An oscillating screw sliding down an incline

A screw that slides down an incline also oscillates from side to side as a pendulum. Experimental results are presented, together with a simple explanation.

The 2023 IYPT (International Young Physicists' Tournament, see www.iypt.org/)included a problem to investigate the motion of a screw as it slides down an incline.Unlike other familiar objects that slide straight down an incline, a screw oscillates from side to side as it slides.The author investigated the problem out of curiosity and found that the screw oscillated as a pendulum while it simultaneously slid down the incline.More complicated motions of a screw on an incline have recently been observed by Tao et al [1] and will be described in a future publication.The author's observations were simpler and more easily explained and may therefore be of interest to high school physics students and teachers.
The supplementary video Screw.movshows the essential features of the experiment.A 27 mm long metal screw was released from rest at the top Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. of a 7 • smooth incline, with its axis aligned across the incline.Since the head of the screw was at the heavy end, the screw rotated on release as it slid down the incline.Several surprising features were observed.The screw sometimes slid to a stop before it reached the bottom of the incline, it drifted from side to side down the incline, and it oscillated like a pendulum.The geometry is shown in figure 1.

Experimental results
Motion of the screw down the incline was filmed with a video camera and analysed with Tracker software to record the x and y coordinates of each end of the screw.Typical results are shown in figure 2 for an incline angle θ = 7 • .The y coordinates in figure 2(b) show that the screw decelerates at an approximately constant rate down the incline, with a = −0.050± 0.001 m s −2 .The deceleration is very small, indicating that the friction force acting up the incline is slightly larger than the gravitational force mg sin θ acting down the incline, which explains why the screw sometimes came to a stop before it got to the bottom of the incline.
Motion of the screw across the incline is more complicated, but at least it is sinusoidal, as shown in figure 2(a).The whole screw translates back and forth across the incline, but the head of the screw oscillates with greater amplitude than the tip end of the screw.The screw behaves as a short pendulum where the top and bottom ends of the pendulum both oscillate in phase and at the same frequency but with different amplitudes.
At any given time, the screw is aligned at an angle α to the y axis, where α is taken to be zero if the head is directly below the tip.The variation of α with time is shown in figure 3. The oscillation frequency in figure 3 is the same as that in figure 2(a), the period being T = 1.1 s.The period is much longer than that of a simple pendulum with a length the same as that of the screw, mainly because the acceleration down the incline is much smaller that g [2].In effect, the screw oscillates as a pendulum on a planet with a small value of g.If the incline was horizontal, the period of oscillation would be infinite.That is, the screw would not oscillate at all.

Discussion
The trajectory of the screw down the incline can be attributed to the fact that the diameter of the head is larger than the diameter of the tip.The effect can be seen by rolling a paper cup or any other cone on a horizontal surface, as shown in figure 4(a).If a cone or a screw is set in motion on a horizontal surface, it rolls around a circular path, as shown in figure 4(b).The head and the tip of a screw each rotate in concentric circles, rather than along straight line paths, as they would if the head and tip were the same diameter.
On the incline, the screw started in position A in figure 4(b) and rolled to position B while simultaneously sliding down the incline.It then rotated back towards A as a pendulum but the oscillation amplitude decreased with time due to sliding friction on the incline.The screw behaved as a pendulum due to the gravitational force on the screw acting down the incline.When the screw was suspended vertically by a short length of cotton thread tied to the tip, it oscillated as a pendulum with a period T = 0.34 s.On the incline, the screw oscillated as a pendulum with a period T = 1.1 s, consistent with the fact that the period T = 2π √ L/g sin θ rather than T = 2π √ L/g.However, the effective length of the pendulum is slightly longer than the actual length of the screw since the screw pivots about a point beyond the tip, as indicated in figure 4(b).
Other effects are likely to be observed at different incline angles and with screws with a different cone angle and with different initial incline angles of the screw.Tao et al [1] found that under some conditions, the oscillation amplitude grew rather than decayed as the screw slid down the incline.

Figure 1 .
Figure 1.Incline geometry and motion of the screw (dashed curve).

Figure 2 .
Figure 2. x and y of the head and tip of the screw vs time.

Figure 3 .
Figure 3. Angular displacement of the screw, α, with respect to the y axis.

Figure 4 .
Figure 4. Rotation of (a) a paper cup and (b) the screw on a horizontal surface.