Physics Complete

Up thrust or the Buoyant Force and Problems with Solutions

Velocity of Efflux and Venturi Meter
Dynamic Lift and Air Foil

Problems on Bernoulli's theorem and Its Applications

Dynamic lift
Venturi meter
Torricelli's theorem
Blood flow and heart attack
Stream line flow
What is pressure ?
Pressure variation with depth
Pascal's Law

Dynamic Lift and Bernoulie's Theorem

When a body is placed in a liquid in the state of rest, it experience a upthrust and it is called static lift. If the resultant force is experienced a body in a fluid obviously in the state of motion then it is called dynamic lift.

Dynamic lift is the force that acts on a body, such as airplane wing, a hydrofoil or a spinning ball, by virtue of its motion through a fluid. In many games such as cricket, tennis, baseball, or golf, we notice that a spinning ball deviates from its parabolic trajectory as it moves through air. This deviation can be partly explained on the basis of Bernoulli’s principle.This can be explained under three different conditions as shown below.

1. Ball moving without spin: The figure a shows the streamlines around a non spinning ball moving relative to a fluid. From the symmetry of streamlines it is clear that the velocity of fluid (air) above and below the ball at corresponding points is the same resulting in zero pressure difference. The air therefore, exerts no upward or downward force on the ball.

2. Ball moving with spin: A ball which is spinning drags air along with it. If the surface is rough more air will be dragged. Figure b shows the streamlines of air for a ball which is moving and spinning at the same time. The ball is moving forward and relative to it the air is moving backwards. Therefore, the velocity of air above the ball relative to it is larger and below it is smaller. The stream lines thus get crowded above and rarified below.

This difference in the velocities of air results in the pressure difference between the lower and upper faces and their is a net upward force on the ball. This dynamic lift due to spinning is called Magnus effect.

Aerofoil or lift on aircraft wing: Figure c shows an aerofoil, which is a solid piece shaped to provide an upward dynamic lift when it moves horizontally through air. The cross section of the wings of an aeroplane looks somewhat like the aerofoil shown in figure c with streamlines around it. When the aerofoil moves against the wind, the orientation of the wing relative to flow direction causes the streamlines to crowd together above the wing more than those below it. The flow speed on top is higher than that below it. There is an upward force resulting in a dynamic lift of the wings and this balances the weight of the plane.

Hydrostatics topics :

Problems on Bernoulli's theorem and Its Applications

Venturi meter
Torricelli's theorem
Blood flow and heart attack
Stream line flow
What is pressure ?
Pressure variation with depth
Pascal's Law

Fluid Flow Venturi-meter

The Venturi-meter is a device to measure the 0f flow speed of incompressible fluid. It consists of a tube with a broad diameter and a small constriction at the middle as shown in figure below. A manometer in the form of a U-tube is also attached to it, with one arm at the broad neck point of the tube and the other at constriction as shown in figure. The manometer contains a liquid of density ρm. The speed v1 of the liquid flowing through the tube at the broad neck area A is to be measured from equation of continuity.

This pressure difference causes the fluid in the U tube connected at the narrow neck to rise in comparison to the other arm. The difference in height h measure the pressure difference.

The principle behind this meter has many applications. The carburetor of automobile has a Venturi channel (nozzle) through which air flows with a large speed. The pressure is then lowered at the narrow neck and the petrol (gasoline) is sucked up in the chamber to provide the correct mixture of air to fuel necessary for combustion. Filter pumps or aspirators, Bunsen burner, atomisers and sprayers used for perfumes or to spray insecticides work on the same principle.

Hydrostatics topics :

Problems on Bernoulli's theorem and Its Applications

Torricelli's theorem
Blood flow and heart attack
Stream line flow
What is pressure ?
Pressure variation with depth
Pascal's Law

Fluid Flow Torricelli’s Law

The word efflux means fluid outflow. Torricelli discovered that the speed of efflux from an open tank is given by a formula identical to that of a freely falling body. Consider a tank containing a liquid of density ρ with a small hole in its side at a height y1 from the bottom .

The air above the liquid, whose surface is at height y2, is at pressure P. From the equation of continuity we have

If the cross sectional area of the tank A2 is much larger than that of the hole (A2 >>A1), then we may take the fluid to be approximately at rest at the top, i.e. v2 = 0. Now applying the Bernoulli equation at points 1 and 2 and noting that at the hole P1 = Pa, the atmospheric pressure

When P >>Pa and 2 g h may be ignored, the speed of efflux is determined by the container pressure. Such a situation occurs in rocket propulsion. On the other hand if the tank is open to the atmosphere, then P = Pa then V₁=(2gh)½

This is the speed of a freely falling body. This equation is known as Torricelli’s law.

Hydrostatics topics :

Up thrust or the Buoyant Force and Problems with Solutions

Velocity of Efflux and Venturi Meter
Dynamic Lift and Air Foil

Problems on Bernoulli's theorem and Its Applications

Blood flow and heart attact
Stream line flow
What is pressure ?
Pressure variation with depth
Pascal's Law

Blood Flow and Heart Attack

Bernoulli’s principle says that the sum of potential,kinetic and pressure energy per unit mass of a incompressible,non viscous fluid remains constant.

Bernoulli’s principle helps in explaining blood flow in artery. The artery may get constricted due to the accumulation of plaque on its inner walls. In order to drive the blood through this constriction a greater demand is placed on the activity of the heart.

The speed of the flow of the blood in this region is raised which lowers the pressure inside and the artery may collapse due to the external pressure. The heart exerts further pressure to open this artery and forces the blood through. As the blood rushes through the opening, the internal pressure once again drops due to same reasons leading to a repeat collapse. This may result in heart attack.

Hydrostatics topics :

Up thrust or the Buoyant Force and Problems with Solutions

Problems on Bernoulli's theorem and Its Applications

Stream line flow
What is pressure ?
Pressure variation with depth
Pascal's Law
Bulk Modulus
Shear modulus
Elastic behavior of Solids

Stream line flow

The flow of the fluid is said to be steady if at any given point, the velocity of each passing fluid particle remains constant in time. This does not mean that the velocity at different points in space is same. The velocity of a particular particle may change as it moves from one point to another.

That is, at some other point the particle may have a different velocity, but every other particle which passes the second point behaves exactly as the previous particle that has just passed that point. Each particle follows a smooth path, and the paths of the particles do not cross each other.

The path taken by a fluid particle under a steady flow is a streamline. It is defined as a curve whose tangent at any point is in the direction of the fluid velocity at that point.

Let us consider the path of a particle as shown in figure below, the curve describes how a fluid particle moves with time. The curve PQ is like a permanent map of fluid flow, indicating how the fluid streams.

No two streamlines can cross, for if they do, an oncoming fluid particle can go either one way or the other and the flow would not be steady. Hence, in steady flow, the map of flow is stationary in time. If we intend to show streamline of every flowing particle, we would end up with a continuum of lines. Consider planes perpendicular to the direction of fluid flow e.g., at three points P, R and Q in figure b.

The plane pieces are so chosen that their boundaries be determined by the same set of streamlines.

In general Av = constant Av gives the volume flux or flow rate and remains constant throughout the pipe of flow. This is called the equation of continuity and it is a statement of conservation of mass in flow of incompressible fluids.

Thus, at narrower portions where the streamlines are closely spaced, velocity increases and its vice versa. From Fig b it is clear that AR > AQ or vR < vQ, the fluid is accelerated while passing from R to Q. This is associated with a change in pressure in fluid flow in horizontal pipes.

Hydrostatics topics :

Up thrust or the Buoyant Force and Problems with Solutions

Problems on Bernoulli's theorem and Its Applications

What is pressure ?
Pressure variation with depth
Pascal's Law
Bulk Modulus
Shear modulus
Elastic behavior of Solids

Hydraulic Lift

Hydraulic lift and hydraulic brakes are based on the Pascal’s law. In these devices fluids are used for transmitting pressure. In a hydraulic lift as shown in figure below two pistons are separated by the space filled with a liquid. A piston of small cross section A1 is used to exert a force F1 directly on the liquid. The pressure P = F/A in the first case.

P is transmitted throughout the liquid to the larger cylinder attached with a larger piston of area A2, which results in an upward force of P × A2. Therefore, the piston is capable of supporting a large force (large weight of, say a car, or a truck placed on the platform) F2 = PA2 =A₂F₁/A₁

By changing the force at A1, the platform can be moved up or down. Thus, the applied force has been increased by a factor of A₂/A₁ and this factor is the mechanical advantage of the device.This is shown with the help of the following diagram.

Hydrostatics topics :

Up thrust or the Buoyant Force and Problems with Solutions

Problems on Bernoulli's theorem and Its Applications

What is pressure ?
Pressure variation with depth
Pascal's Law
Bulk Modulus
Shear modulus
Elastic behavior of Solids

Variation of Pressure with Depth

Pressure is the force acting on a body per unit surface area.It varies with depth as explained below.

Consider a fluid at rest in a container. In figure below point 1 is at height h above a point 2. The pressures at points 1 and 2 are P1 and P2 respectively. Let us Consider a cylindrical element of fluid having area of base A and height h.

As the fluid is at rest the resultant horizontal forces should be zero and the resultant vertical forces should balance the weight of the element. The forces acting in the vertical direction are due to the fluid pressure at the top (P1A) acting downward, at the bottom (P2A) acting upward. mg is weight of the fluid in the cylinder we have (P2 − P1) A = mg

if ρ is the mass density of the fluid, mass of fluid to be m = ρV= ρhA so that P2 − P1= ρgh

Pressure difference depends on the vertical distance h between the points (1 and 2), mass density of the fluid ρ and acceleration due to gravity g. If the point 1 is shifted to the top of the fluid (say water), which is open to the atmosphere, P1 may be replaced by atmospheric pressure (Pa) and we replace P2 by P.

P = Pa + ρgh

The pressure P, at depth below the surface of a liquid open to the atmosphere is greater than atmospheric pressure by an amount ρgh. The excess of pressure, P − Pa, at depth h is called a gauge pressure at that point.

The area of the cylinder is not appearing in the expression of absolute pressure. Thus, the height of the fluid column is important and not cross sectional or base area or the shape of the container. The liquid pressure is the same at all points at the same horizontal level (same depth).

The result is appreciated through the example of hydrostatic paradox. Consider three vessels A, B and C [Fig below] of different shapes. They are connected at the bottom by a horizontal pipe. On filling with water the level in the three vessels is the same though they hold different amounts of water. It is because water at the bottom has the same pressure below each section of the vessel.

Hydrostatics topics :

Up thrust or the Buoyant Force and Problems with Solutions

Problems on Bernoulli's theorem and Its Applications

What is pressure ?
Pascal's Law
Bulk Modulus
Shear modulus
Elastic behavior of Solids

Pascal's Law

the pressure in a fluid at rest is the same at all points if they are at the same height. This can be be demonstrated in a simple way.

The figure below shows an element in the interior of a fluid at rest. This element ABC-DEF is in the form of a right-angled prism. In principle, this prismatic element is very small so that every part of it can be considered at the same depth from the liquid surface and therefore, the effect of the gravity is the same at all these points.

Let us enlarged this element. The forces on this element are those exerted by the rest of the fluid and they must be normal to the surfaces of the element . Thus, the fluid exerts pressures Pa, Pb and Pc on this element of area corresponding to the normal forces Fa, Fb and Fc as shown in fig.ures on the faces BEFC, ADFC and ADEB denoted by Aa, Ab and Ac respectively.

Fb sinθ = Fc, Fb cosθ = Fa (by equilibrium)

Ab sinθ = Ac, Ab cosθ = Aa (by geometry)

Thus, ratio of force per area and hence pressure is same at all points A,B and C.

Hence, pressure exerted is same in all directions in a fluid at rest. Pressure is not a vector quantity. No direction can be assigned to it. The force against any area within (or bounding) a fluid at rest and under pressure is normal to the area, regardless of the orientation of the area.

In a a fluid element in the form of a horizontal bar of uniform cross-section. The bar is in equilibrium. The horizontal forces exerted at its two ends must be balanced or the pressure at the two ends should be equal. Hence for a liquid in equilibrium the pressure is same at all points in a horizontal plane.

If the pressure were not equal in different parts of the fluid, then there would be a flow as the fluid will have some net force acting on it. Thus in the absence of flow the pressure in the fluid must be same everywhere. Wind is flow of air due to pressure differences.

Related posts :

Up thrust or the Buoyant Force and Problems with Solutions

Problems on Bernoulli's theorem and Its Applications

What is pressure ?
Bulk Modulus
Shear modulus
Elastic behavior of Solids
Stress and strain
Stress and Strain Curve
Determination of Young's modules

Mechnical Properties Fluid Pressure

Liquids and gases can flow and are therefore, called fluids. The volume of solid, liquid or gas depends on the stress or pressure acting on it. The difference between gases and solids or liquids is that for solids or liquids the change in volume due to change of external pressure is rather small.

Solids and liquids have much lower compressibility as compared to gases. Shear stress can change the shape of a solid keeping its volume fixed. The key property of fluids is that they offer very little resistance to shear stress; their shape changes by application of very small shear stress. The shearing stress of fluids is about million times smaller than that of solids.

Pressure: Smaller the area on which the force acts, greater is the impact. This concept is known as pressure.

When an object is submerged in a fluid at rest, the fluid exerts a force on its surface. This force is always normal to the object’s surface. If there were a component of force parallel to the surface, the object will also exert a force on the fluid parallel to it; as a consequence of Newton’s third law. This force will cause the fluid to flow parallel to the surface.

Since the fluid is at rest, this cannot happen. Hence, the force exerted by the fluid at rest has to be perpendicular to the surface in contact with it. This is shown in Figure.a.

The normal force exerted by the fluid at a point may be measured. An idealised form of one such
pressure-measuring device is shown in Fig.b. It consists of an evacuated chamber with a spring that is calibrated to measure the force acting on the piston. This device is placed at a point inside the fluid. The inward force exerted by the fluid on the piston is balanced by the outward spring force and is thereby measured.

If F is the magnitude of this normal force on the piston of area A then the average pressure Pav
is defined as the normal force acting per unit area.

Pressure is a scalar quantity.

Related posts :

Up thrust or the Buoyant Force and Problems with Solutions

Problems on Bernoulli's theorem and Its Applications