Statics
CHAPTER – 6
DEFINITIONS
1. Static
Statics deals with the bodies at rest under number of forces, the equilibrium and the conditions of equilibrium.
2. Resultant Force
The net effect of two or more forces is a single force, that is called the resultant force.
3. Moment Arm
The perpendicular distance between the axis of rotation and the line
of the action of force is called the moment arm of the force.
TORQUE
It is the turning effects of a force about an axis of rotation is called moment of force or torque.
FACTORS ON WHICH TORQUE DEPENDS
1. The magnitude of the applied force.
2. The perpendicular distance between axis of rotation and point of application of force.
1. The magnitude of the applied force.
2. The perpendicular distance between axis of rotation and point of application of force.
REPRESENTATION
Torque may be represented as,
Torque = Force * moment arm
T = F * d
Torque may be represented as,
Torque = Force * moment arm
T = F * d
CENTRE OF GRAVITY
The centre of gravity is a point at which the whole weight of the body appears to act.
Centre of Gravity of Regular Shaped Objects
We can find the centre of gravity of any regular shaped body having the following shapes:
1. Triangle: The point of intersection of all the medians.
2. Circle: Centre of gravity of circle is also the centre of gravity.
3. Square: Point of intersection of the diagnonals.
4. Parallelogram: Point of intersection of the diagonals.
5. Sphere: Centre of the sphere.
1. Triangle: The point of intersection of all the medians.
2. Circle: Centre of gravity of circle is also the centre of gravity.
3. Square: Point of intersection of the diagnonals.
4. Parallelogram: Point of intersection of the diagonals.
5. Sphere: Centre of the sphere.
Centre of Gravity of Irregular Shaped Objects
We can find the center of gravity of any irregular shaped object by
using following method. Drill a few small holes near the edge of the
irregular plate. Using the hole A, suspend the plate from a nail fixed
horizontally in a wall. The plate will come to rest after a few moments.
It will be in a position so that its centre of gravity is vertically
below the point of suspension.
Now, suspend a plumb line from the supporting nail. Draw a line AA’ in the plate along the plumb line. The centre of gravity is located somewhere on this line.
Repeat the same process using the second hole B. This gives the line BB’ on the plate. Also repeat this process and use hole C and get line CC’.
The lines AA’, BB’ and CC’ intersect each other at a point. It is our required point, i.e.e the centre of gravity. We can use this procedure with any irregular shaped body and find out its centre of gravity.
Now, suspend a plumb line from the supporting nail. Draw a line AA’ in the plate along the plumb line. The centre of gravity is located somewhere on this line.
Repeat the same process using the second hole B. This gives the line BB’ on the plate. Also repeat this process and use hole C and get line CC’.
The lines AA’, BB’ and CC’ intersect each other at a point. It is our required point, i.e.e the centre of gravity. We can use this procedure with any irregular shaped body and find out its centre of gravity.
EQUILIBRIUM
A body will be in equilibrium if the forces acting on it must be cancel the effect of each other.
In the other word we can also write that:
A body is said to be in equilibrium condition if there is no unbalance or net force acting on it.
In the other word we can also write that:
A body is said to be in equilibrium condition if there is no unbalance or net force acting on it.
Static Equilibrium
When a body is at rest and all forces applied on the body cancel each other then it is said to be in static equilibrium.
When a body is at rest and all forces applied on the body cancel each other then it is said to be in static equilibrium.
Dynamic Equilibrium
When a body is moving with uniform velocity and forces applied on the body
When a body is moving with uniform velocity and forces applied on the body
cancel each other then it is said to be in the dynamic equilibrium.
CONDITIONS OF EQUILIBRIUM
FIRST CONDITION OF EQUILIBRIUM
“A body will be in first condition of equilibrium if sum of all
forces along X-axis and sum of all forces along Y-axis are are equal to
zero, then the body is said to be in first condition of equilibrium.”
( Fx = 0 Fy = 0 )
( Fx = 0 Fy = 0 )
SECOND CONDITIONS OF EQUILIBRIUM
“A body will be in second condition of equilibrium if sum of
clockwise(Moment) torque must be equal to the sum of anticlockwise
torque(Moment), then the body is said to be in second condition of
equilibrium.”
Sum of torque = 0
Sum of torque = 0
STATES OF EQUILIBRIUM
There are following three states of Equilibrium:
1. First State (Stable Equilibrium)
A body at rest is in stable equilibrium if on being displaced, it has the tendency to come back to its initial position.
When the centre of gravity of a body i.e. below the point of suspension or support, then body is said to be in stable equilibrium.
When the centre of gravity of a body i.e. below the point of suspension or support, then body is said to be in stable equilibrium.
2. Second State (Unstable Equilibrium)
If a body on displacement topples over and occupies a new position then it is said to be in the state of unstable equilibrium.
When the centre of gravity lies above the point of suspension or support, the body is said to be in the state of unstable equilibrium.
When the centre of gravity lies above the point of suspension or support, the body is said to be in the state of unstable equilibrium.
3. Third State
If a body is placed in such state that if it is displaced then
neither it topples over nor does it come back to its original position,
then such state is called neutral equilibrium.
When the centre of gravity of a body lies at the point of suspension, then the body is said to be in neutral equilibrium.
When the centre of gravity of a body lies at the point of suspension, then the body is said to be in neutral equilibrium.
Circular Motion and Gravitation
CHAPTER – 7
Centripetal Force
Definition
“The force that causes an object to move along a curve (or a curved path) is called centripetal force.”
“The force that causes an object to move along a curve (or a curved path) is called centripetal force.”
Mathematical Expression
We know that the magnitude of centripetal acceleration of a body in a uniform circular motions is directly proportional to the square of velocity and inversely proportional to the radius of the path Therefore,
a(c) < v2 (Here < represents the sign of proportionality do not write this in your examination and 2 represents square of v)
a(c) < 1/r
Combining both the equations:
a(c) < v2/r From Newton’s Second Law of Motion: F = ma => F(c) = mv2/r
We know that the magnitude of centripetal acceleration of a body in a uniform circular motions is directly proportional to the square of velocity and inversely proportional to the radius of the path Therefore,
a(c) < v2 (Here < represents the sign of proportionality do not write this in your examination and 2 represents square of v)
a(c) < 1/r
Combining both the equations:
a(c) < v2/r From Newton’s Second Law of Motion: F = ma => F(c) = mv2/r
Where,
Fc = Centripetal Force
m = Mass of object
v = Velocity of object
r = Radius of the curved path
Fc = Centripetal Force
m = Mass of object
v = Velocity of object
r = Radius of the curved path
Factors on which Fc Depends:
Fc depends upon the following factors:
Increase in the mass increases Fc.
It increases with the square of velocity.
It decreases with the increase in radius of the curved path.
Fc depends upon the following factors:
Increase in the mass increases Fc.
It increases with the square of velocity.
It decreases with the increase in radius of the curved path.
Examples
The centripetal force required by natural planets to move constantly round a circle is provided by the gravitational force of the sun.
If a stone tied to a string is whirled in a circle, the required centripetal force is supplied to it by our hand. As a reaction the stone exerts an equal force which is felt by our hand.
The pilot while turning his aeroplane tilts one wing in the upward direction so that the air pressure may provide the required suitable Fc.
The centripetal force required by natural planets to move constantly round a circle is provided by the gravitational force of the sun.
If a stone tied to a string is whirled in a circle, the required centripetal force is supplied to it by our hand. As a reaction the stone exerts an equal force which is felt by our hand.
The pilot while turning his aeroplane tilts one wing in the upward direction so that the air pressure may provide the required suitable Fc.
Centrifugal Force
Definition
“A force supposed to act radially outward on a body moving in a curve is known as centrifugal force.”
“A force supposed to act radially outward on a body moving in a curve is known as centrifugal force.”
Explanation
Centrifugal force is actually a reaction to the centripetal force. It is a well-known fact that Fc is directed towards the centre of the circle, so the centrifugal force, which is a force of reaction, is directed away from the centre of the circle or the curved path.
According to Newton’s third law of motion action and reaction do not act on the same body, so the centrifugal force does not act on the body moving round a circle, but it acts on the body that provides Fc.
Centrifugal force is actually a reaction to the centripetal force. It is a well-known fact that Fc is directed towards the centre of the circle, so the centrifugal force, which is a force of reaction, is directed away from the centre of the circle or the curved path.
According to Newton’s third law of motion action and reaction do not act on the same body, so the centrifugal force does not act on the body moving round a circle, but it acts on the body that provides Fc.
Examples
If a stone is tied to one end of a string and it is moved round a circle, then the force exerted on the string on outward direction is called centrifugal force.
The aeroplane moving in a circle exerts force in a direction opposite to the pressure of air.
When a train rounds a curve, the centrifugal force is also exerted on the track.
If a stone is tied to one end of a string and it is moved round a circle, then the force exerted on the string on outward direction is called centrifugal force.
The aeroplane moving in a circle exerts force in a direction opposite to the pressure of air.
When a train rounds a curve, the centrifugal force is also exerted on the track.
Law of Gravitation
Introduction
Newton proposed the theory that all objects in the universe attract each other with a force known as gravitation. the gravitational attraction exists between all bodies. Hence, two stones are not only attracted towards the earth, but also towards each other.
Newton proposed the theory that all objects in the universe attract each other with a force known as gravitation. the gravitational attraction exists between all bodies. Hence, two stones are not only attracted towards the earth, but also towards each other.
Statement
Every body in the universe attracts every other body with a force, which is directly proportional to the product of masses and inversely proportional to the square of the distance between their centres.
Every body in the universe attracts every other body with a force, which is directly proportional to the product of masses and inversely proportional to the square of the distance between their centres.
Mathematical Expression
Two objects having mass m1 and m2 are placed at a distance r. According to Newton’s Law of Universal Gravitation.
F < m1m2 ((Here < represents the sign of proportionality do not write this in your examination)
Also F < 1/r2 (Here 2 represents square of r)
Combining both the equations :
F < m1m2/r2
Removing the sign of proportionality and introducing a constant:
F = G (m1m2/r2)
Two objects having mass m1 and m2 are placed at a distance r. According to Newton’s Law of Universal Gravitation.
F < m1m2 ((Here < represents the sign of proportionality do not write this in your examination)
Also F < 1/r2 (Here 2 represents square of r)
Combining both the equations :
F < m1m2/r2
Removing the sign of proportionality and introducing a constant:
F = G (m1m2/r2)
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