Question

Example 1 A star undergoes some mode of oscillation. Scientists/engineers have hypothesized that the oscillation frequency (cycles per second), o, is dependent on the density p and the radius R and the gravitational constant G which appears in Newton's law of universal gravitation. If you are not familiar with the gravitational constant, read the section on mass and weight in Chapter 4 (Dimensions/units) of the text. Therefore, w has dimensions (T), and P, R, Gare the governing parameters, with dimensions p = [ML-3), R=[ L), and G = [ M LT-). In order to determine the formula, Let w xp'R "Gʻ where the exponents a, b, c are unknown: We can then substitute the dimensions of each parameter in the formula =C PORG where is some dimensionless constant. Substituting dimensions for the variables, we obtain [T4] = [ML]*[L]"[MLPT21" | I1 = MỸ--3a+b+3T-2c. Equating exponents on both sides, we have a-C=0 -3a + b + 3c = 0, - 2c = -1. Solving, we find a = c = %, b = 0, so that the formula becomes «F CGр

Answer 1

99 (1) Given that critical velocity ve Velocity Ve 3 m/s LT accelesation due to gravity g = m/s =) LT' depth =) d=) L o ng by substituting gives (251) = c [ 27119 [ Jb - Constant LT =[ 229.tb = 1. By comparing gives. 20+6 = 1 and -=-al 2013 +6=1 [6=- So, ve = c ['[a' =velo C-s constant (11) r Given drag forcefo - to & Mulde Fo= CM uobde FD~N~ 18m/ 2 =) MLT- M v - 92 m/s LT, dem L By substituting gives. I (ML52) = C [ Mir]" [LI] [JC (ML+2) = 0 [mo zatote a b] By comparing gives, a=1 Laub=-2 t-atb+c=i1 altbtc =) a+b=2 1 +6=2 N ob+c=2 +6=2 Scanned with CamScanner coll

- culod So, ED = cpood (1) Given powes toonsmitted pa to web Aged in Hamacs per unit length =) M/L - MI w angulas velocity a rodls - oft Amplitude =) A =) L Given/ mentioned that the exponent of velocity is 1 powers watt So, P = Cocobare - Corobale - J/ Intan ML73-c [ME]"[71] [1397 > by * % MEPPS.CO[Mazatem b] > 79 m So, 0-1 -a+c=21-b=-3 2) M2273 -1+0=2 Te=3 13,59 So, p=cHW A v 1p=CHIO ASW3 Obc I of Temp - (1) speed of Gas molecule wa Rome Boltzmann constant units kym rg 5 R o dimension - Temp(T) o mkg M. Ibra not I u time m/ =) LTT - 1 (257) = [ { M72 550 [0]b (me (Lt) = c5 29 Mate 729 atb] Scanned this

. -ath=0. By comparing 20=1, atc=0, -20=-. a=Y I +6=0 %+b=0 [C=-12 boy So, &- ckt? TY 2 % ICS Scanned with CamScanner