Newtons second law

NEWTON'S SECOND LAW OF MOTION

Explanation: The acceleration of an object is directly proportional to the force applied to that object, in the direction of the force. This law is expressed by the equation, F = ma, where 'F' stands for the net force that is acting on an object. If this body of mass 'm' has an acceleration 'a' (change in velocity with time), then the net force 'F' acting upon that object is the product of its mass and acceleration. This body also undergoes acceleration during its movement. In simpler words, the acceleration produced by a particular force acting on a body is directly proportional to the magnitude of the force, and inversely proportional to the mass of the body.

Example: You are pushing a table across a frictionless surface. If you want to increase the speed of the table's movement (increase in acceleration), you push harder (increase in force). If the table was replaced with a heavier one (increase in mass), you have to push even more harder than the previous time (increase in force) to speed it up at the same rate as before.

The second law states that the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force.

${\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}={\frac {\mathrm {d} (m\mathbf {v} )}{\mathrm {d} t}}.}$

The second law can also be stated in terms of an object's acceleration. Since Newton's second law is valid only for constant-mass systems,^[20][21][22] m can be taken outside the differentiation operator by the constant factor rule in differentiation. Thus,

${\displaystyle \mathbf {F} =m\,{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=m\mathbf {a} ,}$

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. An application of this notation is the derivation of G Subscript C.

The above statements hint that the second law is merely a definition of ${\displaystyle \mathbf {F} }$, not a precious observation of nature. However, current physics restate the second law in measurable steps: (1)defining the term 'one unit of mass' by a specified stone, (2)defining the term 'one unit of force' by a specified spring with specified length, (3)measuring by experiment or proving by theory (with a principle that every direction of space are equivalent), that force can be added as a mathematical vector, (4)finally conclude that ${\displaystyle \mathbf {F} =m\mathbf {a} }$. These steps hint the second law is a precious feature of nature.

The second law 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 (see below).

Newton's second law is an approximation that is increasingly worse at high speeds because of relativistic effects.

According to modern ideas of how Newton was using his terminology,^[23] the law is understood, in modern terms, as an equivalent of:

The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed.

This may be expressed by the formula F = p', where p' is the time derivative of the momentum p. This equation can be seen clearly in the Wren Library of Trinity College, Cambridge, in a glass case in which Newton's manuscript is open to the relevant page.

Motte's 1729 translation of Newton's Latin continued with Newton's commentary on the second law of motion, reading:

If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

The sense or senses in which Newton used his terminology, and how he understood the second law and intended it to be understood, have been extensively discussed by historians of science, along with the relations between Newton's formulation and modern formulations.^[24]

Impulse

An impulse J occurs when a force F acts over an interval of time Δt, and it is given by^[25][26]

${\displaystyle \mathbf {J} =\int _{\Delta t}\mathbf {F} \,\mathrm {d} t.}$

Since force is the time derivative of momentum, it follows that

${\displaystyle \mathbf {J} =\Delta \mathbf {p} =m\Delta \mathbf {v} .}$

This relation between impulse and momentum is closer to Newton's wording of the second law.^[27]

Impulse is a concept frequently used in the analysis of collisions and impacts.^[28]

Variable-mass systems

Main article: Variable-mass system

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;^[21] that is, the following formula is wrong:^[22]

${\displaystyle \mathbf {F} _{\mathrm {net} }={\frac {\mathrm {d} }{\mathrm {d} t}}{\big [}m(t)\mathbf {v} (t){\big ]}=m(t){\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}+\mathbf {v} (t){\frac {\mathrm {d} m}{\mathrm {d} t}}.\qquad \mathrm {(wrong)} }$

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.^[20] 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^[20]

${\displaystyle \mathbf {F} +\mathbf {u} {\frac {\mathrm {d} m}{\mathrm {d} t}}=m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}}$

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.