Tiêu đề PHYSICS PART- 1.pdf ✅

PHYSICSClass - C2 (Part - I) INDEX Chapter No. Pame of the Chapter Page No. 1. Basic Mathematics 04 − 24 2. Centre of mass and collisions 25 − 73 3. Rotatory Motion 74 − 110 1. BASIC MATHEMATICS SIMPLE ALGEBRA FORMULAE : • a 2 − b 2 = (a − b)(a + b) • (a + b) 2 = a 2 + 2ab + b 2 • (a − b) 2 = a 2 − 2ab + b 2 • (a 2 + b 2 ) = (a + b) 2 − 2ab
• (a + b + c) 2 = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc • (a − b − c) 2 = a 2 + b 2 + c 2 − 2ab − 2ac + 2bc • a 3 − b 3 = (a − b)(a 2 + ab + b 2 ) • a 3 + b 3 = (a + b)(a 2 − ab + b 2 ) QUADRATIC EQUATION : An algebraic equation of second order (highest power of variable is 2 ) is called a quadratic equation e.g. ax 2 + bx + c = 0, a ≠ 0 It has solution for two values of x which are given by x = −b±√b 2−4ac 2a Th quantity b 2 − 4ac, is called discriminant of the equation. BINOMIAL THEOREM : i) The binomial theorem for any positive value of n (x + a) n = x n + nC1ax n−1 + nC2a 2x n−2 + ⋯ . . + nCra rx n−r + ⋯ . . +a n Where ' a ' is constant and nCr = n! r!(n−r)! Here n! = n(n − 1)(n − 2) ... . . .3 × 2 × 1 So, 5! = 5 × 4 × 3 × 2 × 1 = 120 ii) (1 + x) n = 1 + nx + n(n−1) 2! x 2 + n(n−1)(n−2) 3! x 3 + ⋯ ... ... ... ..... For |x| << 1, we can neglect the higher power of x. So, (1 + x) n ≃ 1 + nx Similarity (1 − x) n ≃ 1 − nx (1 + x) −n ≃ 1 − nx
(1 − x) −n ≃ 1 + nx Here n may have any value. P.1. Evaluate √(1. 01) Sol. (1.01) 1 2 = (1 + 0.01) 1 2 = 1 + 1 2 × 0.01 = 1.005 ARITHMETIC PROGRESSION (A.P) : A sequence like a, a + d, a + 2d, is called arithmetic progreesion. Here d is the common difference. i) The n th term of an A.P is given by an = a + (n − 1)d ii) The sum of first n term of an A.P. is given by Sn = n 2 [1 term + last term ] = n 2 (a1 + an ) Here a1 = a and an = a + (n − 1)d ∴ Sn = n 2 [2a + (n − 1)d] GEOMETRIC PROGRESSION (G.P) : The progression like, a, ar, ar2 , ..., is called geometric progression, here r is called geometric ratio or common ratio. i) The n th term of G.P. is given by an = ar n−1 ii) The sum of the first n terms of this G.P. is given by Sn = a(r n − 1) (r − 1) for (r > 0) and Sn = a (1 − r n) (1 − r) for (r < 0) iii) The sum of an infinite term of G.P. for r < 1, is given by S = I st term 1− Geometric ratio or S = a 1−r .
P.2. Find sum of the progression : 1, 1 2 , 1 4 , 1 8 , ... . . ∞. Sol. We have S = a 1−r Here, a = 1, r = 1 2 ∴ S = 1 1 − 1 2 = 2 LOGARITHMS : Definition : Let a,N be two positive real numbers and a ≠ 1. If x is a real number such that a x = N, then x is called the logarithm of N to the base ' a '. It is denoted by loga N. i.e., a x = N ⇒ x = log2 N a x = N is called an exponential form and x = loga N is called a logarithmic form. Example : Exponential form Logarithmic form i) 2 5 = 32 5 = log2 32 ii) 7 3 = 343 3 = log7 343 iii) 5 −2 = 1 25 −2 = log5 1 25 Laws of Logarithms :