Question

Explain why
(a) to keep a piece of paper horizontal, you should blow over it, not under it.
(b) when we try to close a water tap with our fingers fast jets of water gush through the openings between our fingers
(c)the size of the needle of a syringe controls flow rate better than the thumb pressure exerted by a doctor while administering an injection
(d)a fluid flowing out of a small hole in a vessel results in a backward thrust on the vessel
(e) a spinning cricket ball in air does not follow a parabolic trajectory

Answer 1

Hint: (a)According to Bernoulli's theorem, if velocity of a gas increases then its pressure decreases. So, when we blow on the upper side of the paper the pressure there decreases.
(b) The equation of continuity states that area multiplied by velocity is a constant. When we close the water tap the total area of the opening is now reduced.
(c). using the flow rate equation we can explain this situation. Flow rate equation is,
$Q={\displaystyle \frac{\pi r^4\left(\mathrm{\Delta }p\right)}{8\eta l}}$
Where $r$ is the radius and $\mathrm{\Delta }p$ change in pressure, $\eta$ is the viscosity of fluid and $l$ is the length.
(d) According to equation of continuity the fluid flowing out through a small area will have a large velocity
No external forces are acting on the system, so we can apply the law of conservation of momentum in this case.
(e) There is a difference in velocity of air above and below the ball due to its spin. Thich creates a pressure difference according to Bernoulli's principle. This results in a lift.
Step by step solution:
(a) To keep a piece of paper horizontal we need to blow over it, not under it. This can be explained using Bernoulli's theorem.
According to Bernoulli's theorem,
$P+{\displaystyle \frac{1}{2}}\rho v^2+\rho gh=\text{constant}$
Where, $P$ is the pressure ${\displaystyle \frac{1}{2}}\rho v^2$ denotes the kinetic energy per unit volume and $\rho gh$ denotes the potential energy per unit volume.
When density $\rho$ acceleration due to gravity $g$ and height $h$ remains constant if we increase the velocity $v$ of the fluid the pressure $P$ should decrease so that the final value remains constant.
Thus, if velocity of a gas increases then its pressure decreases. So, when we blow on the upper side of the paper the pressure there decreases. Since the pressure under the paper is now greater than the pressure above, the net force will be acting upward, resulting in the paper to stay horizontal.
If you are blowing air under, velocity there is increased so pressure will be reduced in the lower part. Then the pressure above will be greater and thus the net force will be acting downward, resulting in the fall of the paper
(b) The equation of continuity states that area multiplied by velocity is a constant.
$Av=\text{constant}$
Where $A$ is the area and $v$ is the velocity .
Or we can say that area is inversely proportional to velocity. Therefore, when area increases velocity decreases and when area decreases velocity increases. When we close the water tap the total area of the opening is now reduced therefore, the velocity of the water flowing out through the openings of our fingers will increase.
(c)The size of needle of a syringe controls flow rate better than the thumb pressure exerted by a doctor.
The flow rate is given by the equation
$Q={\displaystyle \frac{\pi r^4\left(\mathrm{\Delta }p\right)}{8\eta l}}$
Where $r$ is the radius and $\mathrm{\Delta }p$ change in pressure, $\eta$ is the viscosity of fluid and $l$ is the length.
So, we can see that dependence is much greater for radius than change in pressure. Even a small change in radius of the syringe can increase the flow to a great extent. Since the power of radius is $4$.
(d) When a fluid flows out from a small opening, the vessel receives a backward thrust. According to the equation of continuity the fluid flowing out through a small area will have a large velocity. So, the fluid which was initially at rest now has momentum. Since no external forces are acting on the system, we can apply conservation of momentum in this case. According to which the total momentum is conserved. That is, total initial momentum will be equal to the total final momentum. To conserve momentum the vessel attains a backward velocity or a thrust is exerted on the vessel in the backward direction.
(e) A spinning ball has two types of velocity. Linear velocity due to translatory motion and angular velocity due to rotation.
According to Bernoulli's theorem,
$P+{\displaystyle \frac{1}{2}}\rho v^2+\rho gh=\text{constant}$
so, when velocity increases pressure should decrease and vice versa.
There is a difference in velocity of air above and below the ball due to its spin which creates a pressure difference. The velocity above is greater so pressure there is lower than below. This results in a lift. This effect is known as magnus effect. It prevents the ball from moving in a parabolic path.

Note: The Magnus effect is a lifting force. This effect happens only if the body spins when it travels through a fluid. If the cricket ball was not spinning then this effect will not be present.
We have applied Bernoulli’s theorem under the assumption that friction due to resistive force offered by air also called viscous force is negligible.