Motion of a metal nut sliding around a vertical loop

An experiment is described where a metal nut was allowed to slide from the top to the bottom of a vertical wire loop. An interesting feature is that the normal reaction force on the nut decreases to zero and reverses direction before the nut reaches the bottom of the loop.


Introduction
A common problem involving circular motion is one where an object rolls around a vertical loop.The best known example is provided by loop-theloop apparatus where a ball is injected into the bottom of a circular loop and makes its way to the top if it has sufficient speed at the bottom [1][2][3].A similar problem concerns a small ball that rolls down the outside surface a large ball.Starting from rest, the small ball will lose contact with the large ball at a point where the normal reaction force decreases to zero [4,5].Other authors have investigated the effects of friction when an object slides or rolls along a vertical convex or concave path [6][7][8].
A related experiment is described in the present paper where a metal nut is released at the top of circular wire loop and slides all the way 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.
to the bottom and part way up the other side.The experiment is quite simple, could easily be investigated by students and the theoretical model involves only elementary mechanics.However, the resulting equations require a numerical solution and a good qualitative understanding of the physics involved.
The experimental arrangement is shown in figure 1 and in supplementary video Loop.mov.If a nut of mass m is released from rest at a point near the top of the loop, its speed will increase as it slides towards the bottom.The forces on the nut are its weight, mg, the normal reaction force, N, and a sliding friction force F = µN where µ is the coefficient of sliding friction.Motion of the nut could be calculated easily using conservation of energy if the friction force was zero.In practice, friction prevents the nut from rising up the other side of the loop all the way to the top.The equations of motion are

R Cross
Figure 1.A metal nut sliding around a circular wire loop. and where |N| is the magnitude of N and mv 2 /R is the centripetal force on the nut.Equation ( 1) also applies when a block slides down an incline, but in the present case θ does not remain constant.N can change sign and point radially inwards rather than outwards, but that does not change the direction of the friction force.However, the sign in of µ needs to change if v reverses direction since sliding friction acts in the opposite direction to v. Since v increases with time at the start and cos θ decreases with time, N can decrease to zero when v 2 = gR cos θ.The same relation applies when a small ball rolls down a large ball.The small ball loses contact when N = 0.However, the nut does not lose contact with the wire loop when N = 0 since the wire passes through the hole in the nut.Nevertheless, F decreases to zero when N = 0 then increases again if N changes sign.

Experimental results
A wire loop of diameter 135 mm was constructed with coat hanger wire and each end was bent to insert in a hole in a vertical board to support the loop in a vertical plane.A 5.05 g metal nut was threaded on the loop and released from rest at a point near the top of the loop.The result was filmed at 300 frames s −1 and analysed with Tracker motion analysis software to record the x, y coordinates of the centre of mass of the nut.The Smooth curves were fit to the results in figure 2 using Kaleidagraph software to determine the v x and v y velocity components vs time, plus the velocity v = (v 2 x + v 2 y ) 1/2 and the rotation angle, θ, given by tan θ = x/y.Graphs of v vs t and θ vs t are shown in figure 3. Numerical solutions of equations ( 1) and ( 2) are also shown in figure 3, assuming that v = 0.1 m s −1 and θ = 20 • at t = 0, with µ = 0.24 providing a good fit to the experimental results.
The velocity of the nut is a maximum at t = 0.25 s where θ = 130 • , and decreases to zero at t = 0.41 s where θ = 240 • .In the absence of friction, the velocity of the nut would be a maximum at θ = 180 • and the velocity would decrease to zero near θ = 360 • , depending on the initial starting angle.The starting angle was 20 • in figure 3

Conclusion
The behaviour of the nut is relatively simple and almost intuitively obvious, since the nut behaves in a similar way to a heavily damped pendulum bob, which also rotates in a circular path in the vertical plane.However, it requires a good understanding of circular motion and the effects of friction to explain the results in a quantitative manner.Less obvious is the fact that N changes sign and then oscillates in magnitude, but the result can be understood by examining the magnitude of each term in equation (2) as a function of time.
It is also not obvious that the results in figure 1 lead to the variations in v and θ shown in figure 2, although it is easy to see that the results in figure 2 are consistent with the behaviour of the nut in the supplementary video.The experiment therefore provides students with a challenge to understand and explain the physics involved.

Figure 2 .
Figure 2. x and y coordinates of the nut vs time.
since the nut remained at rest if θ < 20 • .The friction force did not remain constant around the loop since N varied with v and θ.The variation of N with time is shown in figure 4, calculated from the best fit theoretical solution.At the start, N ≈ mg cos θ since v is small, and N points radially outwards.At the end, N = −mg = −0.049N, and points radially inwards.At t = 0.145 s, N = 0 where v 2 = Rg cos θ.

Figure 4 .
Figure 4. Variation of N with time.