Newton's laws of motion are three physical laws that, together, laid the foundation for classical
mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. They have been expressed in several different ways, over nearly three centuries, and can be summarised as follows.
First law: When viewed in an inertial reference frame, an object either remains at rest or
continues to move at a constant velocity, unless acted upon by an external force.
Second law: The vector sum of the external forces F on an object is equal to the mass m of that
object multiplied by the acceleration vector a of the object: F = ma.
Third law: When one body exerts a force on a second body, the second body simultaneously exerts a
force equal in magnitude and opposite in direction on the first body.
Newton's first law
The first law states that if the net force (the vector sum of all forces acting on an object) is zero,
then the velocity of the object is constant. Velocity is a vector quantity which expresses both the
object's speed and the direction of its motion; therefore, the statement that the object's velocity is
constant is a statement that both its speed and the direction of its motion are constant.
The first law can be stated mathematically as
Consequently,
An object that is at rest will stay at rest unless an external force acts upon it.
An object that is in motion will not change its velocity unless an external force acts upon it.
This is known as uniform motion. An object continues to do whatever it happens to be doing unless a
force is exerted upon it. If it is at rest, it continues in a state of rest (demonstrated when a
tablecloth is skilfully whipped from under dishes on a tabletop and the dishes remain in their initial
state of rest). If an object is moving, it continues to move without turning or changing its speed.
This is evident in space probes that continually move in outer space. Changes in motion must be imposed against the tendency of an object to retain its state of motion. In the absence of net forces, a moving object tends to move along a straight line path indefinitely.
Newton placed the first law of motion to establish frames of reference for which the other laws are
applicable. The first law of motion postulates the existence of at least one frame of reference called
a Newtonian or inertial reference frame, relative to which the motion of a particle not subject to
forces is a straight line at a constant speed.[8][12] Newton's first law is often referred to as the
law of inertia. Thus, a condition necessary for the uniform motion of a particle relative to an
inertial reference frame is that the total net force acting on it is zero. In this sense, the first law
can be restated as:
In every material universe, the motion of a particle in a preferential reference frame Φ is determined
by the action of forces whose total vanished for all times when and only when the velocity of the
particle is constant in Φ. That is, a particle initially at rest or in uniform motion in the
preferential frame Φ continues in that state unless compelled by forces to change it.
Newton's laws are valid only in an inertial reference frame. Any reference frame that is in uniform
motion with respect to an inertial frame is also an inertial frame, i.e. Galilean invariance or the
principle of Newtonian relativity.
Newton's second law
The second law states that the net force on an object is equal to the rate of change (that is, the
derivative) of its linear momentum p in an inertial reference frame:
The second law can also be stated in terms of an object's acceleration. Since Newton's second law is
only valid for constant-mass systems,mass can be taken outside the differentiation
operator by the constant factor rule in differentiation. Thus,
where F is the net force applied, m is the mass of the body, and a is the body's acceleration. Thus,
the net force applied to a body produces a proportional acceleration. In other words, if a body is
accelerating, then there is a force on it.
Consistent with the first law, the time derivative of the momentum is non-zero when the momentum
changes direction, even if there is no change in its magnitude; such is the case with uniform circular
motion. The relationship also implies the conservation of momentum: when the net force on the body is zero, the momentum of the body is constant. Any net force is equal to the rate of change of the
momentum.
Any mass that is gained or lost by the system will cause a change in momentum that is not the result of an external force. A different equation is necessary for variable-mass systems.
Newton's second law requires modification if the effects of special relativity are to be taken into
account, because at high speeds the approximation that momentum is the product of rest mass and
velocity is not accurate.
Impulse
An impulse J occurs when a force F acts over an interval of time Δt, and it is given by
Since force is the time derivative of momentum, it follows that
This relation between impulse and momentum is closer to Newton's wording of the second law.
Impulse is a concept frequently used in the analysis of collisions and impacts.
Variable-mass systems
Variable-mass systems, like a rocket burning fuel and ejecting spent gases, are not closed and cannot
be directly treated by making mass a function of time in the second law;that is, the following
formula is wrong:
The falsehood of this formula can be seen by noting that it does not respect Galilean invariance: a
variable-mass object with F = 0 in one frame will be seen to have F ≠ 0 in another frame.The
correct equation of motion for a body whose mass m varies with time by either ejecting or accreting
mass is obtained by applying the second law to the entire, constant-mass system consisting of the body and its ejected/accreted mass; the result is
where u is the velocity of the escaping or incoming mass relative to the body. From this equation one
can derive the equation of motion for a varying mass system, for example, the Tsiolkovsky rocket
equation. Under some conventions, the quantity u dm/dt on the left-hand side, which represents the
advection of momentum, is defined as a force (the force exerted on the body by the changing mass, such as rocket exhaust) and is included in the quantity F. Then, by substituting the definition of
acceleration, the equation becomes F = ma.
Newton's third law
The third law states that all forces between two objects exist in equal magnitude and opposite
direction: if one object A exerts a force FA on a second object B, then B simultaneously exerts a force FB on A, and the two forces are equal and opposite: FA = −FB. The third law means that all forces are interactions between different bodies, and thus that there is no such thing as a unidirectional force or a force that acts on only one body. This law is sometimes referred to as the action-reaction law, with FA called the "action" and FB the "reaction". The action and the reaction are simultaneous, and it does not matter which is called the action and which is called reaction; both forces are part of a single interaction, and neither force exists without the other.
The two forces in Newton's third law are of the same type (e.g., if the road exerts a forward
frictional force on an accelerating car's tires, then it is also a frictional force that Newton's third
law predicts for the tires pushing backward on the road).
From a conceptual standpoint, Newton's third law is seen when a person walks: they push against the
floor, and the floor pushes against the person. Similarly, the tires of a car push against the road
while the road pushes back on the tires—the tires and road simultaneously push against each other. In
swimming, a person interacts with the water, pushing the water backward, while the water simultaneously pushes the person forward—both the person and the water push against each other. The reaction forces account for the motion in these examples. These forces depend on friction; a person or car on ice, for example, may be unable to exert the action force to produce the needed reaction force.
mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. They have been expressed in several different ways, over nearly three centuries, and can be summarised as follows.
First law: When viewed in an inertial reference frame, an object either remains at rest or
continues to move at a constant velocity, unless acted upon by an external force.
Second law: The vector sum of the external forces F on an object is equal to the mass m of that
object multiplied by the acceleration vector a of the object: F = ma.
Third law: When one body exerts a force on a second body, the second body simultaneously exerts a
force equal in magnitude and opposite in direction on the first body.
Newton's first law
The first law states that if the net force (the vector sum of all forces acting on an object) is zero,
then the velocity of the object is constant. Velocity is a vector quantity which expresses both the
object's speed and the direction of its motion; therefore, the statement that the object's velocity is
constant is a statement that both its speed and the direction of its motion are constant.
The first law can be stated mathematically as
Consequently,
An object that is at rest will stay at rest unless an external force acts upon it.
An object that is in motion will not change its velocity unless an external force acts upon it.
This is known as uniform motion. An object continues to do whatever it happens to be doing unless a
force is exerted upon it. If it is at rest, it continues in a state of rest (demonstrated when a
tablecloth is skilfully whipped from under dishes on a tabletop and the dishes remain in their initial
state of rest). If an object is moving, it continues to move without turning or changing its speed.
This is evident in space probes that continually move in outer space. Changes in motion must be imposed against the tendency of an object to retain its state of motion. In the absence of net forces, a moving object tends to move along a straight line path indefinitely.
Newton placed the first law of motion to establish frames of reference for which the other laws are
applicable. The first law of motion postulates the existence of at least one frame of reference called
a Newtonian or inertial reference frame, relative to which the motion of a particle not subject to
forces is a straight line at a constant speed.[8][12] Newton's first law is often referred to as the
law of inertia. Thus, a condition necessary for the uniform motion of a particle relative to an
inertial reference frame is that the total net force acting on it is zero. In this sense, the first law
can be restated as:
In every material universe, the motion of a particle in a preferential reference frame Φ is determined
by the action of forces whose total vanished for all times when and only when the velocity of the
particle is constant in Φ. That is, a particle initially at rest or in uniform motion in the
preferential frame Φ continues in that state unless compelled by forces to change it.
Newton's laws are valid only in an inertial reference frame. Any reference frame that is in uniform
motion with respect to an inertial frame is also an inertial frame, i.e. Galilean invariance or the
principle of Newtonian relativity.
Newton's second law
The second law states that the net force on an object is equal to the rate of change (that is, the
derivative) of its linear momentum p in an inertial reference frame:
The second law can also be stated in terms of an object's acceleration. Since Newton's second law is
only valid for constant-mass systems,mass can be taken outside the differentiation
operator by the constant factor rule in differentiation. Thus,
where F is the net force applied, m is the mass of the body, and a is the body's acceleration. Thus,
the net force applied to a body produces a proportional acceleration. In other words, if a body is
accelerating, then there is a force on it.
Consistent with the first law, the time derivative of the momentum is non-zero when the momentum
changes direction, even if there is no change in its magnitude; such is the case with uniform circular
motion. The relationship also implies the conservation of momentum: when the net force on the body is zero, the momentum of the body is constant. Any net force is equal to the rate of change of the
momentum.
Any mass that is gained or lost by the system will cause a change in momentum that is not the result of an external force. A different equation is necessary for variable-mass systems.
Newton's second law requires modification if the effects of special relativity are to be taken into
account, because at high speeds the approximation that momentum is the product of rest mass and
velocity is not accurate.
Impulse
An impulse J occurs when a force F acts over an interval of time Δt, and it is given by
This relation between impulse and momentum is closer to Newton's wording of the second law.
Impulse is a concept frequently used in the analysis of collisions and impacts.
Variable-mass systems
Variable-mass systems, like a rocket burning fuel and ejecting spent gases, are not closed and cannot
be directly treated by making mass a function of time in the second law;that is, the following
formula is wrong:
The falsehood of this formula can be seen by noting that it does not respect Galilean invariance: a
variable-mass object with F = 0 in one frame will be seen to have F ≠ 0 in another frame.The
correct equation of motion for a body whose mass m varies with time by either ejecting or accreting
mass is obtained by applying the second law to the entire, constant-mass system consisting of the body and its ejected/accreted mass; the result is
where u is the velocity of the escaping or incoming mass relative to the body. From this equation one
can derive the equation of motion for a varying mass system, for example, the Tsiolkovsky rocket
equation. Under some conventions, the quantity u dm/dt on the left-hand side, which represents the
advection of momentum, is defined as a force (the force exerted on the body by the changing mass, such as rocket exhaust) and is included in the quantity F. Then, by substituting the definition of
acceleration, the equation becomes F = ma.
Newton's third law
The third law states that all forces between two objects exist in equal magnitude and opposite
direction: if one object A exerts a force FA on a second object B, then B simultaneously exerts a force FB on A, and the two forces are equal and opposite: FA = −FB. The third law means that all forces are interactions between different bodies, and thus that there is no such thing as a unidirectional force or a force that acts on only one body. This law is sometimes referred to as the action-reaction law, with FA called the "action" and FB the "reaction". The action and the reaction are simultaneous, and it does not matter which is called the action and which is called reaction; both forces are part of a single interaction, and neither force exists without the other.
The two forces in Newton's third law are of the same type (e.g., if the road exerts a forward
frictional force on an accelerating car's tires, then it is also a frictional force that Newton's third
law predicts for the tires pushing backward on the road).
From a conceptual standpoint, Newton's third law is seen when a person walks: they push against the
floor, and the floor pushes against the person. Similarly, the tires of a car push against the road
while the road pushes back on the tires—the tires and road simultaneously push against each other. In
swimming, a person interacts with the water, pushing the water backward, while the water simultaneously pushes the person forward—both the person and the water push against each other. The reaction forces account for the motion in these examples. These forces depend on friction; a person or car on ice, for example, may be unable to exert the action force to produce the needed reaction force.
No comments:
Post a Comment