length .
Frames extracted from a high-speed video of the fall of a rainbow-colored slinky, illustrating the collapse of the top of the slinky, and the continued suspension of the bottom after release of the top. The top end of the slinky takes to reach the bottom.
Data extracted from the video shown in Fig. 2, illustrating the finite time for collapse of the turns of the slinky. Upper panel: position versus time of the top of the slinky (circles, blue online), turn eight of the slinky (+symbols, black online) and turn ten of the slinky (× symbols, red online). Position is negative downwards in this panel. Lower panel: The spacing of turns eight and ten versus time.
Left panel: the model for a hanging slinky, with the slinky represented as a helix with turn spacing matching . Typical slinky parameters are used. The dot in each panel indicates the location of the center of mass, and the light gray (green online) part of the slinky at the bottom is the collapsed section. Right panel: the finite-collapse-time model for the slinky during the fall at time . The top dark gray (blue online) section of the slinky is the section undergoing collapse, above the downward-propagating collapse front indicated by a dashed horizontal line.
The instant-collapse model (Ref. 9) for a falling slinky using parameters typical of a real slinky. Upper panel: position versus time of the slinky top (upper solid curve), center-of-mass (middle solid curve), bottom (lower solid curve), and wave front initiating collapse (dashed curve). Position is negative downwards in this figure. Lower panel: velocity of the slinky top versus time. The total collapse time is shown as the vertical dashed line in both panels.
The gradient , which describes the local slinky extension, versus mass density in the finite-collapse-time model. The tension defined by this profile declines linearly behind the wave front [located at ] from a value matching the tension in the hanging slinky at the front, to the minimum tension value at . Ahead of the front the tension is unchanged from that in the hanging slinky.
The finite-collapse-time model for a falling slinky using the same slinky parameters as in Fig. 5. The collapse of the model slinky is assumed to occur via a linear decay in tension over ten turns of the slinky. Upper panel: position versus time of the slinky top (upper solid curve), center-of-mass (middle solid curve), bottom (lower solid curve), and wave front initiating collapse (dashed curve). Position is negative downwards in this figure. Lower panel: velocity of the slinky top versus time. The total collapse time is shown as the vertical dashed line in both panels.
The finite-collapse-time model applied to slinky A. The upper panel shows position versus time for the slinky top (upper), turn 10 (middle), and slinky bottom (lower), with the observed data represented by symbols and the best-fit model values by curves. The fitting is based on the observed positions of the slinky top. The lower panel shows the velocity of the slinky top versus time. The vertical dashed lines in both panels show the time of release of the slinky (left), which is a model parameter, and the model collapse time (right).
The finite-collapse-time model applied to slinky B. The presentation is the same as in Fig. 8.
An experiment with different methods of suspension of the top of slinky B. In the left-hand image the top is suspended from a string tied across a diameter of the first turn of the slinky. In the right-hand image the string is tied around the first two turns.
Data extracted for the initial fall of slinky B following suspension using the two methods shown in Fig. 10. The circles and squares show results for suspension by one turn and the + and × symbols for suspension by two turns. The upper panel shows the position versus time of the top and of the first turn below the initially tied top section. The lower panel shows the velocities of the top in each case, obtained by differencing the position data (circles for suspension by one turn and + symbols for suspension by two).
Measured data for two real slinkies.
Best-fit model parameters for the slinkies.
Predictions (for best-fit model parameters) and observations for the number of collapsed turns when hanging, and for the fundamental mode frequency.
Article metrics loading...
Full text loading...