Shaskolskaya & Eltsin Problem 14

Physics · mechanics

Problem

If the disk of a stroboscope with two holes performs N revolutions per second, then the total number of flashes per second will be 2N and the period between two flashes will be

t = \frac{1}{2N}

The velocity of the drops at the point illuminated by the stroboscope will be

v = \sqrt{2gh} = \sqrt{2 \times 980 \times 22.5} = 210 \text{ cm/sec}

Since the velocity of the drops is great and the distance between any two of them is small (s = 2 cm), we can assume that each drop traverses this distance with constant velocity. Then the time taken by a drop to move through distance s is

t_1 = \frac{s}{v} = \frac{2}{210} \text{ sec}

If the drops are to appear to be stationary, then

t_1 = t

, i.e.,

\frac{1}{2N} = \frac{2}{210}

Thus

N = \frac{210}{4} = 52.5 \text{ rev/sec}

If the drops have time to traverse a distance which is two, three, etc., times greater than the distance between the drops, then they will also be illuminated at the same point and will appear motionless. Therefore the number of revolutions can be any whole number of times less than 52.5 rev/sec. But the fewer the revolutions, the longer the drops will be illuminated (given a fixed size of hole), and the more noticeable will be each drop's displacement during this time. Therefore the picture will be less clear. 15. We shall see a circle of light where the illuminated apertures succeed one another at intervals of

\frac{1}{16}

sec or less. Since the apertures are 1 cm apart, it is clear that their speed must be

v = \frac{s}{t} = \frac{1 \text{ cm}}{1/16 \text{ sec}} = 16 \text{ cm/sec}

Thus we shall see a circle of light where the linear velocity of the revolving disk

v \geq 16

cm/sec. From the relationship between linear and angular velocity,

v = \omega R

, we have

v \geq 16 \text{ cm/sec}; \quad \omega = 2\pi n = 2\pi \times 30 \text{ rad/sec}

R \geq \frac{16 \times 60}{2\pi \times 30} = 5.1 \text{ cm}

i.e., the illuminated apertures merge into a continuous circle at a distance of

R \geq 5.1

cm from the centre.

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