Friday, 16 February 2018

ANALYTICAL METHOD TO DETERMINE METACENTRIC HEIGHT

We were discussing the basic definition and derivation of total pressurecentre of pressurebuoyancy or buoyancy force and centre of buoyancy in our previous posts.

Today we will see here the analytical method to determine the meta-centric height with the help of this post.

Analytical method to determine the meta-centric height

As we know that meta-centre is basically defined as the point about which a body in stable equilibrium will start to oscillate when body will be displaced by an angular displacement.

 
Let us consider a body which is floating in the liquid. Let us assume that body is in equilibrium condition. Let us think that G is the centre of gravity of the body and B is the centre of buoyancy of the body when body is in equilibrium condition.

In equilibrium situation, centre of gravity G and centre of buoyancy B will lie on same axis which is displayed here in following figure with a vertical line.


Let us assume that we have given an angular displacement to the body in clockwise direction as displayed here in above figure.

Centre of buoyancy will be shifted now towards right side from neutral axis and let us assume that it is now B1.

Line of action of buoyancy force passing through this new position will intersect the normal axis passing through the centre of gravity and centre of buoyancy in original position of the body at a point M as displayed here in above figure. Where, M is the meta-centre.

As we have already discussed the term Meta-centric height i.e. the distance between the meta-centre of the floating body and the centre of gravity of the body. Therefore, MG in above figure will be the meta-centric height.

Angular displacement of the body in clockwise direction will cause a wedge-shaped prism BOB’ on the right side of the axis to go inside the water as displayed in above figure. This wedge will indicate the gain in buoyant force to the right of the axis. This gain in buoyant force will be indicated by dFB acting vertically upward through the centre of gravity of the prism BOB’.

Similarly, there will be one identical wedge-shaped prism AOA’ on the left side of the axis to go outside the water and we have also displayed it in above figure. This wedge will indicate the respective loss in buoyant force to the left of axis. This loss in buoyant force will be indicated by equal and opposite force dFB acting vertically downward through the centre of gravity of the prism AOA’.

There will be two equal and opposite couple acting on the body. Let us find first these two equal and opposite couple.

 
Let us analyze these two forces dFB. These two forces will form one couple which will tend to rotate the body in counter-clockwise direction.

Another equal and opposite couple will be developed due to displacement of centre of buoyancy from B to B1.

Let us consider one small strip of thickness dx at a distance x from the centre O, at right side of the axis, as displayed in above figure.

Area of the strip = x θ dx
Volume of the strip = x θ dx L
Where L is the length of the floating body
Weight of the strip = ρ g x θ L dx

Similarly, we will consider one small strip of thickness dx at a distance x from the centre O, at left side of the axis and we will have the weight of the strip ρ g x θ L dx.

Above two forces i.e. weights are acting in the opposite direction and therefore there will be developed one couple.


We will discuss another term i.e. Conditions of equilibrium of a floating and sub-merged bodies in our next post.
 
Do you have any suggestions? Please write in comment box.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

Also read

Continue Reading

Thursday, 15 February 2018

METACENTRE AND METACENTRIC HEIGHT

We were discussing the basic definition and derivation of total pressurecentre of pressure, buoyancy or buoyancy force and centre of buoyancy in our previous posts.

Today we will see here the basic concept of Meta-centre and meta-centric height with the help of this post.

Meta-centre

Meta-centre is basically defined as the point about which a body in stable equilibrium will start to oscillate when body will be displaced by an angular displacement.

 
We can also define the meta-centre as the point of intersection of the axis of body passing through the centre of gravity and original centre of buoyancy and a vertical line passing through the centre of buoyancy of the body in tilted position.

Let us consider a body which is floating in the liquid. Let us assume that body is in equilibrium condition. Let us think that G is the centre of gravity of the body and B is the centre of buoyancy of the body when body is in equilibrium condition.


In equilibrium situation, centre of gravity G and centre of buoyancy B will lie on same axis which is displayed here in above figure with a vertical line.

Let us assume that we have given an angular displacement to the body in clockwise direction as displayed here in above figure.

Centre of buoyancy will be shifted now towards right side from neutral axis and let us assume that it is now B1.

 
Line of action of buoyancy force passing through this new position will intersect the normal axis passing through the centre of gravity and centre of buoyancy in original position of the body at a point M as displayed here in above figure. Where, M is the meta-centre.

Meta-centric height

Meta-centric height is basically defined as the distance between the meta-centre of the floating body and the centre of gravity of the body.

Therefore, MG in above figure will be termed as meta-centric height.

We will discuss the analytical method to determine the meta-centric height in our next post.
Do you have any suggestions? Please write in comment box.

 

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

Also read

Continue Reading

Wednesday, 14 February 2018

WHAT IS CENTRE OF BUOYANCY?

We were discussing the basic definition and derivation of total pressurecentre of pressure and buoyancy or buoyancy force in our previous posts.

Today we will see here the basic concept of centre of buoyancy with the help of this post. Further, we will see the concept of Metacentre and Metacentric height in our next post.

Centre of buoyancy

 
As we know that when a body is immersed in fluid, an upward force is exerted by the fluid on the body. This force will be equal to the weight of the fluid displaced by the body and this force will be termed as force of buoyancy or buoyancy.

Buoyancy force will act through the centre of gravity of the displaced fluid and that point i.e. centre of gravity of the displaced fluid will be termed as centre of buoyancy.

Therefore we can define the term centre of buoyancy as the point through which the force of buoyancy is supposed to act.

Centre of buoyancy = Centre of gravity of the displaced fluid = Centre of gravity of the portion of the body immersed in the liquid

Let us explain the term centre of buoyancy

Let us consider one vessel as displayed here in following figure. Weight of vessel will be distributed throughout the length of vessel and will act downward over the entire structure of vessel.
But, what do we consider?

We consider that complete weight of the vessel will act downward vertically through one point and that point will be termed as the centre of gravity of that vessel.

 
In similar way, buoyancy force will be supposed to act vertically in upward direction through a single point and that point will be termed as centre of buoyancy.
We will discuss another term i.e. Meta-centre and Meta-centric height in our next post.

Do you have any suggestions? Please write in comment box.

 

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

Also read

Continue Reading

Tuesday, 13 February 2018

DEFINE BUOYANCY AND BUOYANCY FORCE

We were discussing the basic definition and derivation of total pressurecentre of pressure and we have also secured the expression for total pressure and centre of pressure for inclined plane surface submerged in liquid in our previous posts.

Today we will start a new topic i.e. Buoyancy and floatation, in the subject of fluid mechanics, with the help of this post. We will see here the basic concept of Buoyancy, Archimedes principle, Centre of buoyancy, Metacentre, Metacentric height etc.

We will first see here the basic concept of buoyancy or buoyancy force with the help of this post.

 

Buoyancy or buoyancy force

When a body is immersed in fluid, an upward force is exerted by the fluid on the body. This force will be equal to the weight of the fluid displaced by the body and this force will be termed as force of buoyancy or buoyancy.

Let us consider we have one container filled with water as displayed here in following figure. We have one object of weight 7 N. Let us think that we are now immersing the object in to the liquid i.e. water.

Once object will be immersed in the water, some amount of water will be displaced by the object and one upward force will be applied over the object by the water.

Weight of the displaced water will be equal to this upward force which will be exerted by the water on the object. As we can see from above figure that, water of weight 3N is displaced here and one upward force of 3N is exerted by the water over the object.

 

Conclusion for buoyancy force

Buoyancy force is the force which will be exerted on the object by the surrounding fluid. When one object will be immersed in the water, object will push the water and water will push back the object with as much force as it can.

Force of buoyancy = Weight of the displaced fluid
Force of buoyancy = Weight of the object in air – Weight of the object in given water

Positive buoyancy

Force of buoyancy will be greater than the weight of the object. Hence, object will float and this case will be termed as positive buoyancy.

Neutral buoyancy

Force of buoyancy will be equal to the weight of the object. Hence, object will be suspended in the fluid and this case will be termed as neutral buoyancy.

Negative buoyancy

Force of buoyancy will be less than the weight of the object. Hence, object will be sunk and this case will be termed as negative buoyancy.

 
We will see another topic i.e. Centre of buoyancy in our next post.

Do you have any suggestions? Please write in comment box.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

Also read

Continue Reading

Saturday, 10 February 2018

TOTAL HYDROSTATIC FORCE ON CURVED SURFACES

We were discussing the basic definition and derivation of total pressure; centre of pressure and we have also secured the expression for total pressure and centre of pressure for inclined plane surface submerged in liquid in our previous posts.

Today we will be interested here to see the total hydrostatic force on curved surfaces, in the subject of fluid mechanics, with the help of this post.

Let us consider a curved surface AB sub-merged in a static liquid as displayed here in following figure.
Let us consider one small strip area dA at a depth of h from free surface of liquid. We have following data from above figure.

A = Total area of curved surface
ρ = Density of the liquid
g = Acceleration due to gravity

 
Pressure intensity on small area dA = ρ g h

Hydrostatic force on small area dA will be given by following formula as mentioned here.

dF= ρ g h x dA

Direction of this hydrostatic force will be normal to the curved surface and will vary from point to point. Therefore, in order to secure the value of total hydrostatic force we will not integrate the above equation.

We will secure the value or expression for total hydrostatic force on curved surface by resolving the force dF in its two components or we can say that dF force will be resolved in X direction i.e. dFx and in Y direction i.e. dFy.

dFx = dF Sin θ = ρ g h x dA Sin θ
dFy = dF Cos θ = ρ g h x dA Cos θ

Total force in X- direction and in Y- direction will be given as mentioned here.

Let us analyze the above equation

FG will be dA Sin θ or vertical projection of area dA. Therefore, the expression for Fx will be total pressure force on the projected area of the curved surface on the vertical plane.

Fx = Total pressure force on the projected area of the curved surface on the vertical plane

 
EG will be dA Cos θ or horizontal projection of dA. Therefore, the expression for Fy will be the weight of the liquid contained between the curved surface extended up to free surface of liquid.

Fy = Weight of the liquid contained between the curved surface extended up to free surface of liquid


We will discuss the basic concept of buoyancy, in the subject of fluid mechanics, in our next post.

Do you have any suggestions? Please write in comment box.

 

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

Also read

Continue Reading

Tuesday, 6 February 2018

TOTAL PRESSURE AND CENTRE OF PRESSURE FOR INCLINED PLANE SURFACE IMMERSED IN A LIQUID

We were discussing the basic definition and derivation of total pressure and centre of pressure in our previous posts.

Today we will be interested to see here the expression for total pressure and centre of pressure for inclined plane surface submerged in liquid, in the subject of fluid mechanics, with the help of this post.

 
Let us consider a plane surface of arbitrary shape immersed in liquid in such a way that the plane of surface makes an angle θ with the free surface of liquid as displayed here in following figure.


Let us consider that we have following data from above figure.

A = Total area of inclined surface
ħ = Height of centre of gravity of inclined area from free surface
h* = Distance of centre of pressure from free surface of the liquid
θ = Angle made by the surface of inclined plane with free surface of the liquid

Total pressure which is basically defined as the hydrostatic force applied by a static fluid on a plane or curved surface when fluid will come in contact with the surfaces.

 
Total pressure for inclined plane surface submerged in liquid will be given by following formula as mentioned here.

Total pressure = ρ g A ħ

Centre of pressure is basically defined as a single point through which or at which total pressure or total hydrostatic force will act.

Centre of pressure for inclined plane surface submerged in liquid will be given by following formula as mentioned here.
For a vertical plane submerged surface, θ = 90

We will find out the total hydrostatic force on curved surfacesin the subject of fluid mechanics, in our next post.

 
Do you have any suggestions? Please write in comment box.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

Also read

Continue Reading

CENTER OF PRESSURE DERIVATION

We were discussing the basic definition and derivation of total pressure in our previous post. Today we will understand here the basic concept of "Centre of pressure", in the subject of fluid mechanics, with the help of this post. We will also find out here the expression for centre of pressure with the help of this post.

Centre of pressure

Centre of pressure is basically defined as a single point through which or at which total pressure or total hydrostatic force will act.

 
Let us consider that we have one tank filled with liquid e.g. water. Let us consider that there is one object of arbitrary shape immersed inside the water as displayed here in following figure.



Let us consider G is the centre of gravity and P is the centre of pressure. ħ is the height of C.G from free surface of liquid and h* is the height of centre of pressure from free surface of liquid.

Centre of pressure will be determined with the help of following formula as mentioned here.

Derivation of centre of pressure

In order to determine the centre of pressure, we will consider the object in terms of small strips as displayed here in above figure. We will use the concept of “principle of moments” to determine the centre of pressure.

According to the principle of moments, moment of the resultant force about an axis will be equal to the sum of the moments of components about the same axis.

As we have shown above in figure, total hydrostatic force F is applied at centre of pressure P which is at height of h* from the free surface of liquid.

Therefore, let us determine the moment of resultant force F about the free surface of liquid and it will be determined as F x h*.

 
As we have considered here the object in terms of small strips as displayed here in above figure and hence we will determine the moment of force dF acting on small strip about the free surface of liquid.
Moment of force dF = dF x h
Moment of force dF = ρ g h x b dh x h

Let us sum of all moments of such small forces about the free surface of liquid and it will be written as mentioned here.

We have already shown above that moment of resultant force F about the free surface of liquid i.e. F x h* and sum of moments about free surface i.e. ρ g I0
We can also write here 


We will discuss the total pressure and centre of pressure for inclined plane surface submerged in liquid, in the subject of fluid mechanics, in our next post.

Do you have any suggestions? Please write in comment box.

 

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

Also read

Continue Reading

Archivo del blog

Copyright © HKDIVEDI.COM | Powered by Blogger | Designed by Dapinder