Important Formulas And Units {For Class XII & XI}

Important Formulas and Units

Chapter 1: Physical World and Measurement

1. SI unit of luminous intensity is candela (cd).

2. In expressing a physical quantity, we choose a unit and then find that how many times that unit is contained in the given physical quantity:

Physical quantity (Q) = Magnitude × Unit = n × u,

where n is the numerical value and u is the unit. Thus, while expressing definite amount of physical quantity, it is clear that as the unit (u) changes, the magnitude (n) will also change, but their product nu will remain the same: nu = constant or n u n u1 1 2 2= = constant. Therefore,

nu

∝ 1.

3. In science, very large and very small decimal numbers are conveniently expressed in terms of powers of 10, some of which are listed below:

10 10 10 10 1000 101

10 10 100 001

10 10 10 100 101

10

3 3

2 2

= × × = =× ×

=

= × = =

−

−

.

××=

= = =

=

−

100 01

10 10 101

100 1

10 1

1 1

0

.

.

Using powers of 10, we can write the radius of Earth as 6,380,000 m = 6.38 × 106 m.

4. Mean absolute error is the arithmetic mean of the mag-nitudes of absolute errors in all the measurements of the quantity. It is represented by ∆a and is expressed as

∆ ∆ ∆ ∆a

a a an

n= + +| | | | | |.1 2

5. Relative error or fractional error

= Mean absolute error

Mean value= ∆a

am

.

6. Percentage error is expressed as

∆aam

×100%.

7. Error in sum of quantities is expressed as

xa ba b

= ++

×( )%.

∆ ∆100

8. Error in difference of quantities is expressed as

xa ba b

= +−

×( )%.

∆ ∆100

9. Error in division of quantities is expressed as

∆ ∆ ∆xx

aa

bb

= ± +

.

10. Error in quantity raised to some power is expressed as

∆ ∆ ∆xx

na

am

bb

= ± +

.

11. Checking whether a given equation is correct or incor-rect using dimensional analysis is based on the principle of homogeneity. According to this principle, the dimensions of each term on both sides of an equation must be the same.

12. If X A BC DEF= ± +( ) ,2 according to the principle of homo-geneity, we have

[ ] [ ] [( ) ] [ ].X A BC DEF= = =2

Chapter 2: Motion in a Straight Line

1. For one-dimensional motion, the displacement ∆x is the difference between vectors x and x0:

∆ x x x= − 0 .

2. Average velocity is defined as the ratio of the displace-ment of the object to the time interval

vxt

x xt tavg = = −

−∆∆

2 1

2 1

.

3. The average speed involves the total distance covered and is independent of direction. It is given by the relation

MTPL0139 Formula 1-14.indd 1 4/12/2012 1:07:55 PM

2 Important Formulas and Units

stavg

Total distance=∆

4. Instantaneous velocity is given by

vxt

dxdtt

= ∆∆

=∆ →lim .

0

5. Average acceleration is given by

av vt t

vtavg = −

−= ∆

∆2 1

1 2

.

6. The instantaneous acceleration can be expressed in terms of average acceleration as

avt

advdtt t

= ∆∆

= =∆ → ∆ →lim lim( ) .

0 0 avg

7. The equations of motion for constant acceleration a are as follows:

v v at= +0 ;

x x v t at− = +0 021

2;

v v a x x= + −02

02 ( );

x x v v t− = +0 0

12

( ) ;

x x vt at− = −021

2.

8. For a freely falling body, acceleration a = –g.

Chapter 3: Motion in a Plane

1. The position vector r at any time t, in terms of two-

dimensional coordinates x and y is given by r x y= + or i r x i y j

= + ,

where the magnitude r r x y= = +2 2 .

2. The position vector r at any time t, in terms of three-

dimensional coordinates x, y, and z is given by r x y z= + + or r xi y j zk= + + .

where the magnitude r r x y z= = + +2 2 2 .

3. Vector addition: Vector s is the vector sum of vectors

a

and b : s a b= + .

4. Commutative law: a b b a+ = + .

5. Associative law: ( ) ( ). a b c a b c+ + = + +

6. Vector subtraction: d a b a b= − = + −( ).

7. If we know a vector in component notation (ax and ay) and if we want it in magnitude – angle notation (a and q ), to transform it, we can use the following equations:

a a aa

ax yy

x

= + =2 2 and tan .θ

8. Unit vectors in the positive directions of the x, y, and z axes are labeled i j , and k , respectively, where the hat or cap symbol k , is used instead of an overhead arrow as for other vectors. Unit vectors are very useful for expressing other vectors:

a a i a jx y= + ; b b i b jx y= + ,

where the quantities a ix and a jy

are vectors – called the vector components of

a. The quantities ax and ay are

scalars – called the scalar components of a.

9. A third way to add vectors is to combine their components axis by axis:

dx = ax – bx , dy = ay – by and dz = az – bz ,

where d d i d j d kx y z= + + .

10. It follows from triangle law of vectors that if three vectors A, B , and

C and can be represented completely by the

three sides of a triangle taken in order, then their vector sum is zero:

A B C+ + = 0.

11. When a particle moves, the position vector changes – say, from

r1 to

r2 during a certain time interval – then the

particle’s displacement ∆r during that time interval is

∆ r = −r r2 1.

12. If a particle moves through a displacement ∆r in a time

interval ∆t, its average velocity vavg is expressed as

follows:

Average velocity = DisplacementTime interval avg⇒ =

vrt

∆∆

.

13. When a particle’s velocity changes from v1 to

v2 in a time

interval ∆t, its average acceleration aavg during the

time interval ∆t is expressed as follows:

Average acceleration = Change in velocity

Time interval

a⇒a vvg = − =

v v

tv

t2 1

∆ ∆.

14. The velocity is otherwise called as instantaneous velocity, which is given by the limiting value of the average velocity as the time interval approaches zero:

vrt

drdtt

= =→

lim .∆

∆∆0

15. In a circular motion, the distance traveled by a particle in one revolution is just the circumference of the circle (2p r).


3Important Formulas and Units

The time for a particle to go around a closed path exactly once has a special name – the period of revolution or simply the period of the motion. The period is repre-sented with the symbol T, which is expressed as

Tr

v= 2π

.

16. In a circular motion, the total number of revolutions by a particle in a given time is known as the frequency (n ) of revolution. From the definitions we have given for period and frequency, they are related by the expression

ν = 1T

.

Chapter 4: Laws of Motion

1. Gravitational field strength (g) is the force of gravity on a unit of mass, which is a vector quantity. Weight, mass, and gravitational field strength are related as W = mg.

2. Newton’s first law of motion: Every object continues in its state of rest or uniform motion unless made to change by a non-zero net external force: F = ma.

3. Momentum is the product of the mass of an object and its velocity, which is a vector quantity. Momentum (p) of an object of mass (m) with a velocity (

v) is expressed as

p = mv

Momentum is also the “inertia of a body in motion.”

4. The SI unit of momentum is kg m/s.

5. Newton’s second law of motion: The rate of change of momentum of an object is directly proportional to the applied force and takes place in the direction in which the force acts. Thus, if under the action of a force F for time interval ∆t, the velocity of a body of mass m changes from v to v + ∆

v , that is, its initial momentum

p = mv changes by

∆ ∆ p = m v , according to the Newton’s second law of

motion, we have

F

pt

F kpt

∝ =∆∆

∆∆

or ,

where k is the proportionality constant.

6. Impulse of a force is the product of the force and the time interval over which it acts. Impulse is a vector quantity. The impulse (

l ) delivered by a changing force is expressed as

l F t.= avg∆

7. Newton’s third law of motion: Whenever an object applies a force (an action) on a second object, the second object applies an equal and opposite force (a reaction) on the first object Newton’s third law is also defined as to every action there is an equal and opposite reaction and it takes place on two different bodies.

8. Law of conservation of momentum: If there are no external forces acting on a system, the total momentum remains constant, that is, if

Fnet = ∆ =0 0, .p

9. An object is said to be in equilibrium when it has zero acceleration.

10. A gravitational force F

g on a body is a certain type of pull that is directed toward a second body. The weight (

W ) of

a body is equal to the magnitude F

g of the gravitational force on the body.

11. When a body presses against a surface, the surface (even a seemingly rigid one) deforms and pushes on the body with a normal force F

N N that is perpendicular to the surface.

12. Friction is the force applied on the surface of an object when it is pushed or pulled against the surface of another object.

13. Properties of friction:

Property 1: If the body does not move, then the static frictional force f

s and the component of F

that is parallel

to the surface balance each other. They are equal in mag-nitude and f

s is directed opposite that component of F

.

Property 2: The magnitude of f

s has a maximum value fs, max that is given by

fs,max = msFN,

where ms is the coefficient of static friction and FN is the magnitude of the normal force on the body from the sur-face. If the magnitude of the component of F

that is par-

allel to the surface exceeds fs,max, the body begins to slide along the surface.

Property 3: If the body begins to slide along the surface, the magnitude of the frictional force rapidly decreases to a value fk given by

fk = mkFN,

where mk is the coefficient of kinetic friction.

14. The maximum static friction that a body can exert on the other body in contact with it is called limiting friction (Fmax ):

fs <Fmax = msR.

where ms is the coefficient of static friction.

15. The coefficient of static friction is always greater than the coefficient of kinetic friction, that is, ms >mk.

16. The angle of friction (f) is defined as the angle between the normal reaction N and the resultant of the friction force f and the normal reaction:

tan .φ = fN

Since f = mN, tan f = m.

17. The relationship between period T and speed v is given by

vr

T= 2π

,

where 2p r is the circumference of the circle.



18. Magnitude of centripetal acceleration: The centrip-etal acceleration of an object moving with a speed v on a circular path of radius r has a magnitude ac given by

avrc .=

2

19. Magnitude of a centripetal force: The centripetal force is the name given to the net force required to keep an object of mass m, moving at a speed v, on a circular path of radius r, and it has a magnitude of

Fmv

rc .=2

Chapter 5: Work, Energy, and Power

1. For an object of mass m whose speed v is well below the speed of light, the kinetic energy is expressed as

K mv= 12

2.

2. The SI unit of kinetic energy is joule (J).

3. The scalar product of two vectors a and

b is written as

a b⋅ , which is defined as a b ab⋅ = cos ,φ

where a is the magnitude of a, b is the magnitude of

b ,

and f is the angle between a and

b.

4. A dot product can be regarded as the product of two quantities: (a) the magnitude of one of the vectors and (b) the scalar component of the second vector along the direction of the first vector.

5. If the angle q between two vectors is 0°, the component of one vector along the other is maximum, and so also is the dot product of the vectors. If, instead, q is 90°, the component of one vector along the other is zero, and so is the dot product.

6. The relationship that relates work to the change in kinetic energy is known as work – energy theorem – when a net external force does work W on an object, the kinetic energy of the object changes from its initial value (KE0 ) to a final value (KEf), the difference between the two values being equal to the work:

W mv mv= KE KEf 0 f− = −12

12

202 .

7. Work can be expressed as follows: W ≡ F∆scosq.

8. The SI units of work are units of force (N) times units of displacement (m) and are called joules (J):

1 J ≡ 1 N × 1 m ⇒1 J = 1 N m.

9. Work done by a constant force is expressed as W F d=

⋅ .

10. Work done by a gravitational force is expressed as Wg = mgd cosq.

11. The law of force for a spring is called Hooke’s law, which is expressed mathematically as

F kxs ,= −

where k is called the spring constant whose SI unit is the newton per meter (N/m).

12. Work done by a spring force is expressed as

W kx kxs i f .= −12

12

2 2

13. The work done by a variable force is expressed as

W F x dxx

x= ∫ ( ) .

i

f

14. Gravitational potential energy is the energy stored in an object as a result of its position relative to another object to which it is attracted by the force of gravity. Gravitational potential energy, as a function of height h, is mathemati-cally expressed as

V(h) = mgh.

15. Elastic potential energy is the energy stored in an object as a result of a reversible change in shape, which is math-ematically expressed as

V x kx( ) .= 12

2

16. Mass–energy equivalence: An object’s mass m and the equivalent energy E are related by the equation

E = mc 2,

which is the famous Einstein’s equation.

17. If a force does an amount of work W in an amount of time ∆t, the average power due to the force during that time interval is

PW

tavg =∆

.

18. The instantaneous power P is the instantaneous time rate of doing work, which can be expressed as

PdWdtinst = .

19. The SI unit of power is J/s. This unit is used so often that it has a special name, watt (W), named after the scientist James Watt. 1 horse power, another unit of power often used in automobile industry, is equal to 746 W.

20. Instantaneous power is also expressed in terms of force and velocity as

P F v= ⋅

.



Chapter 6: System of Particles and Rotational Dynamics

1. The center of mass of a system of particles is the point that moves as though (1) all of the system’s mass were concentrated there and (2) all external forces were applied there. The center of mass of a system of particles is defined as the point whose position vector is expressed as

r

Mm r

i

n

i icom = ∑=

11

,

where M is the total mass of the system.

2. The cross product of a vector with itself is a null vector.

A A A A n× = ° =( )( )sin .0 0

3. The cross product of two vectors does not obey commu-tative law.

A B B A× ≠ × .

4. The cross product obeys the distributive law. A B C A B A C× + = × + ×( ) .

5. The (instantaneous) magnitude of angular velocity w is expressed as

ω θ= ddt

.

6. The unit of angular velocity is commonly the radian per second (rad/s) or the revolution per second (rev/s).

7. The relationship between angular velocity and linear velocity is expressed as

v = rw.

8. The linear momentum of a particle is a vector quantity that is defined as

p mv= ,

in which m is the mass of the particle and v is its

velocity.

9. If a and

b are parallel or antiparallel,

a b× = 0. The mag-

nitude of a b× , which can be written as | |,

a b× is maxi-

mum when a and

b are perpendicular to each other.

10. The angular momentum of a system of particles is given by

L r p

= × .

11. The torque τ acting on the particle relative to the fixed

point O is a vector quantity, which is defined as τ = ×r F .

12. Conditions for equilibrium: (1) Resultant of all the exter-nal forces ( )Fnet and external torques ( )τnet must be zero. (2) Center of gravity is the location in the extended body where we can assume the whole weight of the body to be concentrated.

13. The gravitational force Fg on a body that effectively acts at

a single point is called the center of gravity of the body.

14. The moment of inertia of a rigid body about an axis is expressed as

I m ri i= ∑ 2 ,

where ri is the perpendicular distance of the i th point of the body from the axis. The kinetic energy of rotation is given by

K I= 12

2ω .

15. The SI unit of moment of inertia is kilogram square meter (kg m2).

16. Theorem of parallel axes: The moment of inertia of a body about any axis is equal to the sum of the moments of inertia of the body about a parallel axis passing through its center of mass and the product of its mass and the square of distance between the two parallel axes:

I I Md= +g .2

17. Theorem of perpendicular axes: The moment of inertia of planar body about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two per-pendicular axes concurrent with perpendicular axis and lying in the plane of the body:

I I Iz x y= + .

18. In terms of dynamics and kinematics, rotation about a fixed axis is analogous to linear motion.

19. The angular acceleration of a rigid body which is rotating about a fixed axis is expressed as

Ia = t.

If the external torque t is zero, the component of angu-lar momentum about the fixed axis Iw of such a rotating body is constant.

20. For rolling motion without slipping vcom = rw, where vcom is the velocity of translation, that is, of the center of mass, r is the radius, and m is the mass of the body. The kinetic energy of such a rolling body is the sum of kinetic ener-gies of translation and rotation:

K I Mv= +12

2 12

2com comω .

21. Law of conservation of angular momentum: If the net resultant external torque acting on an isolated system is zero, the total angular momentum L of system should be conserved.

22. The relation between the arc length s covered by a par-ticle on a rotating rigid body at a distance r from the axis and the displacement q (in radians) is expressed as

s = rq.



Chapter 7: Gravitation

1. Newton proposed a force law that we call Newton’s law of gravitation is defined as every particle attracts other particle with a gravitational force of magnitude

F Gm m

r= 1 2

2 ,

where m1 and m2 are the masses of the particles, r is the dis-tance between them, and G is the gravitational constant (= 6.672 ×10-11 N m2/kg2 or = 6.67 ×10-11 m3/kgs2).

2. The SI unit of gravitational constant is N m2/kg2 and its dimensional formula is [M-1L3T -2].

3. If we are supposed to find the resultant gravitational force acting on the particle m due to a number of masses M1, M2, … Mn, we use the principle of superposition. Let

F F Fn1 2, , be the individual forces due to the masses M1, M2,….Mn, which are given by the law of gravitation, then from the principle of superposition, each of these forces acts independently and uninfluenced by the other bodies. The resultant force

FR can be expressed in vector addition as

F F F F FR n ii

n

= + + + ==∑1 2

1

,

where ∑ is the symbol used for summation.

4. Kepler’s third law of planetary motion: The square of the orbital period of a planet is proportional to the cube of the semi-major axis of the elliptical orbit of the planet, which is also called law of period, is given by

TGM

R22

34=

π

s

,

where T is the period of motion of the planet, R is the radius of the circular orbit of the planet, Ms is the mass of the Sun, and G is the universal gravitational constant (= 6.672 ×10–11 Nm2/kg2). For elliptical orbits, this equation is valid if R is replaced by the semi-major axis (a).

5. When a body of mass m lying on the surface of the Earth of mass ME and radius RE, the exact value of acceleration

due to gravity at an altitude h above the surface of Earth is then given by

gGM

R= E

E

,2

6. Considering Earth be a homogeneous sphere of radius RE and mass ME and a body be taken to a depth d below the free surface of the Earth, then the acceleration due to gravity is gd is given by

g gdRd

E

= −

1 .

7. The gravitational potential energy U of two particles, of masses M and m, separated by a distance r is given by

U = − GMmr

,

The gravitational potential energy decreases when the separation decreases. Since U = 0 for r = ∞, the potential energy is negative for any finite separation and becomes progressively more negative as the particles move closer together.

8. When an isolated system consists of a particle of mass m moving with a speed v in the vicinity of a massive body of mass M, then the total mechanical energy of the particle is given by

E mvGMm

r= −1

22 ,

which implies that the total mechanical energy is the sum of the kinetic and potential energies. The total energy is a constant of motion.

9. The escape velocity, ve , of a body that is projected from the Earth is given by

v gRe ,= 2 E

which has the value of 11.2 km/s.

Chapter 8: Mechanical Properties of Solids

1. If F is the magnitude of the force applied on the body and A is the area of cross-section of the body, the magnitude of stress is given by

StressRestoring force

Area= = F

A.

2. The SI unit of stress is same as that of pressure, that is, N/m2 or pascal (Pa) and its dimensional formula is [ ].ML T1 2− −

3. The restoring force per unit area developed in a body due to the applied tangential force is known as tangential stress or shearing stress. Mathematically, it is expressed as

Shearing stressTangential force

Area= .

4. The ratio of change in configuration to original configuration is called strain. Mathematically, strain is written as

Strain = Change in configuration

Original configuration.

5. If the deforming force acting on an elastic body produces a change in only the length of the body, the change in length per unit original length of the body is known as longitudi-nal strain, which is mathematically exp ressed as

Longitudinal strain =Change in length

Original length= ∆L

L.



6. If the deforming force acting on an elastic body produces a change in the shape of the body without changing its volume, the strain produced in the body is known as shearing strain, which is mathematically expressed as

Shearing strain ( ) .θ = ∆LL

7. If the deforming force acting on an elastic body pro-duces a change in the volume of the body alone, the change in volume per unit original volume of the body is known as volumetric strain, which is mathematically expressed as

Volumetric strain=Change in volume

Original volume= ∆V

V.

8. According to Hooke’s law, within elastic limits, stress is directly proportional to strain,that is, the extension pro-duced in a wire is directly proportional to the load applied to the wire, which can be expressed as

Stress StrainStress Strain,

∝= ×k

where k is a constant of proportionality and is known as the modulus of elasticity. Hooke’s law is applicable to most of the materials but there are certain materials in which the relationship between stress and strain is not linear and they do not obey Hooke’s law.

9. The ratio of the stress to the corresponding strain pro-duced in a body within the elastic limits is called modulus of elasticity or coefficient of elasticity. Modulus of elas-ticity is numerically equal to the ratio of stress and strain and, therefore, it has same dimensions as stress:

Modulus of elasticity = StressStrain

.

10. The ratio of normal stress to the longitudinal strain within the elastic limit is called Young’s modulus of elasticity, which is mathematically expressed as

Y = =Tensile (or compressive) stressLinear strain

σε

.

Greater the Young’s modulus of a material, larger is the elasticity of the material. Therefore, steel is more elastic than copper because Young’s modulus of steel is greater than that of copper.

11. Young’s modulus of the material of a wire is expressed as

YMg r

L LMgLr L

= = = =Linear stressLinear strain

//

σε

ππ

2

2∆ ∆,

where r is the initial radius of the wire, L is the initial length of the wire; πr 2 is the area of cross-section of the wire; M is the mass of the weights in the pan at the bot-tom due to which elongation ∆L is produced in the wire; the force applied by the mass M on the wire is equal to its weight, that is, Mg, where g is the acceleration due to gravity.

12. The ratio of tangential stress to the tangential strain pro-duced in a body within elastic limits is known as shear modulus or modulus of rigidity, which is mathemati-cally expressed as

Shear modulusShearing stressShearing strain

//

( )GF A

x LFL= = = =

σθ ∆ AA x∆

.

13. SI unit of shear modulus is N/m2 or pascals (Pa).

14. The ratio of normal stress to the volumetric strain produced in the body within the elastic limits is called bulk modulus of elasticity, which is mathematically expressed as

BulkmodulusHydroststic stress

Volume strain( )

( / )B

V VV= =

−= −P P

∆ ∆VV,

where the negative sign shows that with increase in pressure P, the volume of the body decreases, that is, if P is positive, ∆V is negative. Hence, for a system in equili-brium, the value of bulk modulus should be positive.

15. The SI unit of bulk modulus is N/m2 or pascals (Pa).

16. The reciprocal of the bulk modulus of a material is called the compressibility of that material and is represented by the symbol k. Compressibility is defined as the fractional change in volume per unit increase in pres-sure. Mathematically, compressibility of a material is expressed as

Compressibility ( )( / )

.kB V V

VV

= =−

= −1 1P P∆

∆

17. When a deforming force is applied at the free end of a sus-pended wire, the ratio of lateral strain and the longitudi-nal strain produced in the wire is called Poisson’s ratio, which is mathematically expressed as

Poisson’s ratio ( )Lateral strain

Longitudinal strain//

σ = =−

∆∆

l lR RR

R ll R

= − ∆∆

,

where l is the initial length and R is the radius of the wire before applying the deforming force and ∆l and ∆R are the increase in length and decrease in radius after the wire is stretched.

18. Applications of elastic behavior of materials: When loaded at the center and supported near its ends, a bar (bridge, buildings etc.) sags by a quantity

δ = Wlbd Y

3

34,

where l is the length of the bar, b is the breadth of the bar, d is the depth of the bar, and Y is the Young’s modulus of the material. (On increasing the depth d of a bar, unless the load is exactly at the right place, the deep bar bends, which effect is known as buckling.)

19. The elastic potential energy stored in a wire is given by

U = × ×12

2Young’smodulus Strain( ) .



Chapter 9: Mechanical Properties of Fluids

1. The uniform density r of a fluid is expressed as

ρ = mV

,

where m is the mass of the of the fluid and V is the volume of the fluid.

2. Density is a scalar property; its SI unit is the kilogram per cubic meter.

3. The pressure at any point in the fluid is the limit of this ratio as the surface area ∆A of the piston, centered on that point, is made smaller and smaller. However, if the force is uniform over a flat area A, pressure of uniform force on flat area is written as

P = FA

,

where F is the magnitude of the normal force on area A. (When we say a force is uniform over an area, we mean that the force is evenly distributed over every point of the area.)

4. The SI unit of pressure is the newton per square meter, which is given a special name, pascal (Pa), which is same as N/m2. Pascal is related to some other common (non-SI) pressure units as follows:

1 atm = 1.01 × 105 Pa = 760 torr.

5. Pascal’s law: A change in the pressure applied to an enclosed incompressible fluid is transmitted undimini-shed to every portion of the fluid and to the walls of its container.

6. The pressure in a fluid varies with depth (h) as per the expression

P = Pa + rgh,

where r is the density of the fluid, when it is uniform.

7. For an incompressible fluid passing any point every sec-ond in a pipe of non-uniform cross-section, the volume is the same in the steady flow, that is,

vA= constant,

where v is the velocity and A is the area of cross-section.

8. Archimedes’ principle: When a body is fully or partially submerged in a fluid, a buoyant force F

b from the sur-

rounding fluid acts on the body. The force is directed upward and has a magnitude equal to the weight mf g of the fluid that has been displaced by the body.

9. When a body floats in a fluid, the magnitude Fb of the buoyant force on the body is equal to the magnitude Fg of the gravitational force on the body.

10. When a body floats in a fluid, the magnitude Fg of the gravitational force on the body is equal to the weight mf g of the fluid that has been displaced by the body.

11. Bernoulli’s principle: As we move along a streamline, the sum of the pressure (P), the potential energy per unit vol-ume (rgy), and the kinetic energy per unit volume (rv2/2) remains a constant:

P + rv2/2 + rgy = constant,

which is basically the conservation of energy applied to non-viscous fluid motion in steady state. There is no fluid that have zero viscosity and hence the above statement is treated true only approximately. The viscosity is similar to friction that converts the kinetic energy to heat energy.

12. Torricelli’s law: “Efflux” means fluid outflow. Torricelli discovered that the speed of efflux from an open tank is expressed by a formula identical to that of a free-falling body:

v1 = 2gh,

when the tank is exposed to the atmosphere, that is, P = Pa ; this equation is known as Torricelli’s law.

13. In a fluid, though shear strain does not require shear stress, when a shear stress is applied to a fluid, the motion is gen-erated which causes a shear strain growing with time. The ratio of the shear stress to the time rate of shearing strain is called coefficient of viscosity (h).

14. The SI unit of coefficient of viscosity is poiseiulle (Pi) or N s/m2 or Pa s.

15. Force to move a layer of viscous fluid with a con-stant velocity: The magnitude of the tangential force

F

required to move a fluid layer at a constant speed v, when the layer has an area A and is located a perpendicular dis-tance y from an immobile surface, is given by

FAvy

=η

,

where h is the coefficient of viscosity.

16. SI unit of viscosity is Pa s. Common unit of viscosity is poise (P).

17. Poiseuille’s law: A fluid whose viscosity is h, flowing through a pipe of radius R and length L, has a volume flow rate Q given by

Q = −πη

R P PL

42 1

8( )

,

where P1 and P2 are the pressures at the ends of the pipe.

18. According to Stokes’s law, viscous force F acting on the sphere varies directly with (1) the coefficient of visco sity h of the fluid, (2) velocity v of the spherical body, and (3) radius r of the spherical body. Stokes’s law– the viscous dragging force – is mathematically expressed as

F av= 6πη ,

which explains the retarding force which is proportional to the velocity.

19. Reynolds number: The onset of turbulence in a fluid is determined by a dimensionless parameter given by

Rvd

e ,= ρη

where d is a typical geometrical length associated with the fluid flow.



20. Surface tension is a property by virtue of which, the free surface of a liquid possesses a tendency to contract so as to acquire a minimum surface area. If F be the force acting and l the length of the imaginary line, then the surface tension is given by

SFl

= .

21. The SI unit of surface tension is N/m. The dimensional for-mula of surface tension is [ML0 T –2].

22. Surface energy: The potential energy per unit area of the surface film is called the surface energy. It is the amount of

work done in increasing the area of a surface film through unity under isothermal conditions:

Surface energyWork done in increasing the surface area

Increase in=

ssurface area.

23. The SI unit of surface energy is N/m and dimension of sur-face energy is [MT -2].

24. The angle between tangent to the liquid surface at the point of contact and solid surface, inside the liquid, is termed as angle of contact and is denoted by q.

Chapter 10: Thermal Properties of Matter

1. In the SI system, temperature is measured on the Kel-vin scale, which is based on the triple point of water (273.16 K). Other temperatures are then defined by use of a constant-volume gas the rmometer, in which a sample of gas is maintained at constant volume so its pressure is proportional to its temperature. We define the tempera-ture T as measured with a gas thermometer to be

TPP

=

→

( . ) lim ,273 160

3

Kgas

where T is in kelvins, and P3 and P are the pressures of the gas at 273.16 K and the measured temperature, respectively.

2. Liquid water, solid ice, and water vapor (gaseous water) can coexist, in thermal equilibrium, at only one set of val-ues of pressure and temperature, called the triple point of water. By international agreement, the triple point of water has been assigned a value of 273.16 K as the stand-ard fixed-point temperature for the calibration of ther-mometers.

3. An ideal gas is an idealized model for real gases that have sufficiently low densities. The condition of low den-sity means that the molecules of the gas are so far apart that they do not interact (except during collisions that are effectively elastic). The ideal gas law expresses the rela-tionship between the absolute pressure, the Kelvin tem-perature, the volume, and the number of moles of the gas, which is given by

PV RT= µ ,

where m is the number of moles and R is the universal gas constant.

4. The Celsius temperature scale is expressed as

T TC . ,= − 273 15

where T is the Kelvin absolute temperature in kelvins and TC is the Celsius scale. The Fahrenheit temperature scale is expressed as

T TF C .= + °95

32

5. All objects change size with changes in temperature. For a temperature change ∆T, a change ∆L in any linear dimension L is given by

∆ ∆L L T= α ,

in which a is the coefficient of linear expansion. The change ∆V in the volume V of a solid or liquid is

∆ ∆V V T= β ,

where, b = 3a is the material’s coefficient of volume expansion.

6. Heat (Q) is energy that is transferred from a higher tem-perature object to a lower-temperature object because of the difference in their temperatures. It can be measured in joules(J), calories(cal), kilocalories(Cal or kcal), or British thermal units(Btu), with

1 cal = 3.968 10 Btu = 4.1868 J.3× −

7. The SI unit for heat is joule (J).

8. If heat Q is absorbed by an object, the object’s tempera-ture change Tf – Ti is related to Q by

Q C T T= −( ),f i

in which C is the heat capacity of the object. If the object has mass m, then

Q sm T T= −( ),f i

where s is the specific heat of the material making up the object. The molar specific heat of a material is the heat capacity per mole, which means per 6.02 × 1023elementary units of the material.

9. Heat absorbed by a material may change the material’s physical state – for example, from solid to liquid or from liquid to gas. The amount of energy required per unit mass to change the state (but not the temperature) of a particular material is its heat of transformation L. Thus,

Q = Lm.

10. The heat of vaporization LV is the amount of energy per unit mass that must be added to vaporize a liquid or that must be removed to condense a gas. For water at its nor-mal boiling or condensation temperature,

LV cal/ g kJ/mol kJ/kg= = =539 40 7 2256. .

11. The heat of fusion LF is the amount of energy per unit mass that must be added to melt a solid or that must be



removed to freeze a liquid. For water at its normal freezing or melting temperature,

LF cal/ g kJ/mol kJ/kg= = =79 5 6 01 33. . .

12. To measure specific heat of a material, we heat a sample to some known temperature Tm, and keep it in a vessel containing water of known mass and temperature Tw , (Tw <T ). We then measure the temperature of the water after equilibrium has been reached. This method is called calorimetry, and vessel in which this energy transfer occurs is called calorimeter.

13. The rate Pcond at which energy is conducted through a slab for which one face is maintained at the higher tem-perature TH and the other face is maintained at the lower temperature TC is

PQt

kAT T

LcondH C ,= = −

where each face of the slab has area A, the length of the slab (the distance between the faces) is L, and k is the thermal conductivity of the material.

14. Radiation is an energy transfer via the emission of elec-tromagnetic energy. The rate Prad at which an object emits

energy via thermal radiation is given by the Stefan– Boltzmann law of radiation,

P ATrad ,= σε 4

where s (= 5.6704 × 10-8W/m2 K4) is the Stefan – Boltz-mann constant, e is the emissivity of the object’s surface, A is its surface area, and T is its surface temperature (in kelvins). The rate Pabs at which an object absorbs energy via thermal radiation from its environment, which is at the uniform temperature Tenv (in kelvins), is

P ATabs env ,= σε 4

15. If T1 is the temperature of the surroundings, and T2 is the temperature of the body, Newton’s law of cooling is stated as the rate of cooling of a body is proportional to the excess temperature of the body over the surroundings:

∆∆

Qt

k T T= − × −( ),2 1

where k is a positive constant depending upon the area and nature of the surface of the body, T2 is the temper-ature of the body, and T1 is the temperature of the sur-rounding medium. The plot between the temperature of the body and time is known as the cooling curve.

Chapter 11: Thermodynamics

1. First law of thermodynamics is the common law of conservation of energy applied to any system in which the energy transfer from the surroundings, or to the sur-roundings, (through heat and work) is taken into account. It states that

∆Q = ∆U + ∆W,

where ∆Q is the heat supplied to the system, ∆W is the work done by the system, and ∆U is the change in internal energy of the system.

2. Specific heat capacity of a substance is expressed as

sm

QT

=

1 ∆∆

where m is the mass of the substance and ∆Q is the heat required to change its temperature by an amount ∆T.

3. The molar specific heat capacity of a substance is expressed as

CQT

= ×1µ

∆∆

,

where m· is the number of moles of the substance [for a solid, the law of equipartition of energy gives C = 3R (where R is the universal gas constant), which agrees with the experiment at ordinary temperatures.

4. Calorie is the old unit of heat. One calorie is the amount of heat required to increase the temperature of 1 g of water from 14.5°C to 15.5°C; 1 cal = 4.186 J.

5. When CP and CV are molar specific heat capacities of an ideal gas at constant pressure and volume, the simple equation for the ideal gas is expressed as

CP and CV = R,

where R is the universal gas constant.

6. The relation between the state variables is called the equation of state. For an ideal gas, the equation of state is expressed as

PV = mRT,

where m· is the number of moles of the substance, R is the universal gas constant, and P, V, and T are the state variables.

7. At temperature T, in an isothermal expansion of an ideal gas from volume V1 to V2, the heat absorbed (Q) is equal to the work done (W) by the gas, which is expressed as

Q W RT= =

µ InVV

2

1

.

8. For an adiabatic process, that is, a system which is insulated from the surroundings and heat absorbed or released is zero, we have

PV γ = constant,

where

γ = CC

P

V

.



9. In an isobaric process, the pressure (P) is fixed, in which case, the work done by the gas is expressed as

W P V V R T T= − = −( ) ( ).2 1 2 1µ

10. The efficiency (h) of a heat engine is defined by

η = WQ1

,

where and W is the work done on the environment in on complete cycle and Q1 is the heat input, that is, the heat absorbed by the system in one complete cycle. According to the first law of thermodynamics, for one complete cycle,

W = Q1 – Q2, therefore,

η = −1 2

1

QQ

.

11. In a refrigerator (or a heat pump), the system extracts heat Q2 from the cold reservoir and discharges Q1 amount of heat to the hot reservoir, with work (W) done on the system. The coefficient of performance of a refrigerator is expressed as

α = =−

QW

QQ Q

2 2

1 2

.

12. Carnot engine: A reversible engine operating between two temperatures – (1) source temperature (T1) and (2) sink temperature (T2) – is called Carnot engine, which consists of two isothermal processes connected by two adiabatic processes. The efficiency of a Carnot engine is given by

η = −1 2

1

TT

.

Chapter 12: Kinetic Theory

1. Newton, Boyle, and many other scientists tried to describe the behavior of gases by considering that gases are made up of very small atomic particles (the size of an atom is about 1 Å = 10–10 m).

2. In solids, the atoms are tightly packed, which are located at a distance of few angstroms (≈2 Å) apart.

3. In liquids, although the distance between the atoms is also approximately 2 Å, the atoms in liquids are not as strongly fixed as in solids, and can move around. This is the reason that the liquids flow.

4. In gases, atoms are located at a distance of tens of angstroms.

5. Boyle’s law: At constant temperature, the volume of a given mass of gas is inversely proportional to pressure, which is expressed as

VP

∝ 1.

6. Charle’s law: When pressure of a gas is constant, the vol-ume of a given mass of gas is directly proportional to its absolute temperature, which is expressed as

VT

= constant.

7. Ideal gas equation is given by

PV = mRT = kBNT,

where m is the number of moles, N is the number of mol-ecules, and R = 8.314 J/mol/K and kB = R/NA = 1.38 × 10–23 J/K. The ideal gas equation is satisfied by real gases only approx-imately that too at low pressures and high temperatures. The gas that follows the ideal gas equation at all possible pressures and volumes is called ideal gas.

8. When a mixture of non-interacting ideal gases with m1 moles of gas 1, m2 of gas 2, and so on, are kept in an enclosed area with volume V, temperature T, and pressure P, the equation of state of mixture is given by

PV = (m1+ m2)RT or P = m1RT/V + m2RT/V + … = P1 + P2 + …,

where P1 = m1RT/V is the pressure the gas 1 that would exert at same V and T if no other gases were present in the enclosure. This phenomenon is called Dalton’s partial pressures, that is, the total pressure of mixture of differ-ent ideal gases is equal to the sum of partial pressures of individual gases of which mixture is made of.

9. For an ideal gas, the relation of kinetic theory is given by

P nmv= 13

2 ,

where n is the density of molecules, m the mass of the molecule, and v 2 is the mean of squared speed. Along with the ideal gas equation, this kinetic theory equation provides the kinetic interpretation of temperature as follows:

12

32

32 21 2

mv k T v vk Tm

= =

=B rms

/B, .

This implies that the temperature of a gas is an amount of the average kinetic energy of a molecule, which is inde-pendent of the nature of the gas or molecule. At a fixed temperature, in a mixture of the gases, heavier molecule has the lower average speed.

10. Law of equipartition of energy is stated as follows: When a system is in equilibrium at absolute tempera-ture T, the total energy is distributed equally in different energy modes of absorption, the energy in each mode being equal to (1/2)kBT. Each translational and rotational



degree of freedom corresponds to one energy mode of absorption and has energy (1/2)kBT. Each vibrational frequency has two modes of energy (kinetic energy and potential energy) with corresponding energy equal to 2 1 2× =/ k T k TB B .

11. The molar specific heat of gases is determined using the law of equipartition of energy and the resultant values are in agreement with the experimental values of specific heats of several gases. The agreement can be improved by including vibrational modes of motion.

12. Translational kinetic energy of the molecules in a gas is given by

E k NT= 32 B ,

which leads to the following equation:

PV E= 23

.

13. The mean free path l is the average distance covered by a molecule between two successive collisions:

l = 12 2n dπ

,

where n is the density and d is the diameter of the molecule.

Chapter 13: Oscillations

1. The SI unit of period is second (s).

2. The total number of repetitions that occur per unit time of a periodic motion is represented by the reciprocal of its period T, which is represented by the symbol n. This quan-tity is called the frequency of the periodic motion:

ν = 1T

.

3. The unit of frequency is s–1 or Hertz (Hz).

4. In a simple harmonic motion, the displacement x(t) of a particle from its equilibrium position is expressed as

x t A t( ) cos( )= +ω φ

where A is the amplitude of the displacement, the quan-tity ( )ω φt + is the phase of the motion, and φ is the phase constant. The angular frequency w is related to the period and frequency of the motion by

ω π πν= =22

T.

5. The SI unit of angular frequency is radians (rad).

6. As functions of time, the particle velocity [n(t)] and accel-eration [a(t)] during SHM, respectively, are expressed as

v t A t

t A t t x t

( ) sin( );

( ) cos( ) ( ) ( ).

= − += − + ⇒ = −

ω ω φα ω ω φ α ω2 2

It is shown that both velocity and acceleration of a body executing SHM motion are periodic functions, having the velocity amplitude v Am = ω and acceleration amplitudea Am ,= ω2 respectively.

7. At any time, a particle executing SHM has potential energy U = (1/2)kx2 and kinetic energy K = (1/2)mn2. If no friction exists, the mechanical energy of the system, that is, E = K + U always remains constant despite the fact that K and U change with time.

8. A particle of mass m oscillating under the influence of a Hooke’s law restoring force given by

F = –kx,

where k is the force constant, exhibits simple harmonic motion with the following angular frequency (w) and period (T ):

ω

π

=

=

km

Tmk

;

.2

9. The unit of force constant is N/m and its dimension is [MT –2].

10. The motion of a simple pendulum moving back and forth through small angles is considered approximately to be simple harmonic and its period of oscillation is expressed as

Tg

= 2π l.

11. In damped oscillations, although the energy of the system is continuously dissipated, the oscillations remain appar-ently periodic.

12. For a SHM, when the damping force is given by

Fd = –bv,

where v is the velocity of the oscillator and b is a damp-ing constant, then the displacement of the oscillator is expressed as

x t A bt m( ) e cos( ’ ),/= +− 2 ω φ

where the angular frequency of the damped oscillator, w ′, is expressed as

ω ’ .= −km

bm

2

24

When the damping constant is small, then ω ω’ ,≈ where w is the angular frequency of the undamped oscillator. The mechanical energy (E) of the damped oscillator is expressed as

E t kA bt m( ) e ./= −12

2



13. When an external force with angular frequency, wd, acts on an oscillating system with natural angular frequency, w, the system oscillates with angular frequency wd.

The amplitude of oscillations is the highest when wd = w, which is a condition known as resonance of the oscillation.

Chapter 14: Waves

1. The displacement relation for a sinusoidal wave propa-gating in the positive x-direction is expressed as

y x t a kx t( , ) sin( ),= − +ω φ

where a is the amplitude of the wave, k is the angular wave number, w is the angular frequency, ( )kx t− +ω φ is the phase, and φ is the phase constant or phase angle.

2. Unit of wavelength is meter (m) and its dimension is [L].

3. The time taken by any element of the medium to move through one complete oscillation is called the period T of oscillation of a wave, which is related to the angular frequency w by

T = 2πω

4. Frequency n of a wave is defined as 1/T, which is related to angular frequency by

ν ωπ

=2

.

5. Speed of a progressive wave:

vk T

= = =ω λ λν.

6. Speed of a transverse wave on a stretched string is set by the properties of the string. The speed on a string with tension T and linear mass density m is expressed as

vT=µ

.

7. Sound wave is a longitudinal mechanical wave which trav-els through solids, liquids, or gases. The speed v of a sound wave in a fluid with bulk modulus (B) and density (r) is expressed as

v = Bρ

.

8. In a metallic bar, the speed of longitudinal waves is given by

vY=ρ

.

9. As B = g P, the speed of sound in gases is expressed as

vP= γ

ρ.

10. Principle of superposition of waves: In the same medium, when two or more waves traverse, the

displacement of any element of the medium is the algebraic sum of the displacements due to each wave:

y f x ttt

n

= −=

∑ ( ).υ1

11. When two sinusoidal waves on the same string show inter-ference, adding or canceling according to the principle of superposition, and if the two are traveling in the same direction and have the same amplitude and frequency but differ in phase by a phase constant φ , the result is a single wave with the same frequency w :

y x t a kx t( , ) cos sin .=

− +

212

12

φ ω φ

In this case, if φ = 0 or an integral multiple of 2p, the waves are exactly in phase and the interference is constructive and if φ π= , they are exactly out of phase and the inter-ference is destructive.

12. A traveling wave, at a rigid boundary or a closed end, is reflected with a phase reversal, but the reflection at an open boundary takes place without any phase change. For an incident wave,

y x t a kx ti ( , ) sin( )= + ω

For the reflected wave at a rigid boundary is

y x t a kx tr ( , ) sin( )= +− ω

For reflection at an open boundary is

y x t a kx tr ( , ) sin( )= + ω

13. Standing waves are produced by the interference of two identical waves moving in opposite directions. For a string with fixed ends, the standing wave is expressed by

y x t a kx t( , ) [ sin ]cos= 2 ω

14. Standing waves are characterized by fixed locations of zero displacement called nodes and fixed locations of maximum displacements called antinodes. The distance between the two consecutive nodes or antinodes is l/2. A stretched string of length L fixed at both the ends vibrates with frequencies given by

ν = =nvL

for n2

1 2 3, , , ,…

in which, the set of frequencies are called the normal modes of oscillation of the system. The oscillation mode with lowest frequency is called the fundamental mode or the first harmonic. The second harmonic is the oscil-lation mode with n = 2, etc. For a pipe of length L with



one end open and other end closed (such as air columns) vibrates with frequencies given by

ν = +

=nvL

for n12 2

0 1 2 3, , , , ,…

in which the set of frequencies are the normal modes of oscillation of such a system. The lowest frequency given by v/4L is the fundamental mode or the first harmonic.

15. When two waves, having slightly different frequencies (n1 and n2) and comparable amplitudes, are superposed, the outcome is called beats; the beat frequency is given by

ν ν νbeat .= −1 2

16. A change in the observed frequency of a wave, when the source and the observer move relative to the medium, is called Doppler effect. For sound, the observed frequency, n, is given in terms of the source frequency, n0 , by

ν ν= ++

0

0v vv vs

,

where v is the speed of sound through the medium, v0 is the velocity of observer relative to the medium, and vs

is the source velocity relative to the medium. In using this formula, velocities in from the direction of observer to source should be treated as positive and those opposite to it should be taken to be negative.


Education

Important Formulas And Units {For Class XII & XI}