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Physics

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01_Unit_and_Dimension.pdf 02_Motion_in_1-D (1).pdf 04_Motion_in_2-D1.pdf 05_Laws_of_Motion_and_Frictions.pdf 06_Work_power_and_Energy.pdf 09_Magnetism.pdf 10__Electromagnetic_Induction.pdf

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  1. MOTION IN 1 DIMENSION (Rectilinear Motion & Relative motion) MECHANICS Kinematics The word kinematics means 'science of motion' branch of the mechanics which deals with study of motion without going into the cause of motion, i.e. force, torque etc. Position Dynamics (or Kinetics) It is branch of mechanics which is concerned about the causes (i.e. force, torque) that cause motion of bodies. If a particle is restricted to move along a given straight line (assumed along x-axis), its position is represented by the x-coordinate relative to a fixed origin. If the particle moves in a plane (let x-y plane) its position is completely known when the x and y coordinates of its position are known with respect to the given coordinate axes ox and oy. o If a body does not change its position as time passes with respect to frame of reference, it is said to Rest : be at rest. Motion : If a body changes its position as time passes with respect to frame of reference, it is said to be in motion T es of motion : One dimensional Motion of a body in a straight line is called one dimensional motion. When only one coordinate of the position of a body changes with time then it is said to be moving one dimensionally. Ex.. (i) Motion of car on a straight road. (ii) Motion of freely falling body. DISTANCE AND SPEED (a) Distance Two dimensional Motion of body in a plane is called two dimensional motion. When two coordinates of the position of a body changes with time then it is said to be moving two dimensionally. Ex. (i) Motion of car on a circular turn. (ii) Motion of billiards ball. Three dimensional Motion of body in a space is called three dimensional motion. When all three coordinates of the position of a body changes with time then it is said to be moving three dimensionally. Ex.. (i) Motion of flying kite. (ii) Motion of flying insect. (i) The total length of actual path traversed by the body between initial and final positions is called distance. It has no direction and is always positive. (iii) Distance covered by particle never decreases. (iv) Its Sl unit is meter (m) and dimensional formula is [MOLITO]. (b) Speed : The rate of distance covered with time is called speed. (c) (d) (i) It is a scalar quantity having symbol D. (ii) Dimension : [ML I T l] (iii) Unit : metre/second (S.l.), cm/second (C.G.S.) Average speed . It is defined as distance travelled by particle per unit time in a given interval of time. (ii) If S is the distance travelled by particle in time interval t, then average speed in that time interval s is Total distance travelled (iii) Average speed Time taken Instantaneous speed : av As At ) It is the speed of a particle at a particular instant of time. When we say "speed", means instantaneous speed. it usually
  2. (ii) The instantaneous speed is average speed for infinitesimally small time interval (i.e., At —->0 ). As ds (iii) Instantaneous speed v = lim At.0At dt DISPLACEMENT AND VELOCITY (a) (b) (c) (d) (e) Displacement (i) The change in position of a body in a certain direction is known as displacement. (ii) The distance between the initial and final position is known as magnitude of displacement. (iii) Displacement of an object may be positive, negative or zero and it is independent of the path followed by the object. (iv) Its Sl unit is meter and dimensional formula is [MOLITO]. (v) Displacement is a vector S drawn from the initial position (A) to the final position (B) Displacement S Actual path Velocity : Velocity is the rate of change of position vector. df dt Unit : ms-I (metre per second) Average velocity : It is defined as the ratio of displacement to time taken by the body Displacement . Average velocity Time taken Ar av At Instantaneous velocity (v) : It is the velocity of particle at any instant of time Ax dx Mathematically, v = Limit < v > = Limit — At dt Uniform velocity : A particle with uniform velocity undergoes equal displacements in equal intervals of time however small the intervals may be. Example 1: Solution : Ram takes path 1 (straight line) to go from P to Q and Shyam takes path 2 (semicircle). p 2 1 4—100 m (a) Find the distance travelled by Ram and Shyam? (b) Find the displacement of Ram and Shyam? (a) Distance travelled by Ram = 100 m Distance traveled by Shyam = (50 m) = 50Ttrn (b) Displacement of Ram = 100 m Displacement of Shyam = 100 m AVERAGE SPEED AND AVERAGE VELOCITY (i) If a body covers distance with speed VI , *With speed 14, then its average speed is av (ii) If a body coves first half distance with speed VI and next half with speed 14, then
  3. (iv) If body covers first one third with speed VI , next one third with speed and remaining one third with 3VlV2V3 speed then v av VIV 2 + V2V3 + V3V1 (v) If a body moves from one point (A) to another point (B) with speed VI and returns back (from B to A) 2VlV2 with speed then average velocity is 0 but average speed = VI + V 2 Example 2: In the example 1, if Ram takes 4 seconds and Shyam takes 5 seconds to go from P to Q, find 2VlV2 Average speed = (Harmonic mean) VI + V 2 (iii) If a body travels with uniform speed VI for time ti and with uniform speed for time 4, then average speed = If t -4 = — then v (tl-t) av 2 VI + V 2 [T = time of journey] (Arithmatic mean) 2 sol. (a) Average speed of Ram and Shyam? (b) Average velocity of Ram and Shyam? = 199m/s = 25 m/s (a) Average speed of Ram 4 = 5—9-E m/s = 1011 m/s Average speed of Shyam 5 100 (b) Average velocity of Ram = m/s = 25 m/s 4 100 Average velocity of Shyam = — m/s = 20 m/s 5 ACCELERATION Time rate of change of velocity is called acceleration. dv a- dt Unit : ms-2 (metre per second2) (a) Average acceleration : The time rate of change of velocity of an object is called acceleration of the object. change in velocity Av v 2 —VI a av total time aao At At The acceleration (b) Instantaneous acceleration acceleration. Mathematically Av dv a = Limit < a > = Limit — At dt at any instant is called instantaneous (c) Uniform acceleration : A particle with uniform acceleration undergoes equal changes in velocity in equal intervals of time, however small the intervals may be. GRAPHICAL SOLUTION OF RECTILINEAR MOTION (a) v-t Curve The area under the v-t curve measures the change in position x. — xo = area under v-t curve - vdt 1 t
  4. (b) (c) (d) Example 3 : Solution : a-t Curve Area under the a —t curve measures the change in velocity 14 — VI = area under (a —t ) curve Characteristics of v-t graph If the graph obtained is a line parallel to x-axis, the acceleration is zero. a = constant (ii) If the graph obtained is an oblique straight line of positive slope, the acceleration is uniform and if it is of negative slope, the retardation is constant. Graphical representation of motion (i) Slope of tangent to position time graph gives velocity. (ii) Slope of tangent to v-t curve gives acceleration. t a = constant (iii) Area enclosed between v-t curve and time axis between an interval of time gives displacement. (iv) Slope of tangent to a-t curve gives rate of change of acceleration (v) Area enclosed between a-t curve and time axis between an interval of time gives change in velocity. The displacement vs time graph of a particle moving along a straight line is shown in the figure. Draw velocity vs time and acceleration vs time graph. x = 4t2 v Hence, velocity-time graph is a straight line having slope i.e. tano = 8. tano = 8 O dt Hence, acceleration is constant throughout and is equal to 8. 8 EQUATIONS OF MOTION General equations of motion :
  5. dx = vdt fdx — fvdt dt dv dv = adt f — fadt a- dt vdv — vdv = adx fvdv — fadx a- dx Equations of motion of a particle moving with uniform acceleration in straight line : u + at ut + —at2 (iii) ve- — {.12 + 2 as xo + —at2 Here u = velocity of particle at t = 0 S = Displacement of particle between 0 to t = x— xo (xo = position of particle at t = 0, x = position of particle at time t) a = uniform acceleration v = velocity of particle at time t Snth = Displacement of particle in nth second Motion under gravity Whenever a particle is thrown up or down or released from a height, it falls freely under the effect of gravitational force of earth. The equations of motion . (i) v = u + gt (ii) h = ho +ut+—gt2 or h (iii) u2 + 2g(h— ho) or (iv) h th ho s = ut + = u2 + 2gs -gt2 where h = vertical displacement, = h vertical displacement in nth second Example -4: A police inspector in a jeep is chasing a pickpocket an a straight road. The jeep is going at its Solution maximum speed v (assumed uniform). The pickpocket rides on the motorcycle of a waiting friend when the jeep is at a distance d away, and the motorcycle starts with a constant acceleration a. Show that the pick pocket will be caught if v > . Suppose the pickpocket is caught at a time t after motorcycle starts. The distance travelled by the motorcycle during this interval is 1 = —at2 s 2 During this interval the jeep travels a distance (ii) 1 —at2 + d = Vt By (i) and (ii), or, 2 v ± v 2 — 2ad a The pickpocket will be caught if t is real and positive. This will be possible ifv2 > 2ad or, v > E.
  6. RELATIVE MOTION IN ONE DIMENSION (i) If two bodies A and B are moving in straight line same direction with velocity VA and VB, then relative velocity of A with respect to Bis VAB = v A— VB. Similarly VBA = VB — VA (ii) If two bodies A and B are moving in straight line in opposite direction then AB = VA + VB (towards B) = VB + VA (towards A) Same concept is used for acceleration also. (iii) If two cars A and B are moving in same direction with velocity v A and VB (VA > VB) when A is behind B at a distance d, driver in car A applies brake which causes retardation a in car A, then minimum (VA -VB)2 (VA -VB)2 value of dto avoid collision is 2a 2a (iv) A particle is dropped and another particle is thrown downward with initial velocity u, then 5. (a) Relative acceleration is always zero (b) Relative velocity is always u. x (c) Time at which their separation is x is — u Two bodies are thrown upwards with same initial velocity with time gap c. They will meet after a time t from projection of first body. RELATIVE MOTION OF TWO PARTICLES When two particles A and B move along the same straight line, denoting by XB/A, the relative position coordinate of B with respect to A, we have = x A +XB/A Denoting by VB/A and aB/A respectively, the relative velocity and the relative acceleration of B with respect to A, we have VB = VA + V aB = aA +aB/A DEPENDENT MOTIONS The position of a particle will depend upon the position of another or several other particles. The motions are then said to be dependent. In figure (a) the position of block B depends upon the position of block A length of rope Differentiating twice with respect to time t VA 2 VB = a A + 2aB = O Example -5: A body is released from a height and falls freely towards the earth. Exactly 1 sec later, Solution another body is released. What is the distance between the bodies 2 sec after the release of the second body if g = 9.8 m/s2? According to given problem, second body falls for 2 s, so that 1 g (2)2 2 The first body has fallen for 2 + 1 = 3 sec.
  7. m = -g (3)2 . Separation between two bodies 2 s after the release of second body d hl m = -22) = 4.9 x 5 = 24.5 m.

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