Classical mechanics questions and answers PDF

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Classical mechanics questions and answers...

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Classical mechanics is an ambitious theory? True The purpose of classical mechanics is to predict the future, reconstruct the past and determine the history of every particle in the universe. The theory of classical mechanics was formulated by Newton in the year 1687 with the purpose of building on the earlier insight of Galileo. What did Isaac Newton constructed which he term as a powerful tool enough to explain the broad way of phenomenon from the objects of the planets to the motion of the tides to the scattering of elementary particles? Mathematical Framework. Before the mathematical framework can be applied the framework needs a single unit called Force. Newtonian mechanics is not the last word in theoretical physics it struggles in extremes which means it is the realm of the very small, the very large and the very fast. The theory which laisses with Newtonian mechanics when the speed of a particle is comparable to the speed of light is Galilean Relativity Developments in the past 300 years has confirmed what is perhaps the most important legacy of newton that the laws of nature are written in the language of Mathematics Classical Mechanics is all about the motion of a particle. An object of insignificant size is known as a particle. In classical mechanics we are interested in the position of a particle. To determine what an object looks like in classical mechanics, the only information you would only give is to specify its position. To describe the position of a particle we need, reference frame.

A set of origin together with a set of axes which we consider it to be Cartesian is known as reference frame. The position of a particle is specified by a vector S with respect to this frame since the particle moves with position dependent on time, which trajectory does the particle describe? X=S(t) The velocity of a particle is defined as v=dx/dt = 𝒙󰇗

The acceleration of a particle is defined as a=dx^2/dt^2= 𝒙󰇘

The derivative of a vector is defined by differentiating each of the component defined by the vector. If a vector x defined as x= (s1, s2 and s3) then what is dx/dt? Ds1/dt, ds2/dt, ds3/dt Geometrically the derivative of a path x(t) lies to which dimension to the path or to which direction to the path? Tangent to the path. The framework which allows us to determine the trajectory of a path in any given situation is termed as Newtonian mechanics. Newtonian framework is also known as the Laws of motion. Classical mechanics considers microscopic particles if and only if the position of the particle is determined. The one in which the particle indeed travel at a constant speed when the force acting on it is vanishes is termed as inertia reference frame. The main difference between newton’s first law and second law of motion is that, inertia frame exits in the first law whilst it may or not exist in the second law. One way to ensure that you are in an inertia frame is to insist that you are left alone yourself and this includes, fly out into the deep space far

from the force of gravity and other influence, turn of your engine and sit there. The earth does not quite provide an inertia frame because, the earth is rotating about its own axes and revolving around the sun. Given an inertia frame S’ in which a particle has coordinate S(t) and another inertia frame is to be constructed from it. That inertia frame is with a particle x, what does x represent? Inertia frame The idea that tells us that left alone an object will naturally comes to rest is Aristotelian idea. Who previous realize the natural state of an object is to travel at constant speed? Galileo Particle in rotating frames do not experience? Inertia frame or do not travel at constant velocity. Which principle explains that when there is one inertia frame S with coordinate S(t) another inertia frame S’ can be constructed using the transformations combinations? Galilean Relativity The three transformations generate a group known as Galilean group. The three-transformation group are; Translation, Rotation and Boost

𝑎 If a matrix R is a 3 x 3 matrix and is expressed as R = 𝑑 𝑔 obeys the S’=RS then

𝑏 𝑒 ℎ

𝑐 𝑓 and 𝑖

Find the value of the transpose of the matrix times the matrix itself? 1 Find the determinant of______ matrix.

-1

Translations, rotations and boosts are related by x’= x +a, x’= Rx and x’=x+ vt respectively.

All laws of physics are the same in any inertia frame and this is known as the Principle of Relativity. Why is it that there is no special speed in the universe? This is because the speed of light is special. Combinations of the three types of Galilean group with the principles of relativity explains translations, rotations and boosts as, there is no special point, there is no special direction and there is no special velocity in the universe respectively. Which of the following about the universe is a fallacy? b. Position is not relative and acceleration is relative From newton’s first law if you are not accelerating then you are in an inertia frame. True The principle of relativity is usually referred to? Einstein Who brought the principle of relativity? Galilei Galileo If two particles are in inertia frame S and S’, with a fixed unit in seconds, then the time that exist between the two bodies differs by T’=t + t0 The existence of time measured equally in all inertia frame is termed as? Absolute time The universe itself breaks several of the Galilean transformations that there was some special time in the universe around 13.7 billion years ago and this can be described as the time of the big bang which means we did not know what happened here. There was one inertia frame in which the background universe was stationary. The background here refers to sea of photons of

temperature 2.7k which for the universe known as Cosmic Microwave Background Radiation. Homogenous is said to be defined as there is no special point in the universe. Azeotropic is said to be defined as there is no special direction in the universe. Newton’s second law of motion is also referred to as equation of motion. A measure of the reluctance of a particle to change its motion when subjected to a given force is termed as inertia mass.

The second law can be described or expressed as m𝒙󰇘 =x (x,𝒙 󰇗 ) Which of the following is true about newton’s second law?

d. it depends on both the position and the velocity of the particle. The second most important fact about newton’s second law is that, It is a second order derivative. The question in the 55 above means that it depends on velocity and the position. Which of the following is not true about the validity of Newtonian mechanics? d. All the particles are travelling at the same time. For a particle moving in one dimension whose force depends on its 𝒙 position, the potential on the particle is given as V(x)=-∫𝒙𝟎 𝒅𝒙′f(x)’ From the question just read to you, the force of the particle can be expressed as f(x)= -dV(x)/dx

The x’ that you heard in question 58 is referred to as dummy variable. The equation of motion in one dimension which depends entirely on its position is expressed as m𝒙󰇘 = -dvx/dx For any force in one dimension which depends entirely on the position there exist a conserve quantity called energy and it can be expressed as E=1/2m𝒙󰇗 2 + Vx If f that is force is equal to change in -v is equal to change in dv/dx and is not a central force relation, there exist a quantity called energy In a uniform gravitation field, a particle is subjected to a constant force f=-mg, the minus sign arises because the force is acting downwards. The potential energy and equation of motion of a particle in a uniform gravitational field is given by V(x)=mgx and 𝑥󰇘 =-g respectively. The velocity and position of a particle in a uniform gravitational field 𝒙󰇗 = u-gt and x =x0 +ut-1/2gt2 respectively. The potential energy of a harmonic oscillator is V(x)=1/2kx2

The force and equation of motion in a harmonic oscillator are f=-kx and 𝒙󰇘 =-kx respectively.

The equation of motion of a harmonic oscillator has a general equation of solution x(t)= acos𝝎𝒕 + 𝒃𝒔𝒊𝒏𝝎𝒕 where 𝝎 = √

𝒌

𝒎

The time taken to complete a full cycle is known as period.

The function of a conserve energy in equation of motion is that it allows us to turn the second order to a first order differential equation. Cubic potential can be expressed as V(x)= m(x3-3x) For a particle staying stationary at position x0 the energy of the particle can be expressed as E= Vx0 The entire interval taken for a particle to travel from a point to an infinity can be expressed as t-t0 = -1/√𝟔 log (𝜺 − 𝜺𝟎 )

The equilibrium potential of a particle in a harmonic oscillator is given as dv/dx=0. For a particle undergoing a simple pendulum, the force on the particle is given as -mg sin𝜃 /L and the equation of motion is given as 𝜽󰇘=-g sin𝜽/L The energy expressed by a pendulum is given as E = 1/2m𝜽󰇗2 L2 mgLcos𝜽.

In a simple pendulum if E that is energy, E > mgL that is potential energy, then the kinetic energy cannot be 0 but if E < mgL then kinetic energy can be 0. True The period of a particle in a simple pendulum is expressed as T = 2𝝅√

𝒍

𝒈

Forces in 3-dimension which take the form F = -∆ 𝑉 are called conservative forces.

Forces in the form F=-∆ 𝑉 = -dV/dr with a unit vector 𝑠 is called central force.

There exists a conserve quantity if and only if the force fit the form F=-

∆ 𝑉 = -dV/dr with a unit vector 𝑠 for it is a conserved quantity is called angular momentum. The quantity that gives us the equation of change of angular momentum is called torque. The four fundamental forces in nature are; strong force, weak force, gravitational force and electromagnetic force. The gravitational potential and a force of a particle moving under gravity is given as -GMm/r and -GMm/r2 respectively. The potential energy of a particle of mass m in the presence of M is called gravitational field. The relationship between orbiting speed and escape velocity of a particle is given as v0√𝟐 .

An object is moving in a parabolic way and another in a circular way, if they a separated by a distance, what is the relationship between these two bodies? √𝟐 Which of the following does not explain escape velocity?

It is a vector quantity it doesn’t explain escape velocity. Which of the following about gravitational mass and inertia mass is not true? Inertia and gravitational mass are not equal. Combination of magnetic and electric fields forms the electromagnetic force. The force experience by a particle with electric charge q is called? Lorentz force.

Which of the following is or are not true about the electric and magnetic fields? Electric and magnetic fields are equal. For time independent fields, there is a conserve quantity called energy. Particles undergoing circular motion in a plane with angular frequency experiences a frequency called cyclotron frequency. Magnetic and electric fields affect a particle in a manner that is proportional to the electric charge. True When energy is conserved in a Lorentz force then both the electric and magnetic force must be time- independent. The velocity dependent force which conserve energy is known as magnetic force. The friction which occurs when two solids are in contact is called dry friction. An ongoing theory to unify the four fundamental forces is called the Grand Unified Theory. The main difference between the force of gravity and coulomb force is that the force of gravity always attracts whereas coulomb force can attract and repel A form of fluid drag with particles moving slowly in very viscous liquid is called linear drag. A form of fluid drag with particles moving fast in less viscous liquids is called quadratic drag. Which of the fluid drag obeys F = −γv. Linear drag Which of the fluid drag obeys F = −γ|v|v . Quadratic drag

Which of the following is or are true? c. for linear friction the coefficient is directly proportional to the radius The type of drag is determined by a dimensionless number called the Reynold’s number which is equal to

𝝆𝒗𝒍 𝜼

.

Which type of drag dominates when the reynold’s number is far far less than one (R1)? Quadratic drag The coefficient of proportionality given by i.e the coefficient of proportionality used for viscosity is termed as dynamic viscosity. The speed of a body falling in a liquid is expressed as v=-

𝒎𝒈

√𝝆 𝐭𝐚𝐧𝐡(√𝝆𝒈 𝒎

𝒕)

With the expression in the question 112 above the particle reaches the maximum speed of v=-√

𝒎𝒈 𝝆

What name will you give to the equation in question 112 and why negative? It is an expression of terminal velocity and negative because the object is falling. The quantity σ = e2n/γ is called conductivity The newton second law involves inertia mass whereas newton laws of gravitation involves gravitational mass. True In classical mechanics, electric and magnetic field function as, it acts as guide to particles which carry a charge.

Everything in a single microscopic force is called friction The relation j = σE gives a relation between current density and electric fields. A particle projected without frictionless experience can expressed in position as ut +1/2gt2 + O (γt /m) The conservation of angular momentum in central force implies that the particle moves in a plane The position of a particle in a polar coordinate can be expressed as x = r 𝒓

The velocity of a particle in a polar coordinate can be expressed as  NB that is theta . before the theta hat 𝒙󰇗 =𝒓󰇗 𝒓 + r 𝜽󰇗 𝜽

The acceleration of a particle in a polar coordinate can be expressed as  𝒙󰇘 = (𝒓󰇘 - r𝜽󰇗2)𝒓 + (r𝜽 + 𝟐𝒓󰇗 𝜽󰇗) 𝜽

The equation of motion of a particle in a circular form can be expressed as m(𝒓󰇘 - r𝜽󰇗2)𝒓 = -dV/dr The direction of angular momentum reduces it to 2-D problem and the magnitude of angular momentum reduces it to 1-D problem The effective potential of a particle in a circular motion is given as Veff (r) = V(r) + ml2 / 2r2 and the extra term ml2 / 2r2 is called angular momentum barrier. The term above stops the particle from moving towards the origin. The real potential energy together with contribution from the angular kinetic energy is effective potential energy.

The smallest value of r that a particle reaches in circular motion is called periapsis The farthest distance reached by a particle in a circular motion is called apoapsis Both the shortest value of r and the farthest distance is called apsides. In the case of motion around the sun the shortest value of r is called perihelion In the case of motion around the sun the farthest distance is called aphelion

For V= -k/r for a particle along an orbit it trajectory is a hyperbola. The effective potential of a particle in a circular orbit is Veff (r) = ml2/r3 For a particle moving in a circular orbit the radial velocity is given as 𝒅𝒖

𝒓󰇗 = -L

𝒅𝜽

For a particle moving in a circular orbit the radial acceleration is given as 𝒓󰇘 = -L2 *u2 *(d2u/d𝜽2 ) The orbital acceleration is described as d2u/d𝜽2 + u = -

𝟏

𝒎𝒍𝟐

𝒖𝟐

F(1/u)

The Kepler problem is the name given to understanding planetary orbits about a star. It is named after the astronomer Johannes Kepler The final expression of radius of orbit is r = r0 / e cos𝜽 + 1

Where r0 = l2/k and e = Al2 /k

The equation r = r0 / e cos𝜃 + 1 describes a conic section The integration constant e is called eccentricity and it determines the shape of orbits.

Consider this shape

b

F1

X a

Given that x = 0.24 and a = 0.58 Find the value of e Ans: 0.414 What does a represent? Ans: Semi-major axis

F2

What does b represent? Ans: semi-minor axis What does x represent? Ans: position to the origin What is the formula for eccentricity when r=0 Ans: e = x/a The eccentricity of the earth as its orbits around the sun is e < 0 or e1 In solar system all planets have eccentricity of e...