Nootan Physics Class 12 85.pdf
Genciana Haggins
Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows:
The three laws of motion were first stated by Isaac Newton in his Philosophi Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), originally published in 1687.[3] Newton used them to investigate and explain the motion of many physical objects and systems. In the time since Newton, new insights, especially around the concept of energy, built the field of classical mechanics on his foundations. Limitations to Newton's laws have also been discovered; new theories are necessary when objects move at very high speeds (special relativity), are very massive (general relativity), or are very small (quantum mechanics).
Newton's laws are often stated in terms of point or particle masses, that is, bodies whose volume is negligible. This is a reasonable approximation for real bodies when the motion of internal parts can be neglected, and when the separation between bodies is much larger than the size of each. For instance, the Earth and the Sun can both be approximated as pointlike when considering the orbit of the former around the latter, but the Earth is not pointlike when considering activities on its surface.[note 1]
Newton's first law expresses the principle of inertia: the natural behavior of a body is to move in a straight line at constant speed. A body's motion preserves the status quo, but external forces can perturb this.
By "motion", Newton meant the quantity now called momentum, which depends upon the amount of matter contained in a body, the speed at which that body is moving, and the direction in which it is moving.[20] In modern notation, the momentum of a body is the product of its mass and its velocity: p = m v , \displaystyle \mathbf p =m\mathbf v \,, where all three quantities can change over time.Newton's second law, in modern form, states that the time derivative of the momentum is the force: F = d p d t . \displaystyle \mathbf F =\frac d\mathbf p dt\,. If the mass m \displaystyle m does not change with time, then the derivative acts only upon the velocity, and so the force equals the product of the mass and the time derivative of the velocity, which is the acceleration:[21] F = m d v d t = m a . \displaystyle \mathbf F =m\frac d\mathbf v dt=m\mathbf a \,. As the acceleration is the second derivative of position with respect to time, this can also be written F = m d 2 s d t 2 . \displaystyle \mathbf F =m\frac d^2\mathbf s dt^2.
The forces acting on a body add as vectors, and so the total force on a body depends upon both the magnitudes and the directions of the individual forces. When the net force on a body is equal to zero, then by Newton's second law, the body does not accelerate, and it is said to be in mechanical equilibrium. A state of mechanical equilibrium is stable if, when the position of the body is changed slightly, the body remains near that equilibrium. Otherwise, the equilibrium is unstable.
A common visual representation of forces acting in concert is the free body diagram, which schematically portrays a body of interest and the forces applied to it by outside influences.[22] For example, a free body diagram of a block sitting upon an inclined plane can illustrate the combination of gravitational force, "normal" force, friction, and string tension.[note 4]
Overly brief paraphrases of the third law, like "action equals reaction" might have caused confusion among generations of students: the "action" and "reaction" apply to different bodies. For example, consider a book at rest on a table. The Earth's gravity pulls down upon the book. The "reaction" to that "action" is not the support force from the table holding up the book, but the gravitational pull of the book acting on the Earth.[note 6]
Various sources have proposed elevating other ideas used in classical mechanics to the status of Newton's laws. For example, in Newtonian mechanics, the total mass of a body made by bringing together two smaller bodies is the sum of their individual masses. Frank Wilczek has suggested calling attention to this assumption by designating it "Newton's Zeroth Law".[32] Another candidate for a "zeroth law" is the fact that at any instant, a body reacts to the forces applied to it at that instant.[33] Likewise, the idea that forces add like vectors (or in other words obey the superposition principle), and the idea that forces change the energy of a body, have both been described as a "fourth law".[note 8]
The study of the behavior of massive bodies using Newton's laws is known as Newtonian mechanics. Some example problems in Newtonian mechanics are particularly noteworthy for conceptual or historical reasons.
If the body is not released from rest but instead launched upwards and/or horizontally with nonzero velocity, then free fall becomes projectile motion.[39] When air resistance can be neglected, projectiles follow parabola-shaped trajectories, because gravity affects the body's vertical motion and not its horizontal. At the peak of the projectile's trajectory, its vertical velocity is zero, but its acceleration is g \displaystyle g downwards, as it is at all times. Setting the wrong vector equal to zero is a common confusion among physics students.[40]
Newton's cannonball is a thought experiment that interpolates between projectile motion and uniform circular motion. A cannonball that is lobbed weakly off the edge of a tall cliff will hit the ground in the same amount of time as if it were dropped from rest, because the force of gravity only affects the cannonball's momentum in the downward direction, and its effect is not diminished by horizontal movement. If the cannonball is launched with a greater initial horizontal velocity, then it will travel farther before it hits the ground, but it will still hit the ground in the same amount of time. However, if the cannonball is launched with an even larger initial velocity, then the curvature of the Earth becomes significant: the ground itself will curve away from the falling cannonball. A very fast cannonball will fall away from the inertial straight-line trajectory at the same rate that the Earth curves away beneath it; in other words, it will be in orbit (imagining that it is not slowed by air resistance or obstacles).[43]
A harmonic oscillator can be damped, often by friction or viscous drag, in which case energy bleeds out of the oscillator and the amplitude of the oscillations decreases over time. Also, a harmonic oscillator can be driven by an applied force, which can lead to the phenomenon of resonance.[45]
A rigid body is an object whose size is too large to neglect and which maintains the same shape over time. In Newtonian mechanics, the motion of a rigid body is often understood by separating it into movement of the body's center of mass and movement around the center of mass.
When Newton's laws are applied to rotating extended bodies, they lead to new quantities that are analogous to those invoked in the original laws. The analogue of mass is the moment of inertia, the counterpart of momentum is angular momentum, and the counterpart of force is torque.
If a third mass is added, the Kepler problem becomes the three-body problem, which in general has no exact solution in closed form. That is, there is no way to start from the differential equations implied by Newton's laws and, after a finite sequence of standard mathematical operations, obtain equations that express the three bodies' motions over time.[52][53] Numerical methods can be applied to obtain useful, albeit approximate, results for the three-body problem.[54] The positions and velocities of the bodies can be stored in variables within a computer's memory; Newton's laws are used to calculate how the velocities will change over a short interval of time, and knowing the velocities, the changes of position over that time interval can be computed. This process is looped to calculate, approximately, the bodies' trajectories. Generally speaking, the shorter the time interval, the more accurate the approximation.[55]
It is mathematically possible for a collection of point masses, moving in accord with Newton's laws, to launch some of themselves away so forcefully that they fly off to infinity in a finite time.[60] This unphysical behavior, known as a "noncollision singularity",[53] depends upon the masses being pointlike and able to approach one another arbitrarily closely, as well as the lack of a relativistic speed limit in Newtonian physics.[61]
Coulomb's law for the electric force between two stationary, electrically charged bodies has much the same mathematical form as Newton's law of universal gravitation: the force is proportional to the product of the charges, inversely proportional to the square of the distance between them, and directed along the straight line between them. The Coulomb force that a charge q 1 \displaystyle q_1 exerts upon a charge q 2 \displaystyle q_2 is equal in magnitude to the force that q 2 \displaystyle q_2 exerts upon q 1 \displaystyle q_1 , and it points in the exact opposite direction. Coulomb's law is thus consistent with Newton's third law.[73]
There is subtle conceptual conflict between electromagnetism and Newton's first law: Maxwell's theory of electromagnetism predicts that electromagnetic waves will travel through empty space at a constant, definite speed. Thus, some inertial observers seemingly have a privileged status over the others, namely those who measure the speed of light and find it to be the value predicted by the Maxwell equations. In other words, light provides an absolute standard for speed, yet the principle of inertia holds that there should be no such standard. This tension is resolved in the theory of special relativity, which revises the notions of space and time in such a way that all inertial observers will agree upon the speed of light in vacuum.[note 12]
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