I. FRICTION
When a body is in motion on a rough surface, or when an object moves through water (i.e., viscous medium), then velocity of the body decreases constantly even if no external force is applied on the body. This is due to friction.
So “an opposing force which comes into existence, when two surfaces are in contact with each other and try to move relative to one another, is called friction”.
Frictional force acts along the common surface between the two bodies in such a direction so as to oppose the relative movement of the two bodies.
(a) The force of static friction fs between book and rough surface is opposite to the applied external force Fext. The force of static friction fs = Fext
r . Book R=N W (a) fs Fext. (b) When Fext r
. exceeds the certain maximum value of static friction, the book starts accelerating and during motion Kinetic frictional force is present.
Book R=N
W (b)
fk Fext.
Body just starts moving
(c) A graph Fext r
. versus | f | shown in figure. It is clear that fs, ,max > fk
=m Ns
fk=m Nk Body starts with acceleration Body is at rest static region kinetic region O |f| (c) (f )s max
IMPORTANT CONCEPTS OF PHYSICS for AIPMT
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Fig.(a) shows a book on a horizontal rough surface. Now if we apply external force Fext.
r
, on the book, then the book will remain stationary if Frext. is not too large. If we increase
. ext
Fr then frictional force f also increase up to (f )s max (called maximum force of static friction or limiting friction) and
s max
(f ) = msN. At any instant when Frext. is slightly greater than (f )s maxthen the book moves and accelerates to the right.
Fig.(b) when the book is in motion, the retarding frictional force become less than, (f )s max
Fig.(c) (f )s maxis equal to mkN. When the book is in motion, we call the retarding frictional force as the force of kinetic friction fk.
Since fk<(f )s max, so it is clear that, we require more force to start motion than to maintain it against friction.
By experiment one can find that (f )s maxand fk are proportional to normal force N acting on the book (by rough surface) and depends on the roughness of the two surfaces in contact.
Note :
(i) The force of static friction between any two surfaces in contact is opposite to Fext.
r
and given byfs£msN and
s max s
(f ) = m N (when the body just moves in the right direction).
where N = W = weight of book and ms is called coefficient of static friction, fs is called force of static friction and (f )s max is called limiting friction or maximum value of static friction.
(ii) The force of kinetic friction is opposite to the direction of motion and is given by fk = mkN
where mk is coefficient of kinetic friction.
(iii) The value of mk and ms depends on the nature of surfaces and mk is always less then ms.
Friction on an inclined plane : Now we consider a book on an inclined plane & it just moves or slips, then by definition
mg cos q mg=W q m gs in q q Bo max ( )fs = msR
Now from figure, fs,max =mgsinq and R = mg cosq Þ ms= tanq or q = tan–1(m
s)
where angle q is called the angle of friction or angle of repose Some facts about friction :
(1) The force of kinetic friction is less than the force of static friction and the force of rolling friction is less than force of kinetic friction i.e.,
fr < fk < fs or mrolling < mkinetic < mstatic hence it is easy to roll the drum in comparison to sliding it. (2) Frictional force does not oppose the motion in all cases,
infact in some cases the body moves due to it.
A B
Fext
In the figure, book B moves to the right due to friction between A and B. If book A is totally smooth (i.e., frictionless) then book B does not move to the right. This is because of no force applies on the book B in the right direction. Laws of limiting friction :
(i) The force of friction is independent of area of surfaces in contact and relative velocity between them (if it is not too high).
(ii) The force of friction depends on the nature of material of surfaces in contact (i.e., force of adhesion). m depends upon nature of the surface. It is independent of the normal reaction.
(iii) The force of friction is directly proportional to normal reaction i.e., F µ N or F = mn.
While solving a problem having friction involved, follow the given methodology
Check (a) Fapp (b) Limiting friction (f )l If F < fapp app l
Body does not move and F = frictional force If F = f
Body is on the verge of movement if the body is initially at rest Body moves with constant velocity
app l
Rolling Friction :
The name rolling friction is a misnomer. Rolling friction has nothing
Cause of rolling friction : When a body is kept on a surface of another body it causes a depression (an exaggerated view shown in the figure). When the body moves, it has to overcome the depression. This is the cause of rolling friction.
Rolling friction will be zero only when both the bodies incontact are rigid. Rolling friction is very small as compared to sliding friction. Work done by rolling friction is zero
II. THERMAL EQUILIBRIUM AND LAW OF THERMODYNAMICS
Thermal Equilibrium
Two systems are said to be in thermal equilibrium with each other if they have the same temperature.
Zeroth Law of Thermodynamics
If objects A and B are separately in thermal equilibrium with a third object C then objects A and B are in thermal equilibrium with each other.
FIRST LAW OF THERMODYNAMICS
First law of thermodynamics gives a relationship between heat, work and internal energy.
(a) Heat : It is the energy which is transferred from a system to surrounding or vice-versa due to temperature difference between system and surroundings.
(i) It is a macroscopic quantity.
(ii) It is path dependent i.e., it is not point function. (iii) If system liberates heat, then by sign convention it is
taken negative, If system absorbs heat, it is positive. (b) Work : It is the energy that is transmitted from one system
to another by a force moving its points of application. The expression of work done on a gas or by a gas is
2 1
V V
W =
ò
dW=ò
PdVwhere V1 is volume of gas in initial state and V2 in final state. (i) It is also macroscopic and path dependent function. (ii) By sign convention it is +ive if system does work (i.e.,
expands against surrounding) and it is – ive, if work is done on system (i.e., contracts).
(iii) In cyclic process the work done is equal to area under the cycle and is negative if cycle is anti-clockwise and +ive if cycle is clockwise (shown in fig.(a) and (b)).
3
•
(c) Internal energy : The internal energy of a gas is sum of internal energy due to moleculer motion (called internal kinetic energy UK) and internal energy due to molecular configuration (called internal potential energy UP.E.) i.e., U = UK + UP.E. ……(1)
(i) In ideal gas, as there is no intermolecular attraction, hence 3 2 K n U=U = RT ……(2)
(for n mole of ideal gas)
(ii) Internal energy is path independent i.e., point function. (iii) In cyclic process, there is no change in internal energy
(shown in fig.)
i.e., dU = Uf – Ui = 0
Þ Uf = Ui
(iv) Internal energy of an ideal gas depends only on temperature eq.(2).
First law of thermodynamics is a generalisation of the law of conservation of energy that includes possible change in internal energy.
First law of thermodynamics “If certain quantity of heat dQ is added to a system, a part of it is used in increasing the internal energy by dU and a part is use in performing external work done dW
i.e., dQ dU dW= + ÞdU =dQ dW
-The quantity dU (i.e., dQ – dW) is path independent but dQ and dW individually are not path independent.
Applications of First Law of Thermodynamics (i) In isobaric process P is constant
so =
ò
2 = -1 V V PdV P(V2 V1) dW so dQ = dU + dW = n CP dT(ii) In cyclic process heat given to the system is equal to work done (area of cycle).
(iii) In isothermal process temperature T is constant and work done is 1 2 V V e V V Log nRT PdV dW 2 1
ò
= =Since, T = constant so for ideal gas dU = 0
Hence, 1 2 e V V Log nRT dW
dQ= = (for ideal gas)
(iv) In isochoric process W = 0 as V = constant
It means that heat given to system is used in increasing internal energy of the gas.
(v) In adiabatic process heat given or taken by system from surrounding is zero i.e., dQ = 0
(
1 2)
nR dU dW T T 1 é ù = - = -ê - ú g -ë û 1 1 2 2 (P V P V ) 1 é - ù = ê g - ú ë ûIt means that if system expands dW is +ive and dU is –ive (i.e., temperature decrease) and if system contracts dW is –ive and dU is +ive (i.e., temperature increase).
THERMODYNAMIC PROCESSES
(i) Isothermal process : If a thermodynamic system is perfectly conducting to surroundings and undergoes a physical change in such a way that temperature remains constant throughout, then process is said to be isothermal process.
T = constant
V P
For isothermal process, the equation of state is PV = nRT = constant, where n is no. of moles.
For ideal gas, since internal energy depends only on temperature. 2 2 1 1 0 V V V V dV dU dQ dW PdV nRT V = Þ = =
ò
=ò
or 2 2 10 1 1 loge 2.303 log V V dQ nRT nRT V V = =(ii) Adiabatic process : If system is completely isolated from the surroundings so that no heat flows in or out of it, then any change that the system undergoes is called an adiabatic process.
V P
For ideal gas, dQ = 0
dU = mCVdT (for any process)
2 2 1 1 V V V V K dW PdV dV Vg =
ò
=ò
(where PVg = K = constant) 2 2 1 1 1 1 2 1 ( ) 1 1 1 1 PV PV K Vg Vg g - - g æ ö -= ç - ÷= - è ø-(iii) Isobaric process : A process taking place at constant pressure is called an isobaric process. In this process dQ = n CpdT, dU = n CVdT and dW = P(V2–V1)
(iv) Isochoric process : A process taking place at constant volume is called isochoric process.
In this process, dQ = dU =n CVdT and dW = 0
(v) Cyclic process : In this process the inital state and final state after traversing a cycle (shown in fig.) are same. In cyclic process, dU = 0 = Uf – Ui and dW = area of cycle = area (abcd)
III. STATIONARY OR STANDING WAVES
When two progressive waves having the same amplitude, velocity and time period but travelling in opposite directions superimpose, then stationary wave is produced.
Let two waves of same amplitude and frequency travel in opposite direction at same speed, then
y1 = A sin (wt –kx) and y2 = A sin (wt + kx)
By principle of superposition
y = y1 + y2 = (2A cos kx) sin wt ...(i) s
y A sin ωt=
It is clear that amplitude of stationary wave As vary with position (a) As = 0, when cos kx = 0 i.e., kx = p/2, 3p/2...
i.e., x = l/4, 3l/4...[as k = 2p/l]
These points are called nodes and spacing between two nodes is l/2.
(b) As is maximum, when cos kx is max i.e., kx = 0, p , 2p, 3p i.e., x = 0, ll/2, 2l/2....
It is clear that antinode (where As is maximum) are also equally spaced with spacing l/2.
(c) The distance between node and antinode is l/4 (see figure)
Antinode Antinode
segment 1 segment 2 segment 3
Node x l/2 l/4 o 2A Keep in Memory
1. When a string vibrates in one segment, the sound produced is called fundamental note. The string is said to vibrate in fundamental mode.
l 2
3. If the fundamental frequency be n then 20 n , 30 n , 40 n ...0 are respectively called second third, fourth ... harmonics respectively.
4. If an instrument produces notes of frequencies 4
3 2 1,n ,n ,n
n .... where ν1<ν2 <ν3 <ν4..., then n is2 called first overtone, n is called second overtone, 3 n is4 called third overtone ... so on.
5. Harmonics are the integral multiples of the fundamental frequency. If n0 be the fundamental frequency, then nn is0 the frequency of nth harmonic.
6. Overtones are the notes of frequency higher than the fundamental frequency actually produced by the instrument. 7. In the strings all harmonics are produced.
Stationary Waves in an Organ Pipe :
In the open organ pipe all the harmonics are produced.
In an open organ pipe, the fundamental frequency or first harmonic is 0
2
v n =
l, where v is velocity of sound and l is the
length of air column [see fig. (a)] (a) l (b) l 2 l = l , 1 2l = l 2 2l = l , 2 2l = l (c) l 2 3l = l , 3 L 3 = l
Similarly the frequency of second harmonic or first overtone is [see fig (b)], 01 2
2
v
n =
l
Similarly the frequency of third harmonic and second overtone is, [(see fig. (c)] n02 = 32vl Similarly 03 4 , 04 5 2 2 v v n = n = l l...
In the closed organ pipe only the odd harmonics are produced. In a closed organ pipe, the fundamental frequency (or first harmonic) is (see fig. a)
4 c v n = l (a) l (b) l (c) l 4 l = l 4 3l = l 4 5l = l
Similarly the frequency of third harmonic or first overtone (IInd
harmonic absent) is (see fig. b)
l 4 v 3 nc2= Similarly l l 4 v 7 n , 4 v 5 nc3= c4= ... End Correction
It is observed that the antinode actually occurs a little above the open end. A correction is applied for this which is known as end correction and is denoted by e.
(i) For closed organ pipe : l is replaced by l+ e where e = 0.3D, D is the diameter of the tube.
(ii) For open organ pipe: l is replaced by l + 2e where e = 0.3D
In resonance tube, the velocity of sound in air given by
(
2 1)
2ν v= l -l
where n = frequency of tuning fork, ll = 1st resonating length,
l2 = 2nd resonating length.
IV. OHM’S LAW AND ELECTRICAL RESISTANCE When a potential difference is applied across the ends of a conductor, a current I is set up in the conductor.
According to Ohm’s law “Keeping the given physical conditions such as temperature, mechanical strain etc. constant, the current (I) produced in the conductor is directly proportional to the potential difference (V) applied across the conductor”. i.e., I µV or I =KV ... (1) where K is a constant of proportionality called the conductance of the given conductor.
Alternatively, V µI or V = RI ... (2) where the constant R is called the electrical resistance or simply resistance of the given conductor.
From above two eqs. it is clear that R = 1/K.
If a substance follows Ohm’s law, then a linear relationship exists between V & I as shown by figure 1. These substance are called Ohmic substance. Some substances do not follow Ohm’s law, these are called non-ohmic substance (shown by figure 2) Diode valve, triode valve and electrolytes, thermistors are some examples of non-ohmic conductors.
I V Ohmic conductor or linear conductor Fig. 1 q Non-linear conductor or non-ohmic conductor Fig. 2 V I
Slope of V-I Curve of a conductor provides the resistance of the conductor
slope = tan q =V I
The SI unit of resistance R is volt/ampere = ohm (W) Electrical Resistance
On application of potential difference across the ends of a conductor, the free e–s of the conductor starts drifting towards
the positive end of the conductor. While drifting they make collisions with the ions/atoms of the conductor & hence their motion is obstructed. The net hindrance offered by a conductor to the flow of free e–s or simply current is called electrical resistance.
It depends upon the size, geometry, temperature and nature of the conductor.
Resistivity : For a given conductor of uniform cross-section A and length l, the electrical resistance R is directly proportional to length l and inversely proportional to cross-sectional area A
i.e., Rµ l
A or R= lA
r or RA ρ =
l
r is called specific resistance or electrical resistivity. Also, = m2
ne
r t
The SI unit of resistivity is ohm - m.
Conductivity(s) : It is the reciprocal of resistivity i.e. s = 1 r . The SI unit of conductivity is Ohm–1m–1 or mho/m.
Ohm’s law may also be expressed as, J = sE
where J = current density and E = electric field strength Conductivity, ne2
m t
s = where n is free electron density, t is relaxation time and m is mass of electron.
(i) The value of r is very low for conductor, very high for insulators & alloys, and in between those of conductors & insulators for semiconductors.
(ii) Resistance is the property of object while resistivity is the property of material.
(in W m) (i) Silver 1.6 × 10–8 (ii) Copper 1.7 × 10–8 (iii) Aluminium 2.8 × 10–8 (iv) Tungsten 5.2 × 10–8 (v) Platinum 10.6 × 10–8 (vi) Manganin 42 × 10–8 (vii) Carbon 35 × 10–6 (viii) Germanium .46 (ix) Silicon 2300 (x) Glass ~ 1013 (xi) Mica ~ 2 × 1015 COMMON DEFAULT Since R R A = rl Þ µl
It is incorrect to think that if the length of a resistor is doubled its resistance will become twice.
If you look by an eye of physicist you will find that when l change, A will also change. This is discussed in the following article. Case of Reshaping a Resistor
On reshaping, volume of a material is constant. i.e., Initial volume = final volume
or, Ai li = Af lf ... (i) where li, Ai are initial length and area of cross-section of resistor
and lf, Af are final length and area of cross-section of resistor.
If initial resistance before reshaping is Ri and final resistance after reshaping is Rf then i i i i f f f f i f R A A R A A r = = ´ r l l l l ... (ii)
From eqs. (i) and (ii), 2
2 f i f i R R R l l l Þ µ ÷÷ ø ö çç è æ =
This means that resistance is proportional to the square of the length during reshaping of a resistor wire.
Also from eqs. (i) and (ii), 2
2 i f f i A 1 R A A R R Þ µ ÷÷ ø ö çç è æ =
This means that resistance is inversely proportional to the square of the area of cross-section during reshaping of resistor. Since A = p r2 (for circular cross-section)
4 r
1 Rµ \ where r is radius of cross section.
Effect of Temperature on Resistance and Resistivity Resistance of a conductor is given by Rt = R0 (1 + aDt)
Where a = temperature coefficient of resistance and Dt = change in temperature
where a = temperature coefficient of resistivity and
where DT = t2 – t1 = change in temperature
The value of a is positive for all metallic conductors. \ r2 > r1
In other words, with rise in temperature, the positive ions of the metal vibrate with higher amplitude and these obstruct the path of electrons more frequently. Due to this the mean path decreases and the relaxation time also decreases. This leads to increase in resistivity.
Please note that the value of a for most of the metals is 1 K 1 273
-For alloys : In case of alloys, the rate at which the resistance changes with temperature is less as compared to pure metals. For example, an alloy manganin has a resistance which is 30-40 times that of copper for the same dimensions.
Also the value of a for manganin is very small » 0.00001°C–1. Due
to the above properties manganin is used in preparing wires for standard resistance (heaters), resistance boxes etc.
Please note that eureka and constantan are other alloys for which r is high. These are used to detect small temperature, protect picture tube/ windings of generators, transformers etc.
For semiconductors : The resistivity of semi-conductors decreases with rise in temperature. For semi conductor the value of a is negative. 2 m ne r = t
With rise in temperature, the value of n increases. Please note that
t decreases with rise in temperature. But the value of increase in n is dominating for the value of r in this case.
For electrolytes : The resistivity decreases with rise in temperature. This is because the viscosity of electrolyte decreases with increase in temperature so that ions get more freedom to move.
For insulators : The resistivity increases nearly exponentially with decrease in temperature. Conductivity of insulators is almost zero at 0 K.
Superconductors : There are certain materials for which the resistance becomes zero below a certain temperature. This temperature is called the critical temperature. Below critical temperature the material offers no resistance to the flow of e–s.
The material in this case is called a superconductor. The reason for super conductivity is that the electrons in superconductors are not mutually independent but are mutually coherent. This coherent cloud of e–s makes no collision with the ions of
super-conductor and hence no resistance is offered to the flow of e–s
For example, R = 0 for Hg at 4.2 K and R = 0 for Pb at 7.2 K. These substances are called superconductors at that critical temperature. Superconductors are used (a) in making very strong electromagnets, (b) to produce very high speed computers (c) in transmission of electric power (d) in the study of high energy particle physics and material science.
