Revision Notes on Rotational Motion:-

• Rigid Body:- A rigid body consists of a number of particles confined to a fixed geometrical shape and size in such a way that the distance between any pair of particles always remains constant.
• Cross Product (Vector Product):- Cross product of two vectors  and 1is defined as a single vector  whose magnitude is equal to the product of their individual magnitudes and the sine of the smaller angle between them and is directed along the normal to the plane containing  and .2

  Here  is the unit vector in a direction perpendicular to the plane containing  and .3

• Centripetal force:- The force, acting along the radius towards the center, which is essential to keep the body moving in a circle with uniform speed is called centripetal force. It acts always along the radius towards the center. A centripetal force does no work.

  F = mv²/r = mrω²

• Centrifugal force:- Centrifugal force is the fictitious force which acts on a body, rotating with uniform velocity in a circle, along the radius away from the center. Magnitude of centrifugal force is,

 F = mv²/r

• Centripetal and centrifugal forces are equal in magnitude and opposite in direction. They cannot be termed as action and reaction since action and reaction never act on same body. 

• Moment of Inertia (Rotational Inertia ‘I’)  :- Moment of Inertiaof a body, about a given axis, is defined as the sum of the products of the masses of different particles constituting the body and the square of their distances from the axis of rotation. It depends upon two factors,

(i) Mass of body

(ii) Distribution of mass about the axis of rotation

(iii) Moment of inertia of a body should always be referred to as about a given axis, since it depends upon distribution of mass about that axis.

(iv) It does not depend upon the state of motion of rotating body. It is same whether the body is at rest, rotating slowly or rotating fast about the given axis.

                    I = ∑mr²

• Rotational Kinetic Energy:- Kr = ½Iω² = ½ mr²ω²

            So, I = 2Kr/ω²

• Radius Gyration:- Radius of gyration of a body about a given axis is that distance, at which if whole of the mass of the body were concentrated, it would have same moment of inertia as that of body.

           I = MK²

           So, K = √I/M

Again, Radius of gyration of a body about a given axis is defined as the square root of the mean of the squares of distances of various particles of the body from the axis of rotation.

           So, K = √[r1²+ r2²+ r3²+…./n]

• Center of mass for two particle system:-

          (a) xCM=(m1x1+ m2x2)/(m1+ m2)

          (b)  vCM = (m1v1+ m2v2)/(m1+ m2)

          (c)   aCM = (m1a1+ m2a2)/(m1+  m2)

          (d) vCM = dxCM/dt

          (e) aCM = dvCM/dt = d²xCM/dt²

• System of mass for many particle system:-

           xCM = ∑mixi / ∑mi

• Perpendicular axes theorem:- It states that the moment of inertia of a plane lamina, about an axis perpendicular to the plane lamina, is equal to the sum of the moments of inertia of the lamina about two mutually perpendicular axes lying in the plane of  lamina and intersecting each other at the point where the perpendicular axis passes through the body.

             I = Ix+Iy

• Parallel axes theorem:- It states that moment of inertia of a body, about an axis, is equal to the sum of the moment of inertia of the body about a parallel axis through its center of gravity and the product of the mass of body and the square of the distance between the two axes.

              I = Ig+Mh²

Here, Ig is the moment of inertia of the body about an axis through its center of gravity G.

• Moments of Inertia of Various Objects:-

        

• Moment of inertia of a particle having mass m:-I = mr²
• Moment of inertia  of a thin rod about an axis passing through its center and perpendicular to its length:-

?            I =Ml²/12

• Moment of inertia of a ring about an axis passing through its center and perpendicular to its plane:-

(a) About one of its diameters:- Id = ½ (MR²) 

(b)About a tangent

(i) Tangent lying in the plane of ring:-I = 3/2 (MR²)

(ii) Tangent perpendicular to the plane of ring:-I = 2MR²

• Moment of inertia of a solid disc:-

(a) About an axis passing through its center and perpendicular to its plane:- I = ½ MR²

(b) About one of its diameters:- Id = ¼ (MR²) 

(c) About a tangent:-

(i) Tangent lying in the plane of disc:-I = 5/4 (MR²)

(ii) Tangent perpendicular to the plane of disc:- I = 3/2 (MR²)

• Moment of inertia of an annular disc:-

(a) About an axis passing through the center and perpendicular to the plane:-

(i) For a solid disc:- I = ½ MR²

(ii) For ring:- I = MR²

(b) About any of its diameter:-

(i) For a solid disc:- Id = ¼ (MR²) 

(ii) For ring:- Id = ½ MR²

(c)  About a tangent:-

(i) Tangent lying in the plane of disc:-

(1)For a solid disc:-I = 5/4 MR²

(2)For a ring:-I = 3/2 MR²

(ii) Tangent perpendicular to the plane of the disc:-

(1) For a solid disc:- I = 3/2 MR²

(2) For a Ring:- I = 2MR²

• Torque (τ) in vector form:-

                

• Relation Between Moment of inertia (I) and Torque (?) [Roatational Law of Accelaeration]:-

?? = Iα

Here α is the angular acceleration.

• Newton’s law in rotational motion:-

(a) First Law:- It states that everybody continues in its state of rest or of uniform rotational motion about a given axis unless it is completed by some external unbalanced torque to change that state.

(b) Second Law:- It states that the rate of change of angular momentum of a body is directly proportional to the impressed torque and takes place in the direction of torque. Mathematically,  = Iα.

(c) Third Law:- It states that to every torque there is an equal and opposite torque.

• Angular Momentum (L):-

           

• Moment of Inertia (I) and Angular momentum (L):-

            

• Law of conservation of angular momentum:- The net angular momentum of an isolated system (no external torque), always remains constant.?

                

                 I1ω1=I2ω2

• Motion of a point mass attached to a string would over a cylinder capable of rotating about its axis of symmetry:-

              Tension, T = mg/[1+(mR²/I)]

• Motion of a body rolling down an inclined plane without slipping:-

The maximum allowed angle for rolling without slipping.

(a) A cylinder rolling down the plane,

θmax = tan^-1(3µ)

(b) A sphere rolling down the inclined plane,

θmax = tan^-1[7/2 (µ)]

(c) A ring rolling down the inclined plane,

θmax = tan^-1(2µ)

Here µ is the coefficient of sliding friction.

• Angular impulse:- 
• Rotational work done:-  W =τ?θ =τavθ
• Rotational power:-  
• Kinetic Friction:-Kinetic friction is the force of friction which comes into play between two surfaces when there is some relative motion between them. The magnitude of force of kinetic friction fk (kinetic frictional force) is proportional to the normal force N.

?So fk = µk N

Here µk is the coefficient of kinetic friction.

Thus, coefficient of kinetic friction, µk = fk/N  

• Rolling Friction:-Force of friction which comes into play, between two surfaces, while one is rolling over the other is called rolling friction.  Rolling friction is similar to kinetic friction.

So,fr = µr N

Here µr is the coefficient of rolling friction and N is the normal force.

Thus, coefficient of rolling friction, µr = fr/N  

for notes of thermal property: 1

for notes on laws of motion  :2

Revision Notes on Gravitation and Projectile: 3
Revision Notes on Circle:4
straight lines revision notes:5
hydrocarbons revison notes :6
some basic concepts of chemstry revision notes:7
si units  : 8
M.s dhoni great captian: 9
virat kholi is a great cricketer:10

   written by arpan ruhil

Revision Notes on Thermal Properties of Matter:-

• Heat:- Heat is the agent which produces in us the sensation of warmth and makes bodies hot. It is form of energy. The part of thermal energy which flows from one body to the other due to temperature difference is called heat.

• Nature of heat:-

(a) The weight of a body remains the same weather it is heated or cooled.

(b) Heat flows from higher to lower temperature

(c) In any exchange of heat, heat lost by the hot body is equal to the heat gained by the cold body.

(d) Substances generally expand when heated

(e) A certain amount of heat known as latent heat is required to change the state of a body from solid to liquid or from liquid to gas without any change in temperature.

• Thermal Energy:- In accordance to dynamical theory of heat the sum total of translational, vibrational and rotational energies of the molecules of a system is called the thermal energy of the system .
• Unit of Heat:-

(a) Calorie (cal):- It is the amount of heat required to raise the temperature of 1 gram of water through 1ºC.

(b) Kilocalorie (kcal):- It is the amount of heat required to raise the temperature of 1 kilo gram of water through 1ºC.

• Temperature:- It is defined as the degree of hotness of a body.

• Zeroth Law of Thermodynamics:-

?It states that the two systems (A and B) which are separately in equilibrium with a third system (C) must also be in equilibrium with each other.

• Absoluter Zero of Temperature:-

(a) Charle’s law:- Vt = V0(1+ t/273)

(b) Gay Lussac’s law:- Pt = P0(1+ t/273)

(c) Absolute zero of temperature is defined as the temperature at which a gas has zero volume and exerts zero pressure. It is that temperature at which molecular motion ceases.

(d) C∝√T, C = √[c1² + c2² +…….+ cn²]/n

• Absolute gas scale or absolute scale of temperature:- It is that scale of temperature whose zero (i.e. 0ºK) = -273ºC

        A centigrade degree is exactly equal to the absolute or Kelvin’s degree.

• Conversion of temperature from one scale to another:-

C/100 = (K-273)/100 = (F-32)/180 = Re/80 = (Rα-492)/180

Here C, K, F, Re and Rα are respectively, the temperatures of same both on centigrade, Kelvin, Fahrenheit, Reaumer and Rankin scale, respectively.

• F = [(9/5)C ]+32
• K = C+273
  
• Linear  Expansion (longitudinal expansion):- 

When the expansion due to heating takes place only along one direction, the expansion is said to be one dimensional and linear.

• Coefficient of linear expansion (α):- Coefficient of linear expansion of the material of a rod is defined as the change in length per unit length, at 0^ºC, per degree centigrade rise of temperature.

α = lt-l0/l0t

• Expansion in two dimensions (Superficial expansion):-

?When the thermal expansion of a body is confined to a plane, it is to be two dimensional expansion or superficial expansion.

• Coefficient of superficial expansion (β):- It is defined as the change in area of the surface per unit area at 0ºC, per degree centigrade rise of temperature.

β = St-S0/S0t

• Expansion in three dimensions (Cubical expansion/volume expansion):- When thermal expansion of the body takes place in space, it is said to be three dimensional expansion or cubical expansion.

• Coefficient of cubical expansion (γ):- Coefficient of cubical expansion is defined as the change in volume per unit volume, at 0ºC, per degree celsius rise of temperature .

γ = Vt-V0/V0t

• Relation between expansion coefficients:-

(a) Relation between α and β:-

β = 2α

(b) Relation between α and γ:-

γ= 3α

(c) Relation between β and γ:-

γ = 3/2 β

(d) α : β : γ = 1:2:3

• Thermal expansion of liquids:-

(a) Co-efficient of apparent expansion (γa):- The coefficient of apparent expansion of a liquid is defined as the apparent  (or observed) increase in volume, per unit volume of the liquid at 0ºC per degree celcius rise of temperature.

γa = apparent increase in volume/(original volume at 0ºC) × (rise of temperature)

(b) Co-efficient of real expansion (γr):- The coefficient of real expansion of a liquid is defined as the real increase in volume, per unit volume of the liquid at 0ºC per degree centigrade rise of temperature.

 γa= real increase in volume/(original volume at 0ºC) × (rise of temperature)

• Work and Heat:-

Whenever heat is conserved into work or work into heat, the quantity of energy disappearing in one form is equivalent to the quantity of energy appearing in the order.

W∝H  or W = JH

Joule’s mechanical equivalent of heat is defined as the amount of work required to produce a unit quantity of heat.

J = W/H

Value of J:- J = 4.2×10⁷ erg cal^-1 = 4.2 J cal^-1

• Specific heat capacity or specific heat (c):-

Specific heat capacity of a material is defined as the amount of heat required to raise the temperature of a unit mass of material through 1ºC.

c = Q/mΔT

Unit:- kcal kg^-1K^-1 or J kg^-1K^-1

Dimension:- M⁰L²T^-2K^-1

• Molar specific heat capacity(C):-

? Molar specific heat capacity of a substance is defined as the amount of heat required to raise the temperature of one gram molecule of the substance through one degree centigrade.

(a) C = Mc  (Here M is the molecular weight of the substance)

(b) C = 1/n (dQ/dT)

• Heat Capacity or Thermal Capacity:-

It is defined as the amount of heat required to raise the temperature of body through 1ºC.

Q = mcΔT

If ΔT = 1ºC, Q = heat capacity = mc

Unit:- kcal K^-1 or JK^-1

• Water Equivalent:-

Water equivalent of a body is defined as the mass of water which gets heated through certain range of temperature by the amount of heat required to raise the temperature of body through same range of temperature.

w = mc

Water equivalent of a body is equal to the product of its mass and its specific heat.

• Latent Heat:- When the state of matter changes, the heat absorbed or evolved is given by: Q = mL. Here L is called the latent heat.

(a)Specific latent heat of fusion (Lf):-

Specific latent heat of fusion of a substance is defined as the amount of heat required to convert 1 gram of substance from solid to liquid state, at the melting point, without any change of temperature.

(b) Specific latent heat of vaporization (Lv):-

Specific latent heat of vaporization of a substance is defined as the amount of heat required to convert 1 gram of liquid into its vapours at its boiling point without any rise of temperature.

Dimensional formula:- M⁰L²T^-2

Unit:- kg cal kg^-1  or J kg^-1

• Triple point of water = 273.16 K
• Absolute zero = 0 K = -273.15ºC
• For a gas thermometer, T = (273.15) P/Ptriple (Kelvin)
• For a resistance thermometer, Re = R0[1+αθ]

S. No.	Physical quantity	Symbol	Dimensions	Units
				SI	CGS
1.	Heat	Q	[ML2T–2]	Joule	calorie
2.	Specific–heat	c	[L2T–2q–1]	J/kg K	cal/gm C°
3.	Molar sp. heat	C	[ML2T–2q–1m–1]	J/mol K	cal/mol C°
4.	Latent heat	L	[L2T–2]	J/kg	cal/gm
5.	Thermal capacity	Tc	[ML2T–2q–1]	J/K	cal/C°
6.	Water–equivalent	W	[M]	kg	gm

  

written by arpan ruhil