Съдържание

IPhO 1976, Теория и експеримент — Задача 1. Problem 1

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Всички материали от IPhO 1976

Problems of the 9th International Physics Olympiads (Budapest, Hungary, 1976)

Условие

A hollow sphere of radius $R = 0.5\ \mathrm{m}$ rotates about a vertical axis through its centre with an angular velocity of $\omega = 5\ \mathrm{s^{-1}}$ . Inside the sphere a small block is moving together with the sphere at the height of $R/2$ (Fig. 6). ( $g = 10\ \mathrm{m/s^2}$ .)

a) What should be at least the coefficient of friction to fulfill this condition?

b) Find the minimal coefficient of friction also for the case of $\omega = 8\ \mathrm{s^{-1}}$ .

c) Investigate the problem of stability in both cases,

α) for a small change of the position of the block,

β) for a small change of the angular velocity of the sphere.

Отговори

Решение

Part of the sphere's cross-section with the vertical axis; the radius to the block makes angle α with the axis; force vectors at the block: N along the radius toward the centre, S along the wall upwards, mg downward, and the resultant mω²Rsinα horizontal toward the axis along a dashed line. — Figure 7

a) The block moves along a horizontal circle of radius $R\sin\alpha$ . The net force acting on the block is pointed to the centre of this circle (Fig. 7). The vector sum of the normal force exerted by the wall $N$ , the frictional force $S$ and the weight $mg$ is equal to the resultant: $m\omega^2 R\sin\alpha$ .

The connections between the horizontal and vertical components:

m\omega^2 R\sin\alpha = N\sin\alpha - S\cos\alpha,

mg = N\cos\alpha + S\sin\alpha.

The solution of the system of equations:

S = mg\sin\alpha\left(1-\frac{\omega^2 R\cos\alpha}{g}\right),

N = mg\left(\cos\alpha+\frac{\omega^2 R\sin^2\alpha}{g}\right).

The block does not slip down if

\mu_a \ge \frac{S}{N} = \sin\alpha\cdot\frac{1-\dfrac{\omega^2 R\cos\alpha}{g}}{\cos\alpha+\dfrac{\omega^2 R\sin^2\alpha}{g}} = \frac{3\sqrt{3}}{23} = \mathbf{0.2259}.

In this case there must be at least this friction to prevent slipping, i.e. sliding down.

Cross-section of the sphere with the vertical axis; the radius to the block makes angle α with the axis; forces N along the radius, S along the wall downwards, mg downward, and the horizontal resultant mω²Rsinα along a dashed line. — Figure 8

b) If on the other hand $\dfrac{\omega^2 R\cos\alpha}{g} > 1$ some friction is necessary to prevent the block to slip upwards. $m\omega^2 R\sin\alpha$ must be equal to the resultant of forces $S$ , $N$ and $mg$ . Condition for the minimal coefficient of friction is (Fig. 8):

\mu_b \ge \frac{S}{N} = \sin\alpha\cdot\frac{\dfrac{\omega^2 R\cos\alpha}{g}-1}{\cos\alpha+\dfrac{\omega^2 R\sin^2\alpha}{g}} = \frac{3\sqrt{3}}{29} = \mathbf{0.1792}.

c) We have to investigate $\mu_a$ and $\mu_b$ as functions of $\alpha$ and $\omega$ in the cases a) and b) (see Fig. 9/a and 9/b):

Fig. 9/a: graph of μ_a against α up to 90°, value 0.5 marked on the vertical axis; solid curve for ω = 5/s dipping below zero then rising steeply; dashed curves for ω < 5/s (left) and ω > 5/s (right); a dotted vertical line marks the equilibrium angle with small circles on the curves. — Figure

Fig. 9/b: graph of μ_b against α up to 90°, value 0.5 marked on the vertical axis; solid curve for ω = 8/s rising to a maximum near 0.5 and falling; dashed curves for ω > 8/s and ω < 8/s; a dotted vertical line marks the equilibrium angle with small circles on the curves. — Figure

In case a): if the block slips upwards, it comes back; if it slips down it does not return. If $\omega$ increases, the block remains in equilibrium, if $\omega$ decreases it slips downwards.

In case b): if the block slips upwards it stays there; if the block slips downwards it returns. If $\omega$ increases the block climbs upwards, if $\omega$ decreases the block remains in equilibrium.

A sphere of radius R rotating about a vertical axis (rotation arrow below); a small block (black dot) on the inner surface lies on a dashed horizontal circle; its height above the bottom of the sphere is marked R/2. — Figure 6

Literature

[1] W. Gorzkowski: Problems of the 1st International Physics Olympiad
Physics Competitions 5, no2 pp6-17, 2003

[2] R. Kunfalvi: Collection of Competition Tasks from the Ist through XVth International Physics Olympiads 1967-1984
Roland Eötvös Physical Society in cooperation with UNESCO, Budapest, 1985

[3] A Nemzetközi Fizikai Diákolimpiák feladatai I.-XV.
Eötvös Loránd Fizikai Társulat, Középiskolai Matematikai Lapok, 1985

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