# Popular Science Monthly/Volume 83/August 1913/Bernoulli's Principle and its Application to Explain the Curving of a Baseball

 BERNOULLI'S PRINCIPLE AND ITS APPLICATION TO EXPLAIN THE CURVING OF A BASEBALL
By Dr. S. LEROY BROWN Ph.D.,

UNIVERSITY OF TEXAS

WHEN a liquid or a gas is flowing through a horizontal pipe and encounters a constriction in the pipe, there is a higher velocity of the fluid and a lower pressure in the constriction than in the larger section of the pipe. At first thought, this is contrary to what one would expect, for the crowding of the fluid into a smaller section would apparently raise the pressure. Closer analysis, however, shows that places in the pipe where the velocity of the fluid is greater must be places of lower pressure and at places where the velocity of the fluid is less, the pressure must be greater.

Consider a definite mass of water as m in Fig. 1. When this piece of water moves from position A to position B, its velocity is increased

since the velocity of the water in the smaller section of the pipe must be larger than in the larger section if the same amount of water per second which flows through the larger section is to go through the smaller section. Since the velocity of this mass of water is increased (mass m is accelerated) there must be an unbalanced force acting on it. This unbalanced force is furnished by a higher pressure at position A than at position B. That is, the pressure behind the moving mass m is greater than in front of it and, consequently, the velocity is increased. As the piece of water leaves the neck in the pipe, the pressure in front of it is greater than the pressure behind it and it slows down to the lower velocity in the larger section of the pipe.

The generalization of the above described phenomenon is, that places in a fluid where the velocity is relatively greater are places of lower pressure and places where the velocity of the fluid is relatively smaller are places of higher pressure. This generalization (first established by John Bernoulli) is called Bernoulli's principle and its application to explain many paradoxical results is interesting.

Fig. 2 shows how the weight of a marble may be held up by blowing through the tube. The high velocity of the air over the top of the marble causes a lower pressure than there is under the marble where the air has a comparatively low velocity and this difference in pressure exerts an upward force which is sufficient to balance the weight of the marble.

A light ball may be held in midair by a stream of air flowing just above it, as shown in Fig. 3. Just above the ball is a region of high

 ⁠Fig. 2. Fig. 3.

velocity and low pressure, while under the ball is a low velocity and high pressure region and therefore the force of gravity on the ball is balanced by the difference in pressure.

The difference between the higher pressure in the larger section and the lower pressure (higher velocity of water) in the smaller section of a water pipe is indicated by the manometer in Fig. 4. The pressure in the larger section of the pipe is greater than the pressure in the smaller section by an amount equal to the pressure exerted by a column of mercury h high. If the areas of the larger and smaller sections are known, the rate at which the water is flowing through the pipe (cubic feet per second or gallons per second) can be determined from the difference in pressure which is indicated by the manometer. This method of measuring rates of discharge is used in the Venturi water meter, which is not essentially different from the arrangement shown in Fig. 4.

A baseball moving through the air is the same as air moving past the baseball as far as the forces which the air exert on the ball are concerned. A ball thrown straight (without rotating) through the air can

be pictured as air moving past the ball with the same velocity on all sides of the ball which is shown by the equal density of stream lines above and below the ball in Fig. 5. According to Bernoulli's principle,

there are equal pressures (equal velocities) of the air on all sides of the ball and it does not curve.

If the ball is rotating as it moves through the air, its spin will increase the velocity of the air past the ball on one side and retard the velocity of the air past the ball on the opposite side as is indicated in Fig. 6 by many stream lines on one side and few on the other. The higher pressure (low velocity) on the one side pushes the ball toward the low pressure (high velocity) region and it curves as shown by the heavy arrow in Fig. 6. If the ball had been rotating in an opposite

direction, the low pressure region would have been on the other side of the ball and it would have curved in the opposite direction.

In order to show this difference in pressure on the sides of a rotating ball as it is thrown through the air, or in practise as the air is driven past the ball, the author has devised the following demonstration. The air is driven past the ball by a centrifugal blower and the pressures on two opposite sides of the ball are indicated by manometers as shown in Fig. 7.

When the ball is not rotating, the velocity of the air on the two sides of the ball is the same (shown by equal density of stream lines in top view section of Fig. 8) and the manometers indicate equal pressures on the two sides of the ball (end view of Fig. 8). This is equivalent to the ball going through the air without rotating and without curving to either side as shown by the heavy arrow. However, the pressure on either side of the ball is less than the pressure in the still air outside the tube which directs the air past the ball; that is, the high velocity regions near the ball are low-pressure regions.

When the ball is rotating as shown in Fig. 9 the friction against the surface of the ball accelerates the flow of air past it on one side and retards the air stream on the other side; that is, the stream lines are more dense on one side (shown in top view of Fig. 9) and the manometers indicate unequal pressures on the two sides of the ball (shown

 Fig. 8. ⁠Fig. 9. Fig. 10.

in end view of Fig. 9). This is equivalent to the bail going through the air with a rotary motion and the difference in pressure on the two sides curves it as shown by the heavy arrow.

If the rotation of the ball is reversed, the lower-pressure region (higher velocity region) is on the other side of the ball. This is equivalent to the ball going through the air with a rotary motion as shown in Fig. 10 and the difference in pressure on the two sides of it causes it to curve as shown by the heavy arrow.

1. When a fluid flows from a region of low velocity to a region of high velocity the pressure decreases but the reverse, that when a fluid flows from a region of high velocity to a region of low velocity the pressure increases, is not always true. For example, the friction of the water against the sides of the tube in Fig. 1 might be sufficient to decrease the velocity of the water as it flows out of the neck without the pressure increasing.