Dynamics is the branch of mechanics concerned with forces that change or produce the motion of bodies.
In ancient timer, most philosophers believed that a body moved with uniform velocity due to some external agent. They also thought that if there were no external agent the body would naturally come to rest. Galileo was the first to show that some external force was necessary to change the velocity of a body but that no external force was necessary to maintain the velocity of a body. This principle was adopted by Newton in his first law of motion.
Newton's Laws of Motion:
First Law: "Every body continues in its state of rest or of uniform motion in a straight line
unless it is acted upon by a net external force".
This law enables us to define inertia and force. We can conclude that if the net external force on an object is zero, the acceleration of the object is zero.
Inertia: If the net external force is zero, a body at rest continues to be at rest and a body in motion continues to move with uniform velocity. This property called inertia.
Examples:
1. When a coin placed over a card kept over a tumbler. Now, if we flip the card quickly away with a finger, the coin resting back at its original place due to inertia of rest drops into the tumbler.
2. When a person jumps out of a moving bus, leans forward in the direction of the motion of the bus, as his feet, suddenly coming in contact with the ground, are brought to rest whereas the upper position of his body continues to be in motion. This due to inertia of motion.
3. A person sitting in a bus falls backwards, when the bus starts suddenly as the lower part of his body in contact with the bus moved forward. With the bus whereas the upper position of his body tends to remain at rest.
Force: We define force as that which changes the state of rest of a body or of its uniform motion in a straight line i.e. force is that which over comes the inertia of a body.
Momentum: It is defined as the product of mass and velocity of a body.
∴ Momentum = Mass × Velocity
Newton's' Second Law of Motion: The rate of change of momentum of a body is directly proportional to the impressed force and takes place in the direction of the force.
∴ F = ma
A force is that which acting on the body produces an acceleration in it.
Newton's Third Law of Motion: "For every action there is an equal and opposite reaction".
No action takes place in the absence reaction. Thus, in nature, forces always occur in pairs. One action and other reaction. The forces of action and reaction are simultaneous.
Examples:
1. If a rubber ball is striking agaist the wall, it comes back due to the reaction of the wall.
2. While rowing a boat, the water is pushed back by oars and due to reaction of water, the boat moves forward.
3. While walking, we push our foot against the ground, the ground in turn exerts an equal and opposite force.
Uniform Circular Motion: When a body is moving in a circle, with constant speed it is said to execute uniform circular motion. The velocity of the body is constantly changing as the direction of motion is undergoing change every instant of time. However the magnitude of the velocity remains constant.
consider a particle moving along a circle of radius 'r' in the anti clock-wise direction. With constant velocity 'v'. At every point on the path the linear velocity of the particle is along the tangent at the point. As show in the following diagram. Let O be the centre of the circle.
Let the particle be at A on the circle at some instant time. The line joining O and A is called the radius vector. Let the particles move from A to B in time 't' seconds. During this time interval. 't' sec the radius vector OA rotates through an angle θ, the new position of the radius vector being OB.
Then AB is the linear displacement of the particle and θ is the angular displacement.
This angular displacement is the angle described by the rotating radius vector of a body in circular motion, in a given time. Angular displacement is generally expressed in radians.
Radian is defined as the angle subtended at the centre of a circle by an arc of the circle nwhose length is equal to the radius of the circle.
Centripetal Force:
The force which continuously defects a particle from its straight line path and makes it travel along a circular path is called centripetal force.
Centripetal force is necessary for the uniform circular motion of a body.
The earth moves around the sun because of gravitational force of attraction between them. The gravitational force here acts as the centripetal force.
The magnitude of the centripetal force in F = mv2/r
F = mrω2 ( v = r ω)
Eg: The domestic churner for separating butter and butter milk from curd. The churner sets the curd into rotatory motion with good speed and the lighter particles of butter collect at the churner.
Centrifuge is a machine used to separate particles of higher mass from those of lower mass in a given mixture.
In ancient timer, most philosophers believed that a body moved with uniform velocity due to some external agent. They also thought that if there were no external agent the body would naturally come to rest. Galileo was the first to show that some external force was necessary to change the velocity of a body but that no external force was necessary to maintain the velocity of a body. This principle was adopted by Newton in his first law of motion.
Newton's Laws of Motion:
First Law: "Every body continues in its state of rest or of uniform motion in a straight line
unless it is acted upon by a net external force".
This law enables us to define inertia and force. We can conclude that if the net external force on an object is zero, the acceleration of the object is zero.
Inertia: If the net external force is zero, a body at rest continues to be at rest and a body in motion continues to move with uniform velocity. This property called inertia.
Examples:
1. When a coin placed over a card kept over a tumbler. Now, if we flip the card quickly away with a finger, the coin resting back at its original place due to inertia of rest drops into the tumbler.
2. When a person jumps out of a moving bus, leans forward in the direction of the motion of the bus, as his feet, suddenly coming in contact with the ground, are brought to rest whereas the upper position of his body continues to be in motion. This due to inertia of motion.
3. A person sitting in a bus falls backwards, when the bus starts suddenly as the lower part of his body in contact with the bus moved forward. With the bus whereas the upper position of his body tends to remain at rest.
Force: We define force as that which changes the state of rest of a body or of its uniform motion in a straight line i.e. force is that which over comes the inertia of a body.
Momentum: It is defined as the product of mass and velocity of a body.
∴ Momentum = Mass × Velocity
Newton's' Second Law of Motion: The rate of change of momentum of a body is directly proportional to the impressed force and takes place in the direction of the force.
∴ F = ma
A force is that which acting on the body produces an acceleration in it.
Newton's Third Law of Motion: "For every action there is an equal and opposite reaction".
No action takes place in the absence reaction. Thus, in nature, forces always occur in pairs. One action and other reaction. The forces of action and reaction are simultaneous.
Examples:
1. If a rubber ball is striking agaist the wall, it comes back due to the reaction of the wall.
2. While rowing a boat, the water is pushed back by oars and due to reaction of water, the boat moves forward.
3. While walking, we push our foot against the ground, the ground in turn exerts an equal and opposite force.
Uniform Circular Motion: When a body is moving in a circle, with constant speed it is said to execute uniform circular motion. The velocity of the body is constantly changing as the direction of motion is undergoing change every instant of time. However the magnitude of the velocity remains constant.
consider a particle moving along a circle of radius 'r' in the anti clock-wise direction. With constant velocity 'v'. At every point on the path the linear velocity of the particle is along the tangent at the point. As show in the following diagram. Let O be the centre of the circle.
Then AB is the linear displacement of the particle and θ is the angular displacement.
This angular displacement is the angle described by the rotating radius vector of a body in circular motion, in a given time. Angular displacement is generally expressed in radians.
Radian is defined as the angle subtended at the centre of a circle by an arc of the circle nwhose length is equal to the radius of the circle.
 | |||
| l (length of arc) = r | l = rθ | one radian = 57°18' |
Angular Velocity (ω):
The rate of angular displacement is known as angular velocity. Hence the angle described in one second by a rotating vector gives the angular velocity of the body. It is denoted by ω (omega).
∴ ω = θ/t
The time taken for one complete rotation is known as time period denoted by T. The angle descirbed by the radius vector for one complete revolution is 2π radians.
Hence angular velocity (ω) = 2π/T ------ (1)
Linear speed v = 2πr/T =
or v = 2π/T × r
or v = ωr (from (1))
∴ Linear velocity = Radius × angular velocity.
Translational and Rotational Motion:
(i) When a body moves without rotation or vibration it is said to describe translationalmotion.
Translational motion can be a straight line or a curve. The path need not necessarily be a straight line.
(ii) When a body turns round a fixed axis we say the body is describing rotatory motion. The particles which are in rotatory motion, will have same angular velocity.
A body may possess both translational and rotational motions simultaneously as in the case of a bicycle wheel and earth motion round the sun.
Centripetal Acceleration:
The acceleration of a particle moving in a circle with uniform speed is always directed towards the centre of the path and is known as centripetal acceleration. a = v2/r
The force which continuously defects a particle from its straight line path and makes it travel along a circular path is called centripetal force.
Centripetal force is necessary for the uniform circular motion of a body.
The earth moves around the sun because of gravitational force of attraction between them. The gravitational force here acts as the centripetal force.
The magnitude of the centripetal force in F = mv2/r
F = mrω2 ( v = r ω)
Eg: The domestic churner for separating butter and butter milk from curd. The churner sets the curd into rotatory motion with good speed and the lighter particles of butter collect at the churner.
Centrifuge is a machine used to separate particles of higher mass from those of lower mass in a given mixture.
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