Centre of gravity
Center of gravity :-
center of gravity (CG) of an object is the point at which weight is evenly dispersed and all sides are in balance.
A human's center of gravity can change as he takes on different positions, but in many other objects, it's a fixed location.
Gravitation:-
The mutual attractive force of bodies due to which they attract each other is called gravitation.
1.Gravity:-
The attractive force of the earth due to which it attracts all bodies towards its centre is called gravity.The value of gravity varies from place to place on the ground surface. Its general value is 9.81 m/s2.
Centroid:-
Different geometrical shapes such as the circle, triangle and rectangle are plane figures having only 2-dimensions.They are also known as laminas. They have only area, but no mass.
The centre of gravity of these plane figures is called as the Centroid. It is also known as the geometrical centre.
The method of finding out the centroid of a plane figure is the same as that of finding out the centre of gravity of a body. If the lamina is assumed to have uniform mass per unit area, then the centroid is also the centre of gravity in a uniform gravitational field.
Methods to calculate centre of gravity:-
1 By geometrical consideration.
2 By moments.
Principle :
The total moment of a weight about any axis
The total moment of a weight about any axis
= The sum of the moments of the various parts about the same axis.
3 By graphical method.
The first two methods are generally used to find out the centre of gravity or centroid, as the third method can become tedious.
Centre of gravity by geometrical consideration:-
1 The centre of gravity of a circle is its centre.
2 The centre of gravity of a square, rectangle or a parallelogram is at the points where its diagonals meet each other. It is also the middle point of the length as well as the width.
3 The centre of gravity of a triangle is at the point where the medians of the triangle meet.
4 The centre of gravity of a right circular Cone is at a distance of from its base.
5 The centre of gravity of a hemisphere is at a distance of from its base.
6 The centre of gravity of a segment of a sphere of radius h is at a perpendicular distance of from the centre of the sphere.
7 The centre of gravity of a semicircle is at a perpendicular distance of from its centre.
8 The centre of gravity of a trapezium with parallel side 'a' and 'b' is at a distance of measured from the base 'b'.
9 The centre of gravity of a cube of side L is at a distance of from every face.
10 The centre of gravity of a Sphere of diameter 'd' is at a distance of from every point.
Equilibrium:-
A body is said to be in equilibrium if the resultant of all the forces acting on a body is zero and if there is no turning moment.
There are three states of equilibrium
1 Stable equilibrium
2 Unstable equilibrium
3 Neutral equilibrium
1 Stable equilibrium:-
A body is said to be in a stable equilibrium if it returns to its original position when slightly displaced. (The C.G. isas low as possible).
E.g : 1 A cone resting on its base
2 A ball on a concave surface
3 Funnel resting on its base.
2 Unstable equilibrium:-
A body is said to be in an unstable equilibrium if it does not return to its original position when slightly displaced.
Its centre of gravity falls taking it away from its original position. (CG is at high points)
E.g: 1 A cone resting on its tip
2 A ball on convex surface
3 Funnel standing on its tube end.
3 Neutral equilibrium:-
A body is said to be in a neutral equilibrium if on being slightly displaced, it takes a new position similar to its original one. The centre of gravity remains undisturbed. (CG is neither raised or lowered)
Eg: 1 A cone resting on its side
2 A ball on flat surface
3 Funnel resting on its side.
Conditions for stable equilibrium:-
• The CG should be as low as possible.
• It should have a broad base.
• The vertical line passing through the CG should fall within the base.
Conditions of equilibrium:-
A body is said to be in a state of equilibrium under the action of forces when there is no motion of rotation or translation of the body. There are three conditions of equilibrium of a body which are given below:
i Algebraic sum of the horizontal components of all the forces acting on the body must be zero.
∑ H = 0
ii Algebraic sum of the vertical components of all the forces acting on the body must be zero.
∑V = 0
iii Algebraic sum of the moments of all the forces acting on the body must be zero.
∑M = 0
Torque or twisting moment of a couple is given by the product of force applied and the arm of the couple (i.e. Radius). In fact, moment means the product of “force applied” and the
“perpendicular distance of the point and the line of the force”.
Some example of equilibrium in daily life
1. The lower decks of the ships are loaded with heavy cargoes. This makes the centre of gravity of the whole ship lower and its equilibrium becomes more stable.
2. A man carrying a bucket full of water in one hand extends his opposite arm and bends his body towards it.
3. While carrying load on back the man bends forward so that his and the load’s centre of gravity falls on his
feet, if he walks erect, he will fall backward.
4. While climbing a mountain, a man bends forward and bends backward while descending so that the centre
of gravity of his load falls on his feet.
5. a double decker, more passengers are accommodated in the lower deck and less on the upper so that the centre of gravity of the bus and the passengers is kept low to eliminate any chance of turning.
Area of cut out regular surfaces - circle, segment and sector of circle
Circle :-
It is the path of a point which is always equal from its centre is called a circle.
r = radius of the circle
d = diametre of the circle
Area of the circle = πr^2
(or) = π/4 d^2 unit^2
Circumference of the circle = 2πr (or) πd unit
Sector of a circle :-
The area bounded by an arc is called the sector of a circle.
In the figure given ABC is the sector of a circle.
r = radius of the circle
θ= Angle of sector in degrees
Area of sector ABC=
πr^2 x θ/360 unit^2
Area of sector =
(Length of arc of sector × radius) ÷ 2 unit2
Length of the arc L = 2πr × θ /360°
unit
Perimeter of the sector = L + 2r unit
r = radius
Segment of a circle :-
When a circle is divided into two by drawing a line, the bigger part is called segment of the circle and the smaller part is also called segment of the circle.
Area of the smaller segment
= Area of the sector - Area of ABC
Area of the greater segment
= Area of the circle - Area of smaller segment
Semi Circle :-
• A semi circle is a sector whose central angle is 180°.
Length of arc of semi circle =
L = 2πr × 180/360 = 2πr × 1/2
= πr unit
Area of semi circle = π r^ 2÷ 2 unit2
Perimeter of a semi circle =
2πr/2 + 2r
= πr +2r unit
= r(π+2) unit
Quadrant of a circle :-
• A quadrant of a circle is a sector whose central angle is 90°.
Length and area of a quadrant of a circle
L= 2πr × 90/360 =2πr ×1/4
= πr/2
Area of quadrant of a circle = πr^2/4
unit2
Perimeter of a quadrant =
=2πr/4 + 2r
= πr/2 + 2r
= r(π/2+2) unit
5 The centre of gravity of a hemisphere is at a distance of from its base.
6 The centre of gravity of a segment of a sphere of radius h is at a perpendicular distance of from the centre of the sphere.
7 The centre of gravity of a semicircle is at a perpendicular distance of from its centre.
8 The centre of gravity of a trapezium with parallel side 'a' and 'b' is at a distance of measured from the base 'b'.
9 The centre of gravity of a cube of side L is at a distance of from every face.
10 The centre of gravity of a Sphere of diameter 'd' is at a distance of from every point.
Equilibrium:-
A body is said to be in equilibrium if the resultant of all the forces acting on a body is zero and if there is no turning moment.
There are three states of equilibrium
1 Stable equilibrium
2 Unstable equilibrium
3 Neutral equilibrium
1 Stable equilibrium:-
A body is said to be in a stable equilibrium if it returns to its original position when slightly displaced. (The C.G. isas low as possible).
E.g : 1 A cone resting on its base
2 A ball on a concave surface
3 Funnel resting on its base.
2 Unstable equilibrium:-
A body is said to be in an unstable equilibrium if it does not return to its original position when slightly displaced.
Its centre of gravity falls taking it away from its original position. (CG is at high points)
E.g: 1 A cone resting on its tip
2 A ball on convex surface
3 Funnel standing on its tube end.
3 Neutral equilibrium:-
A body is said to be in a neutral equilibrium if on being slightly displaced, it takes a new position similar to its original one. The centre of gravity remains undisturbed. (CG is neither raised or lowered)
Eg: 1 A cone resting on its side
2 A ball on flat surface
3 Funnel resting on its side.
Conditions for stable equilibrium:-
• The CG should be as low as possible.
• It should have a broad base.
• The vertical line passing through the CG should fall within the base.
Conditions of equilibrium:-
A body is said to be in a state of equilibrium under the action of forces when there is no motion of rotation or translation of the body. There are three conditions of equilibrium of a body which are given below:
i Algebraic sum of the horizontal components of all the forces acting on the body must be zero.
∑ H = 0
ii Algebraic sum of the vertical components of all the forces acting on the body must be zero.
∑V = 0
iii Algebraic sum of the moments of all the forces acting on the body must be zero.
∑M = 0
Torque or twisting moment of a couple is given by the product of force applied and the arm of the couple (i.e. Radius). In fact, moment means the product of “force applied” and the
“perpendicular distance of the point and the line of the force”.
Some example of equilibrium in daily life
1. The lower decks of the ships are loaded with heavy cargoes. This makes the centre of gravity of the whole ship lower and its equilibrium becomes more stable.
2. A man carrying a bucket full of water in one hand extends his opposite arm and bends his body towards it.
3. While carrying load on back the man bends forward so that his and the load’s centre of gravity falls on his
feet, if he walks erect, he will fall backward.
4. While climbing a mountain, a man bends forward and bends backward while descending so that the centre
of gravity of his load falls on his feet.
5. a double decker, more passengers are accommodated in the lower deck and less on the upper so that the centre of gravity of the bus and the passengers is kept low to eliminate any chance of turning.
Area of cut out regular surfaces - circle, segment and sector of circle
Circle :-
It is the path of a point which is always equal from its centre is called a circle.
r = radius of the circle
d = diametre of the circle
Area of the circle = πr^2
(or) = π/4 d^2 unit^2
Circumference of the circle = 2πr (or) πd unit
Sector of a circle :-
The area bounded by an arc is called the sector of a circle.
In the figure given ABC is the sector of a circle.
r = radius of the circle
θ= Angle of sector in degrees
Area of sector ABC=
πr^2 x θ/360 unit^2
Area of sector =
(Length of arc of sector × radius) ÷ 2 unit2
Length of the arc L = 2πr × θ /360°
unit
Perimeter of the sector = L + 2r unit
r = radius
Segment of a circle :-
When a circle is divided into two by drawing a line, the bigger part is called segment of the circle and the smaller part is also called segment of the circle.
Area of the smaller segment
= Area of the sector - Area of ABC
Area of the greater segment
= Area of the circle - Area of smaller segment
Semi Circle :-
• A semi circle is a sector whose central angle is 180°.
L = 2πr × 180/360 = 2πr × 1/2
= πr unit
Area of semi circle = π r^ 2÷ 2 unit2
Perimeter of a semi circle =
2πr/2 + 2r
= πr +2r unit
= r(π+2) unit
Quadrant of a circle :-
• A quadrant of a circle is a sector whose central angle is 90°.
Length and area of a quadrant of a circle
L= 2πr × 90/360 =2πr ×1/4
= πr/2
Area of quadrant of a circle = πr^2/4
unit2
Perimeter of a quadrant =
=2πr/4 + 2r
= πr/2 + 2r
= r(π/2+2) unit
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