物理実験 III －データ処理－ - cc.u-ryukyu.ac.jp

1 III [1] (1) y=ax+b y=axi+b yi (xi; yi) y=ax+b a b S=X(yi axi b)2 @S@a= 2 Xxi(axi+b yi) = 2X(ax2i+bxi xiyi) = 0@S@b= 2X(axi+b yi) = 0 aXx2i+bXxi=XxiyiaXxi+Nb=XyiN 2 a b a=NPxiyi PxiPyiNPx2i (Pxi)2b=Px2iPyi PxiPxiyiNPx2i (Pxi)2(2) y=ax2+bx+c a b c S=X(yi ax2i bxi c)2 @S@a= 2Xx2i(ax2i+bxi+c yi) = 2X(ax4i+bx3i+cx2i x2iyi) = 0@S@b= 2 Xxi(ax2i+bxi+c yi) = 2X(ax3i+bx2i+cxi xiyi) = 0@S@c= 2X(ax2i+bxi+c yi) = 0aXx4i+bXx3i+cXx2i=Xx2iyiaXx3i+bXx2i+cXx i=XxiyiaXx2i+bXxi+Nc=Xyi1 a b c N Excel (1)

2 (2) (3) - - - 9 (3) (4) (5) 2 a b - EXCEL [2] a; b xi a; b yi yi 5 yi ax+b y 8 y y=s1NX(axi+b yi)2 N= 2 N N 2 y y=s1N 2X(axi+b yi)2 a; b a=NPxiyi PxiPyiNPx2i (Pxi)2b=Px2iPyi PxiPxiyiNPx2i (Pxi)2 a a=N(x1y1+x2y2+:::) Pxi(y1+y2+::::)NPx2i (Pxi)2=(Nx1 Pxi)y1+ (Nx2 Pxi)y2+::::NPx2i (Pxi)2 a a=[(Nx1 Pxi)2 2y+ (Nx2 Pxi)2 2y+::::]1=2 NPx2i (Pxi)22=[(Nx1)2 2Nx1 Pxi+ (Pxi)2+ (Nx2)2 2Nx2 Pxi+ (Pxi)2+::::]1=2 NPx2i (Pxi)2 y a= ysNNPx2i (Pxi)2 b b b=(Px2i (Pxi)x1)y1+ (Px2i (Pxi)x2)y2+:::::NPx2i (Pxi)2 b=[N(Px2i)2 2Px2i(Pxi)2+ (Pxi)2Px2i]1=2 NPx2i (Pxi)2 y b= yvuutPx2iNPx2i (Pxi)2 y=a( a)x+b( b)a ab b [3] x1; x2; x3; x4; ::: 1 2 3.

3 R=f(x1; x2; x3; x4::::) R R R=s(@f@x1)2 21+ (@f@x2)2 22+ (@f@x3)2 23+ (@f@x4)2 24+:::: @ x1; x2; x3; x4; ::: R R R=f(x1; x2; x3; :::::) R xi xi xi R R R=@f@x1 x1+@f@x2 x2+@f@x3 x3+:::::::+@f@xn xn ( R)2= (@f@x1)2( x1)2+ (@f@x2)2( x2)2+::::::::+ 2@f@x1@f@x2 x1 x2+:::::::::3 xi NXk( Rk)2= (@f@x1)2 NXk( x1k)2+ (@f@x2)2 NXk( x2k)2+::::::::+ 2@f@x1@f@x2 NXk( x1k x2k) +::::::::: xi xj PNk( x1k x2k) 0 PNk( xjk)2 N ( R)2= (@f@x1)2( x1)2+ (@f@x2)2( x2)2+:::::::: x y R=f(x; y; z; :::::) x x x.

4 R R (1) R=ax by cz ::::::: R=a x b y c z ::::::::orj Rj<ja xj+jb yj+jc zj+:::::::: p q L=p+ 0:001q L L L= p+ 0:001 qorj Lj<j pj+ 0:001j qj p q 1=1000 L (2) R=f(x; y; z; :::::) R=@f@x x+@f@y y+@f@z z+::::::: R=xmynzo:::::::: R= (ynzo:::::::)mxm 1 x+ (xmzo:::::::)nyn 1 y+ (xmyn:::::::)ozo 1 z+:::::::::orj Rj<j(ynzo:::::::)mxm 1 xj+j(xmzo:::::::)nyn 1 yj+j(xmyn:::::::)ozo 1 zj+:::::::::4 d l W e Young E E= 4lW=( d2e) E E E=E( ll+ WW 2 dd ee)orj Ej<E(j llj+j WWj+ 2j ddj+j eej) l W d e 1% E E E=E (1 + 1 + 2 + 1)% = 5% [4] X i=xi X =sPni(xi X)2n=sPni 2in X Xm m (xi i= 1;2;3.)

5 ::::n) Xm m Xm=1nXxi=1n(x1+x2+:::::::+xn) 2m= (@Xm@x1)2 21+ (@Xm@x2)2 22+::::::::+ (@Xm@xn)2 2n=1n2( 21+ 22+:::::::+ 2n) i ( 1= 2=::::= n= ) 2m=1n2n m= pn X Xm m X i=xi X =sPni 2in=sPni(xi X)2n X Xm i=xi X= (xi Xm) + (Xm X) n 2i= (xi Xm)2+ (Xm X)2+ 2(xi Xm)(Xm X)51nnXi=1 2i=1nnXi(xi Xm)2+1nnXi(Xm X)2+2(Xm X)nnXi(xi Xm) 2=1nnXi(xi Xm)2+ 2m+ 0 2 X Xm Xm m Xm=Pxi=n (xi Xm) 3 Pni(xi Xm) 0 2 2m=1nnXi(xi Xm)2 m= pnn 2 2=nXi(xi Xm)2 =sPni=1(xi Xm)2n 1=sPni 2in 1 m=vuutPni=1(xi Xm)2n(n 1)=vuutP 2in(n 1)

6 I=xi Xm Xm Xm m X i=xi X xi Xm i=xi Xm i=xi X Xm X n X Xm m i m Z z1; z2; z3; :::::; zn Zm 1=z1 Z; 2=z2 Z; 3=z3 Z; ::::::: n=zn Z zi Zm 1=z1 Zm; 2=z2 Zm; 3=z3 Zm; ::::::: n=zn Zm m Zm zi P i= 0 Z Zm 2 =Zm Z=(z1 1) + (z2 2) +::: (z1 1+z2 2+:::)n=1nnXi=1( i i) =1nnXi=1 i=1n( 1+ 2+ 3+:::)6 N 1; 2; 3; ::::::: 1k, 2k, 3k,:::::::, nk(k= 1;2;3; ::::; N) Zm m k=1nnXi=1 ikk= 1;2;3.

7 N 1=Zm1 Z=1nnXi=1 i1 2=Zm2 Z=1nnXi=1 i2:::::: N=ZmN Z=1nnXi=1 iN Zm N Zm m k m=sPNk=1 2kN 2m=1 NNXk=1 2k k 2m=1 NNXk=11n2( 1k+ 2k+ 3k+::::::::+ nk)2=1Nn2(NXk=1nXi=1 2ik+ 2 NXk=1 NXi;j=1 ik jk)=1Nn2 NXk=1nXi=1 2ik=1Nn2 NXk=1n 2=Nn 2Nn2= 2nn 2=nXi=1 2i( )(1Nn2 NXk=1nXi=1 2ik=1Nn2(nXi=1 2i1+nXi=1 2i2+nXi=1 2i3+:::::nXi=1 2iN) ik m= pn Zm m Z Zm m Z 1=pn Zm n m Zm Z zi Z Z zi Z i i=zi Z i=zi Zm =Zm Z i= i+Zm (Zm ) i i i= i+ 7 2i nXi 2i=nXi 2i+ 2 nXi i+nXi 2=nXi 2i+n 2(nXi i=nXi(zi Zm) = 0) i=zi Zm P i= 0 n 2 2 2k 2m n 2=nXi 2i+n 2m=nXi 2i+ 2(n 2=nXi 2i( ) 2m=1 NNXk 2k( ) m= pn))

8 =sPni 2in 1 m=vuutP 2in(n 1) Zm m Zm zi i [5] m (x) (x) =Ae h2x2A: A AZ1 1e h2x2dx= 1 x y D=Z1 1e h2x2dxZ1 1e h2y2dy=Z1 1Z1 1e h2(x2+y2)dxdy=I2 xy (r; ) r2=x2+y2dxdy=drrd D=Z2 0Z10e h2r2drrd = 2 Z10re h2r2dr= 2 [ 12h2e h2r2]10= h2=I2 I=p hAp h= 1A=hp (x) =hp e h2x28 =s 21+ 22+:::::+ 2nn 2= 21+ 22+:::::+ 2nn E(x) P(x) E(x) =x1P(x1) +x2P(x2) +x3P(x3) +::::::::=Z1 1xf(x)dx 2 2=1nZ1 1n 2f( )d =Z1 1 2f( )d =hp Z1 1 2e h2 2d =hp Z1 1 e h2 2d =hp ( [ 12h2e h2 2]1 1+12h2Z1 1e h2 2d ) =hp 12h2p h=12h2 2 =p =1p2hh=1p2 (x) x=m (x) =1p2 e (x m)22 2 (x) S=Zm+ m (x)dx= 0.

9 683 m m+ 2 3 (x) f(x) f(x) =1p2 e (x m)22 2 f0(x) = (x m)p2 3e (x m)22 2f00(x) =(x m)2 2p2 5e (x m)22 2 x=m 1 f0(x) = 0 x=m f00(x) = 0 m 2 x 1 +1 1 Z+1 1f(x)dx= 1orZ+10f(x)dx=129 E(x) E(x) =m E(x) =Z+1 1xf(x)dx=Zxp2 e (x m)22 2dx=1p2 Z(z+m)e z22 2dz=1p2 ([ 2e z22 2]1 1+mp h) =1p2 mp p2 =mx m=z dx=dz V(x) V(x) =Z+1 1(x m)2f(x)dx=Z(x2 2mx+m2)f(x)dx=E(x2) E(x)2=E(x2) m2=Zx2p2 e (x m)22 2dx m2=1p2 Z( z+m)2e z22dz m2=1p2 2Zz2e z22dz+ 2 mZ1p2 ze z22dz+m2Z1p2 e z22dz m2=1p2 2Zz2e z22dz+2 mp2 [ e z22]1 1+m2 m2= 2p2 Zz(ze z22)dz= 2p2 [z( e z22)]1 1+ 2Z1p2 e z22dz= 2 m 1 x x=1nXxix=71 + 72 + 72 + 73 + 715= 71:8 x x x=s1n(n 1)X(xi x)2 x=s15 (5 1)[(71 71:8)2+ (72 71:8)2+ (72 71:8)2+ (73 71:8)2+ (71 71:8)2]= x=x x= 71:8 0:3 = 71.

10 8(3) 2 d d h h V= (d=2)2h V @V@d= 42dh=2dV@V@h= 4d2=1hV [@] V=s(@V@d)2 2d+ (@V@h)2 2h=s(2d)2 2d+ (1h)2 2h V10 3 l l l l[cm] i=l l 10 10 10 10 10 10 10 10 10 10 2l= 2i= l=vuutP 2in(n 1)=s0:56810 (10 1)= 0:0794(n: )l= 143:61 0:08[cm] m m m 4 1614 : (1) (2) (3) (4) (1) (2) 11 y=ax+b a b N x[%] yx2ixiyi(axi+b yi) 10 10 10 10 10 10 10 10 6 Pxi=280 Pyi= 2i= 3:282 10 5 a b a=NPxiyi PxiPyiNPx2i (Pxi)2b=Px2iPyi PxiPxiyiNPx2i (Pxi)2 a b a=8 396:165 280 11:11268 14000 (280)2= 1:7200 10 3b=14000 11:1126 280 396:1658 14000 (280)2= 1:3288 y= 1:720 10 3x+ 1:329 a b y=s1N 2X(axi+b yi)2=s18 2(3:282 10 5) = 2:339 10 3 a= ysNNPx2i (Pxi)2= (2:339 10 3) s88 14000 (280)2= 3.