# HW1 Solution

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HW1 Solution

1. Use dimensional analysis to derive the formula for the period of a physicalpendulum (circular disk) as shown Fig. 1, where R is the radius of the disk, mis the mass, and g is the gravitational acceleration at the surface of earth.Solution:影 響 物 理 擺 週 期 tp 的 因 素 可 能 有 質 量 m , 半 徑 R , 重 力 加 速 度 gL[tp] : [T] ; [m]: [M] ; [R ]: [L] ; [g ]: [ 2 ]Ttpm L gx y zx 2T M L LT yzMxLyzT2zP x y 0 z 02z 1z12;y12t pLgt p kLg(k 為 常 數 , 由 實 驗 測 )

2. Solutiond(cos x) d(sin x)sin x- cos xd d cosx ( a) cotx dx dx2dx dx sinx sin x-sin xsin x - cos xcos x 1 - -csc2 2sin xsin x2xd d 1 1 d sin x( b) (sec x) (cos x) tan xsecx2 2dx dx cos x cos x dx cos xd d 1 1 d cosx( c) (csc x) (sin x) cot xcscx2 2dx dx sin x sin x dx sin x

ddx2( d) ( x sin 2x 1) let u 2x1ddx2 2( x sin 2x 1) ( x sin u)ddx2 2sin u ( x ) x (sin u)sin (2 )ddxu x2x cosu dxdddxdu 1 2 2x1dx22xsin u x cos u (2x1)2x cosu2xsin u(2)2 2x1 2xsin 2x1x2cos 2x12x1

d x x d d(ln u) du 1 du( f ) ln let u ; lnu dx x 2x 2dx du dx u dxd x d 1 d x ln lnudx x 2 dx x dx x 2x 2d d( x 2) ( x) x ( x 2)x 2dx dx2x ( x2)x 2 x 2 x 2 x x x x2( 2) ( 2)d22( x1) 2( e) e let u 2( x 1)dxd d due e e 22( x 1) 4( x 1)edx du dx2 22( x1) u u 2( x1)

3. Mary and Sally are in foot race (Fig.2). When Mary is 50 m from the finish line, shehas a speed of 4.0 m/s and is 5.0 m behind Sally, who has a speed of 5.0 m/s. Duringthe remaining portion of the race, Sally accelerates at a constant rate of 0.2 m/s 2 to thefinish line. What constant acceleration does Mary now need during the remainingportion of the race, if she wishes to cross the finish line side-by-side with Sally ?v m s x mM24.0 / ;M50v 5.0 m / s ; x 45 m ; a 0.2 m / sS2 2SS設 Marry跟 S ally到 達 終 點 的 時 間 為 t , 則1 2xSvSt f aStf21 2 245 5 tf (0.2) tf 5t f 0..1 tf22-525 4(0.1)(-45)0.1t f 5t f-45 0 tf 7.8 (s) 或 -57.8 (s)2(0.1)1t v t a t22f 7.8 (s) 代 入 xMM fM f( 時 間 不 能 為 負 )1 50 4t50 4(7.8)2 f250 4 tf atf a 0.62( m / s )21 2 1 2tf(7.8)2 2f50 mFig. 2

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