Fundamental principle of counting,
Multiplication rule,
If an operation can be performed in m different ways and a second operation can be done in n different ways, then the two operations can be performed in m*n ways. The operations are depending on each other.
Addition rule,
If a operation can be performed in m different ways and another operation independant of the first is performed in n different ways, then either of the two operations are performed in m+n ways.
Factorial notation :
The continued product of first n natural numbers is called n factorial. It is denoted by n!
n!=n(n-1)!
Factorial of negative numbers are not defined.
Permutations
Each of the different arrangements which can be made by taking some or all of a number of things is called permutation.
Number of permutations of n things taken r at a time is nPr=n!/[{n-r}!]
Note :
a. Permutations of n things taken all at a time is nPn=n!
b. Pn=n!
Permutations under conditions :
When all objects are distinct,
a. The no. Of permutations of n different objects taken r at a time when a particular object is to be always included in each arrangement is
r!{(n-1)C(r-1)}.
b. Number of permutations of n different things taken all at a time when m things always come together is m!(n-m+1)!
c. Number of permutations of n different things taken all at a time when m specified things never come together is n!-(n-m+1)!m!
When all objects are not distinct,
The no. Of permutations of n things taken all at a time, p of one kind, q of another kind, and r of the third kind, while the rest of n-(p+q+r) are all different, is
n!/(p+q+r)!
Circular Permutation:
The no. Of circular permutation of n different things taken all at a time is (n-1)!, provided that clockwise and anti-clockwise orders are different. If same, then it (n-1)!/2.
Note : Circular permutation of n things taking r at a time when clockwise and anti-clockwise orders are different is nPr/r and if same then nPr/2r.
Combination :
The different groups or selection that can be made by some or all of a number of given things without reference to the order of the things in each group is called combination.
It is denoted by nCr.
nCr=n!/{(n-r)!r!}.
Note : In a combination only selection is made while in permutation both selection and order is essential.
Properties of combination:
a. nCr=nC(n-r)
b. nCr+nC(r-1)=(n+1)Cr
c. If n is odd then the greatest value of nCr is nC{(n+1)/2} or nC{(n-1)/2}
and if n is even then it is nC(n/2).
d. nCx=nCy, then x=y or x+y=n.
e. nC0+nC1+.....+nCn=2^n
f. nCn+(n+1)Cn+...+(2n-1)Cn = 2nC(n+1)
g. nCr=n{(n-1)C(r-1)}/r
h. nC0+nC2+nC4+... = nC1+nC3+nC4=2^(n-1)
h. Restricted combination,
The number of combinations of n different things taking r at a time when :
i] p things are always included = (n-p)C(r-p)
ii] p things are always excluded = (n-p)Cr
iii] p things are included and q things excluded = (n-p-q)C(r-p).
i. The no. Of combinations of n things taking some or all atleast once at a time is = 2^n - 1.
j. Total selection of some or all out of p+q+r+t items where p,q,r and each distinct and t are different is = {(p+1)(q+1)(r+1)2^t}-1.
Division into groups :
a. The number of ways in which n objects can be split into three groups containing r, s, t objects; r, s, t are distinct and r+s+t=n is given by nCr.(n-r)Cs.(n-r-s)Ct = n!/(r!s!t!)
b. If n objects are to be divided into l groups containing p objects each, m object containing q objects each then ways of group formation,
=n!/{(p!)^l*(q!)^m*l*m} = x
permutations of these = x*{(l+m)!}
Here, lp+mq=n.
Note :
1] if order of group is not important, the number of ways in which mn things can be arranged equally into m groups is
mn!/[{(n!)^m}*m!]
2] if order of group is not important, the number of ways of arranging mn things equally into m groups is (mn!)/{(n!)^m}.
Arrangement into groups :
i. The no. Of ways in which n different things can be arranged into r different groups is r(r-1)(r-2)...(r+n-1) or n!*(n-1)C(r-1) according as blank groups are allowed or not.
ii. The no. Of ways of distributing n things into r parcels
= n!*coefficient of x^n in (e^x-1)^r
= r^n-rC1(r-1)^n+...+(-1)^(r-1)[rC(r-1)].
iii. The no. Of ways of arranging n identical things into r groups is (n+r-1)C(r-1) or (n-1)C(r-1) according as blank groups are admissible or not.
Dearrangements :
If n things are arranged in a row then the no. Of ways in which they can be dearranged so that no one is at the original place is n!(1! - 1/2! + 1/3! +...+(-1)^n*1/n!)
Number of rectangles and squares :
1] no. Of rectangles of any size in a square of n*n is (1^3+2^3+...+n^3) and no. Of squares of any size is (1^2+2^2+...+n^2)
2] In a of rectangle of n*p size, no. Of rectangle of any size is = np(n+1)(p+1)/4 and no. Of squares = summation of (n+r-1)(p+1-r), r=from 1 to n.
Tuesday, February 15, 2011
Monday, February 14, 2011
NEWTON'S LAWS OF MOTION
The British scientist and mathematician Sir Issac Newton was one of the greatest scientists of all time. A leader of scientific thought in England, he worked out how the universe was held together, discovered the secrets of light and colour and invented calculus. However, the great man had a few weaknesses. His work was often affected by his furious temper and his inability to take critisism from other scientists.
Newton's laws of motion
1st law, the law of inertia,
If the resultant force acting on a particle is zero, then the acceleration is 0.
2nd law,
The resultant force acting on a particle is proportional to the rate of change of momentum of the particle and the resultant force is in the direction of the change in momentum.
F=k.dp/dt
k=1,
F=dp/dt
=d(mv)/dt
=mdv/dt+vdm/dt
=ma+vdm/dt
If mass is constant,
F=ma.
Principal of conservation of momentum,
If the resultant force on a particle during a time period is zero then their linear momentum remains constant in that period.
Impulse,
Impulse on a particle due to a force is defined as multiplication of the force and duration of the action on the particle.
Impulse momentum theorem,
According to impulse momentum theorem impulse on a body due to a resultant force is change in its momentum.
3rd law,
If a body exerts a force on another body then the 2nd body also exerts a force on the 1st body. If any one of them is called action, the other is called reaction force. The action and reaction forces are equal in magnitude but opposite in direction. Action and reaction always act on two different bodies. They do net cancel.
Reference Frame-
It is the coordinate frame about which the position of a body with respect to some force is defined.
Inertial refence frame,
is the frame of reference that follows the law of inertia or Newton's 1st law.
Non-inertial reference frame,
is the frame of reference that doesnot follow the law of inertia.
Note,
a. A frame accelerated w.r.t. a reference frame is a non-inertial frame itself and vice-versa.
b. The earth is not perfectly an inertial frame due its rotation about its axes and revolution around the Sun.
c. In a non-inertial frame, Newton's 1st law can be applied if a special force called pseudo or false force is considered.
Pseudo force,
In an accelerating reference frame,i.e.,in a non-inertial reference frame, Newton's 1st law explains that the body in the frame experience a special acceleration and its corresponding force due to the accelerating frame and this force could be added to explain Newton's 2nd law of motion in this frame.
This special force is known as pseudo force.
Pseudo force doesnot act as action and reaction pair and thus donot follow Newton's third law as no force opposes it.
Newton's laws of motion
1st law, the law of inertia,
If the resultant force acting on a particle is zero, then the acceleration is 0.
2nd law,
The resultant force acting on a particle is proportional to the rate of change of momentum of the particle and the resultant force is in the direction of the change in momentum.
F=k.dp/dt
k=1,
F=dp/dt
=d(mv)/dt
=mdv/dt+vdm/dt
=ma+vdm/dt
If mass is constant,
F=ma.
Principal of conservation of momentum,
If the resultant force on a particle during a time period is zero then their linear momentum remains constant in that period.
Impulse,
Impulse on a particle due to a force is defined as multiplication of the force and duration of the action on the particle.
Impulse momentum theorem,
According to impulse momentum theorem impulse on a body due to a resultant force is change in its momentum.
3rd law,
If a body exerts a force on another body then the 2nd body also exerts a force on the 1st body. If any one of them is called action, the other is called reaction force. The action and reaction forces are equal in magnitude but opposite in direction. Action and reaction always act on two different bodies. They do net cancel.
Reference Frame-
It is the coordinate frame about which the position of a body with respect to some force is defined.
Inertial refence frame,
is the frame of reference that follows the law of inertia or Newton's 1st law.
Non-inertial reference frame,
is the frame of reference that doesnot follow the law of inertia.
Note,
a. A frame accelerated w.r.t. a reference frame is a non-inertial frame itself and vice-versa.
b. The earth is not perfectly an inertial frame due its rotation about its axes and revolution around the Sun.
c. In a non-inertial frame, Newton's 1st law can be applied if a special force called pseudo or false force is considered.
Pseudo force,
In an accelerating reference frame,i.e.,in a non-inertial reference frame, Newton's 1st law explains that the body in the frame experience a special acceleration and its corresponding force due to the accelerating frame and this force could be added to explain Newton's 2nd law of motion in this frame.
This special force is known as pseudo force.
Pseudo force doesnot act as action and reaction pair and thus donot follow Newton's third law as no force opposes it.
SIMPLE HARMONIC MOTION
SIMPLE HARMONIC MOTION [SHM]
Oscillatory motion is the to and fro motion of particles about a fixed point called the origin of motion. Its equation is written as f(x)=a0+a1x+a2x^2+a3x^3...
Most of the oscillatory motions are governed by forces acting on them and they are called SHM when there is linear dependance of f(x) on the displacement of particle,i.e., only a1 in the f(x) equation is non-zero. The forces are written as F=-kx. Negativity occurs as the force acts opposite to the displacement.
From newton's second law, F=ma. Thus, comparing, k=m, x=a=d2x/dt2.
Thus, m.d2x/dt2=-kx
d2x/dt2=-kx/m=-k'x
k'=k/m. Thus the double derivative of the function gives a similar type of equation ( F(x)=-kx and F"(x)=-k'x ).
This relation is shown in sine and cosine function. The equation is hence written as
x=x'cos(wt + o).
x'=any multiplicative constant, w=angular velocity.
To determine the meaning of constants we differentiate the equation two times.
dx/dt=-wx'sin(wt+o)
d2x/dt2=-w^2x'cos(wt+o)
Thus, recalling the previous eqn., d2x/dt2=-kx/m,
-w^2x'cos(wt+o)=-kx'cos(wt+o)/m
and, if w is a constant, w^2=k/m.
When x=maximum,i.e., x=x', (wt+o) = 0. And at t=o, (wt+o)=o.
Thus,o = initial phase or epoch or phase constant.
Velocity of SHM = v =dx/dt=-x'wsin(wt+o)
v=w(x'^2-x^2)^1/2.
Acceleration of SHM=a=d2x/dt2=-x'w^2cos(wt+o).
Time period of oscillation is given by,
T=2(pie)/w.
The function 'x' thus repeats itself after 'T'.
Since w^2=k/m,
thus,
T=2(pie)(m/k)^1/2
m/k=inertia factor/elasticity factor.
Energy in SHM,
v=w(x'^2-x^2)^1/2.
Thus, Kinetic energy=1/2mv^2=1/2mw^2(x'^2-x^2).
Potential energy can be derived by integrating, dU=F.dx.
PE=1/2mw^2x^2.
Total energy=KE+PE=1/2mw^2x'^2, which is constant for constant m.
Spring cases:
a)for one spring, attached to a support and the other end to a block of mass m over a frictionless surface when displaced by x,
restoring force F=-kx,
ma=-kx
a+kx/m=0
a+w^2x=0
w^2=k/m
thus, T=2(pie)(m/k)^1/2
b)for a series of spring connected end to end,
equivalent spring constant = k', where,
1/k'=1/k1+1/k2+1/k3+...+1/kN.
w^2=k'/m
T=2(pie)/w as usual.
c) a series of springs attached to the block of mass m from the wall,
k'=k1+k2+k3+...+kN.
T calculated as usual.
d) when 2 spring systems of constant k and k' attaches two sides of a block of mass m to adjacent walls, k"=k+k' and T is calculated by the general formula.
e) when a block of mass M hanging from the ground is attached to a spring of mass m, and length l, connecting a wall.
Mass per unit length of spring=m/l
If M displaces by j, an element located at a distance s in the spring is displaced by js/l and its velocity is
v=s/l.(dj/dt)
Hence, KE of spring=1/2mv^2
and by integrating, we get KE of spring=1/6m(dj/dt)^2
KE of block=1/2M(dj/dt)^2
Total KE=1/2(M+m/3)(dj/dt)^2
PE=integral k.jdj=1/2kj^2
Hence total energy of system=PE+KE
=1/2(M+m/3)(dj/dt)^2+1/2kj^2
as we are in the domain of a conservative field,
dE/dt=0
d2j/dt2=-kj/(M+m/3)
we get,
w={k/(M+m/3)}^1/2
and,
T=2(pie)/w.
Oscillatory motion is the to and fro motion of particles about a fixed point called the origin of motion. Its equation is written as f(x)=a0+a1x+a2x^2+a3x^3...
Most of the oscillatory motions are governed by forces acting on them and they are called SHM when there is linear dependance of f(x) on the displacement of particle,i.e., only a1 in the f(x) equation is non-zero. The forces are written as F=-kx. Negativity occurs as the force acts opposite to the displacement.
From newton's second law, F=ma. Thus, comparing, k=m, x=a=d2x/dt2.
Thus, m.d2x/dt2=-kx
d2x/dt2=-kx/m=-k'x
k'=k/m. Thus the double derivative of the function gives a similar type of equation ( F(x)=-kx and F"(x)=-k'x ).
This relation is shown in sine and cosine function. The equation is hence written as
x=x'cos(wt + o).
x'=any multiplicative constant, w=angular velocity.
To determine the meaning of constants we differentiate the equation two times.
dx/dt=-wx'sin(wt+o)
d2x/dt2=-w^2x'cos(wt+o)
Thus, recalling the previous eqn., d2x/dt2=-kx/m,
-w^2x'cos(wt+o)=-kx'cos(wt+o)/m
and, if w is a constant, w^2=k/m.
When x=maximum,i.e., x=x', (wt+o) = 0. And at t=o, (wt+o)=o.
Thus,o = initial phase or epoch or phase constant.
Velocity of SHM = v =dx/dt=-x'wsin(wt+o)
v=w(x'^2-x^2)^1/2.
Acceleration of SHM=a=d2x/dt2=-x'w^2cos(wt+o).
Time period of oscillation is given by,
T=2(pie)/w.
The function 'x' thus repeats itself after 'T'.
Since w^2=k/m,
thus,
T=2(pie)(m/k)^1/2
m/k=inertia factor/elasticity factor.
Energy in SHM,
v=w(x'^2-x^2)^1/2.
Thus, Kinetic energy=1/2mv^2=1/2mw^2(x'^2-x^2).
Potential energy can be derived by integrating, dU=F.dx.
PE=1/2mw^2x^2.
Total energy=KE+PE=1/2mw^2x'^2, which is constant for constant m.
Spring cases:
a)for one spring, attached to a support and the other end to a block of mass m over a frictionless surface when displaced by x,
restoring force F=-kx,
ma=-kx
a+kx/m=0
a+w^2x=0
w^2=k/m
thus, T=2(pie)(m/k)^1/2
b)for a series of spring connected end to end,
equivalent spring constant = k', where,
1/k'=1/k1+1/k2+1/k3+...+1/kN.
w^2=k'/m
T=2(pie)/w as usual.
c) a series of springs attached to the block of mass m from the wall,
k'=k1+k2+k3+...+kN.
T calculated as usual.
d) when 2 spring systems of constant k and k' attaches two sides of a block of mass m to adjacent walls, k"=k+k' and T is calculated by the general formula.
e) when a block of mass M hanging from the ground is attached to a spring of mass m, and length l, connecting a wall.
Mass per unit length of spring=m/l
If M displaces by j, an element located at a distance s in the spring is displaced by js/l and its velocity is
v=s/l.(dj/dt)
Hence, KE of spring=1/2mv^2
and by integrating, we get KE of spring=1/6m(dj/dt)^2
KE of block=1/2M(dj/dt)^2
Total KE=1/2(M+m/3)(dj/dt)^2
PE=integral k.jdj=1/2kj^2
Hence total energy of system=PE+KE
=1/2(M+m/3)(dj/dt)^2+1/2kj^2
as we are in the domain of a conservative field,
dE/dt=0
d2j/dt2=-kj/(M+m/3)
we get,
w={k/(M+m/3)}^1/2
and,
T=2(pie)/w.
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