Indirect theory notes tủ tài liệu training

INDIRECT THEORY NOTES Dimensions and Units A quantity can have unit without dimensions e.g angle and solid angle But a quantity cannot have dimensions without units Some pairs which have same dimensions : Torque and work (b) Angular momentum and Planck’s constant (c) Latent heat and gravitational potential (d) Momentum and impulse One and two dimensional motion: If two ends of a train accelerating uniformly crosses an observer with a velocity v1 and v2, then the middle point of the train will cross the observer with a velocity v = v1 + v 2 If a body travels two equal distances with different speeds v1 and v2, its average speed is v(average) 2v1 v = v1 + v If a body travels three equal distances with speeds v1, v2 and v3, its average speed is v(average) = 3v v v v1 v + v v + v v1 If a body falls freely, the distance covered by it in each subsequent second starting from first second will be in the ratio 1:3:5:7 etc That is S1:S2:S3: Sn = 1:3:5:7 (2n-1) This means that the distance travelled during the nth second Sn is proportional to 2n-1 If a body is thrown vertically up with an initial velocity u, it takes u/g second to reach maximum height and u/g second to return, if air resistance is negligible The total time body remains in air when it is thrown vertically up with a velocity u is equal to The maximum height reached by a body projected vertically up with an initial velocity u is 10 If air resistance acting on a body is considered, the time taken by the body to reach maximum height is less than the time to fall back the same height 2u g u2 2g ⎡θ⎤ 11 Two vectors of equal magnitude F inclined at an angle θ, has a vector sum of magnitude 2Fcos ⎢ ⎥ ⎣2⎦ Two vectors of equal magnitude F inclined at an angle θ, has a vector difference of magnitude is ⎡θ⎤ 2Fsin ⎢ ⎥ ⎣2⎦ 12 If two equal vectors have a vector sum of magnitude equal to either of them, the angle between them is 1200 13 If two equal vectors have a vector difference of magnitude equal to either of them, the angle between them is 600 14 If magnitude of sum of two vectors is equal to magnitude of their difference, the angle between them is 90o 15 If T is the time of flight, h maximum height, R horizontal range of a projectile, α its angle of projection, then the relations among these quantities h= gT (1) gT2= 2R tanα (2) Rtanα = 4h (3) The equation (1) is true for a body projected vertically up and also projected at an angle Physics for IIT-JEE Screening Test 16 For a given initial velocity, to get the same horizontal range, there are two angles of projection α and 90-α 17 The equation to the parabola traced by a body projected horizontally from the top of a tower of height y, with a velocity u is y=gx2/2u2, where x is the horizontal distance covered by it from the foot of the tower 18 In a uniform circular motion, velocity and acceleration are constants only in magnitude Their directions change 19 In a uniform circular motion, the kinetic energy of the body is a constant 20 In a uniform circular motion, the work done by centripetal force in moving from any point in the circle to any other point will be zero The centripetal force F is along the radius and the displacement s (velocity) is along the tangent This makes W =F.s = Fscos900 = 21 The minimum velocity for a body at the lowest point to complete a vertical circle of radius r is 5rg The minimum velocity at highest point then is rg 22 The difference in tension at the highest and the lowest point in a vertical circle is 6mg, i.e the weight of the body times Force and motion 23 Newton’s second law is general The first and third laws can be deduced from it 24 Not all forces produce acceleration Only unbalanced forces 25 The weight of a body W = mg, but is measured by the resistance, reaction or tension If the resistance, reaction or tension becomes zero, the body feels weightlessness When a body falls freely, only force of gravity acts on it There is no resistance, reaction or tension Hence, the body feels weightlessness actual weight − apparent weight , where the weights is in N If ‘a’ is mass positive lift is moving down, and if it is negative the lift is moving up 26 The acceleration of a lift a = 27 A force is pseudo if (1) We cannot know where it is coming from, (2) It cannot be classified into one of the four forces in nature and (3) The force disappears when we change the frame of reference (observer) 28 The frictional force acting on a body on an inclined plane is µsmgcosα , where α is the angle of the plane 29 Acceleration of a body when it slides down a rough inclined plane = g sinα - µs gcosα 30 Retardation produced by friction on a car in horizontal motion = µsg 31 The maximum safe speed of a cyclist in a curve of a radius r is µ s rg 32 A car overturns in a curve of radius r, if its speed is greater than arg/ h , where 2a is the distance between wheels, h the height of centre of gravity from road Work, Power, Energy 33 If E is kinetic energy and p momentum of a body, then p2 = 2mE 34 When kinetic energy of a body is made n times, its momentum increases to √n times 35 When momentum of a body is made n times, its kinetic energy increases to n2 times 36 If a light body and heavy body have equal momentum, then light body has greater kinetic energy 37 If a light body and a heavy body have equal kinetic energy, then heavy body has greater momentum 38 If a body moves with constant power, distance travelled by it (s) in a time (t) is related by the equation s ∝ t3/2 39 If a body moves with constant power, its velocity (v) is related to distance travelled (x) by the formula v∝ x1/3 Physics at a Glance 40 When water flows through a uniform tube with a constant velocity v, the power to maintain the flow is proportional to cube of velocity Rotational motion and Moment of inertia 41 A rigid body can rotate about a number of axes Its moment of inertia is minimum, when the axis passes through centre of mass 42 The angular displacement of a body (which rotates with uniform angular acceleration) in the 1st , 2nd , 3rd second etc will be in the ratio 1:3:5:7 2n-1 This means θn ∝ 2n-1 43 The acceleration of a body rolling down an inclined plane of angle θ is g sin θ + (k / r ) , where k is radius of gyration and r the radius of the body 44 The kinetic energy of rotation Iω = τθ, where τ is torque and θ the angular displacement 45 The power spent to rotate a body with a constant angular velocity ω is P = τω, where τ is torque 46 If L is angular momentum, E kinetic energy of rotation, I moment of inertia, then L2 = 2EI 47 For a rigid body of radius r and radius of gyration k, rolling forward, if E is the total energy, R kinetic energy of rotation, T kinetic energy of translation, the following equations hold T r2 = 2 (1), E k +r R k2 = 2 (2), E k +r T r2 = R k (3) Gravitation 48 The gravitational field due to a solid sphere of radius R, at a point distant r is proportional to r2 for r>R (i.e for point outside) It is proportional to r, for r 1) If it is L immersed slightly and released, it will execute simple harmonic motion of period T = T = 2π ng ⎡ dg dL ⎤ 137 The loss or gain of a seconds pendulum ( or clock) in a dT = 43200 ⎢ − ⎥ where dL is the L⎦ ⎣g increase in length, dg is increase in value of acceleration due to gravity g Give + sign to dg or dL if it is increase, - sign if it is decrease If the answer carries positive sign, it is gain If it carries negative sign it is loss Electrostatics/Preliminaries 138 Charge is a scalar It is added algebraically (with positive and negative signs) 139 Coulomb’s law is obeyed for all distances except that in nuclear range Nuclear forces not obey inverse square (Coulomb’s) law 140 The coulomb force between two point charges depends only on the charges, their separation and the medium It is independent of other charges present 141 The dielectric constant of a conductor is infinite 142 ε has two units C2N-1m-2 and Fm-1 Its value is 8.85x10-12 and for rough calculations it can be taken as x10-12 143 The charge of electron in SI is equal to -1.6x10-19 C In CGS electrostatic units (esu) it is equal to -4.8x10-10 In CGS electromagnetic units (emu) it is equal to -1.6x10-20 144 The specific charge of electron is nearly 1.7x1011 C/kg Electrostatic field 145 The electrostatic field has two units NC-1 and Vm-1 146 The number of lines of force coming out of a unit positive charge is = 1.11 x 1011 ε0 147 If a charge q coulomb is placed at the centre of cube of side a, the number of lines of force coming out through one side of a cube is q/6ε0 This follows from Gauss’s theorem 148 If a cube is placed in uniform electric field the net flux through it will be zero This also follows from Gauss’s theorem 149 The electric field (E) due to a line of charge is proportional to 150 The electric field (E) due to a point charge is proportional to r r2 Physics at a Glance 151 The electric field (E) due to a dipole is proportional to r3 152 The electric field (E) due to a uniformly charged flat sheet is constant at all points This means it does not depend on distance It is proportional to r0 153 The electric field (E) due to a charged solid sphere at any point inside the sphere is directly proportional to r, where r is the distance from its centre 154 The electric field is uniform in a region; if (a) the number of lines of force crossing unit area normally, is same at all points and (b) the lines of force are parallel The first condition (a) makes the magnitude of the field to be the same, while the second condition (b) makes the direction of the field to be the same at all points 155 To find the direction of electric field at a point, imagine a unit positive charge at the point Find the magnitude of force on it This gives the magnitude of field Find the direction of motion of that charge This gives the direction of electric field Electrostatic potential 156 Electric potential (V) due to a point charge at a distance r is proportional to 157 Electric potential (V) due to a dipole at a distance r is proportional r r2 158 In a uniform electric field a dipole experiences only a torque In a non uniform electric field a dipole experiences both a force and a torque 159 When a wire connects two charged conductors, charges flow from higher potential to lower potential until the potentials are equal and not from higher charge to lower charge Capacitance 160 When n identical capacitors each of capacitance C are connected in parallel, the effective capacitance is CP = nC If n identical capacitors each of capacitance C are in series, the effective capacitance is C Cs = n 161 If n identical capacitors are joined in parallel and in series the ratio of effective capacitance will be Cp C = n2 and S = Cp CS n 162 If three capacitors of capacitance C1, C2 and C3 are connected in series to a potential difference V, the potential across the capacitors in the respective order will be in the ratio C2 C3 : C3 C1 : C1 C2 163 N capacitors are connected in parallel to a potential difference V If those capacitors are now joined in series without disturbing their charges, the potential difference across the combination will be nV 164 When a conductor of capacity C is charged to a potential V, the work done by the battery is CV2 and energy of the conductor is equal to CV2 This means half of the energy of the battery appears heat during charging 165 If a wire connects two conductors of capacitance C1 and C1, charged to potentials V1 and V2, charges flow from higher potential to lower potential In the process, some electrostatic energy is lost The C1C2 loss of electrostatic energy is given by (V1 − V2 ) C1 + C2 166 If two condensers of capacitance C1 and C2, charged to potentials V1 and V2 are connected in parallel, the loss of electrostatic energy will be same as given in the previous equation Physics for IIT-JEE Screening Test 10 General 167 If electric field at a point is zero, electric potential at the point need not be zero This means electric potential can exist without electric field For e.g in figure 1, the electric field at the centre point O is zero, but the electric potential is not zero 168 If electric potential at a point zero, electric field need not be zero This means electric field can exist +q +q O Fig.1 +q -q O Fig.2 without electric potential For example in fig 2, the electric potential at the centre point O is zero, but the electric field at the point is not zero 169 The following table gives how various quantities change when a dielectric slab is introduced between the plates of a parallel plate condenser: Quantities: Q:charge, C capacitance,V potential,U electrostatic energy, E electric field How the quantity changes Quantity When battery is connected When battery is disconnected Q increases constant C increases increases V constant decreases U increases decreases E constant decreases 170 In a charged conductor, there is outward mechanical force acting This force is given by σ2 N/m2, 2ε where σ is the surface density of charge (charge per unit area) 171 The capacity of the earth is nearly 700 µF 172 The approximate electric field above which air becomes ionised is equal to 3x106 N/C About charged drops 173 When n identical drops each charged to a potential V coalesce, the potential of the resulting drop will be n2/3V 174 When n identical drops each of capacity C coalesce, the capacity of the resulting drop will be n1/3C 175 When n identical drops each carrying a charge q coalesce, charge of the resulting drop will be nq 176 When n identical drops each of surface density of charge σ coalesce, the surface density of charge of the resulting drop will be n1/3 σ 177 When n identical drops each of potential energy U coalesce, the potential energy of resulting drop = n5/3U 178 A charged soap bubble always expands whether positively or negatively charged Ohm’s law and applications: 179 The number of electrons crossing when A of current flows through a conductor is 6.25 x 1018 per second 180 One way of writing Ohm’s law is j = σE, where j is current density, σ conductivity and E, electric field Physics at a Glance 11 181 Ohm’s law cannot be applied to circuits which contain diodes, transistors, capacitors etc 182 Ohm’s law and Kirchoff’s loop theorem can be applied to AC voltages only for their instantaneous values They cannot be applied to rms values 183 The electrons inside a metal have two velocities (1) random velocity, which is of the order of 105 m/s, and (2) drift velocity, which is of the order of mm/s Of these current is produced only by drift motion 184 If E is the emf of a cell and V potential difference measured across a resistor R connected in series H G (E − V)R with the cell, the internal resistance of the cell B = V E F 185 A fuse wire should have low melting point and high resistance 186 If there is only one battery in the circuit, problems involving solution of D current by Kirchoff’s law can be done with one equation For this the current is judiciously chosen for the convenience of division A Resistance of a cube C Fig.1 B 187 Resistance of a cube made of 12 identical resistors each of R Ω (fig 1) between body diagonal points (AG) = (5/6)R 188 Resistance of the cube (fig 1)) between any adjacent corners (AB) = (7/12)R 189 Resistance of the cube between face diagonal (fig 1) points (AF) = (3/4)R 190 If two bulbs of different powers are connected in series to a source, the bulb having less power will glow brighter 191 If two bulbs of different powers are connected to the same source in parallel, the bulb with greater power will be brighter R R R R 192 When a uniform wire of resistance R is stretched to n A times its original length, (note: its volume remains constant), it resistance comes n2R 193 When a uniform wire of conductance ‘c’ is stretched to R B n times the original length, its conductance becomes c n2 A 194 Two important properties of a metal to make a standard B resistor are low temperature coefficient and low linear expansivity R R R R Fig 3(a) R R R R To α R R R To α Fig.3(b) ⎡ + 1⎤ ⎥ ⎢⎣ ⎥⎦ 195 Resistance of infinite network shown in top fig.3(a) between A and B = R ⎢ ⎡ − 1⎤ ⎥ ⎢⎣ ⎥⎦ 196 Resistance between A and B of infinite network shown in fig 3(b) is = R ⎢ 197 In a potentiometer, if the EMF of the driving cell (cell connected in the main circuit) is increased, the balancing length will decrease