prática 1: medidas e incertezas

Prática 1: medidas e incertezas

Prof. Diogo Boito

Lab Fis A Eng Amb — 2020

Prof. Diogo BoitoLab Fis A Eng Amb — 2020


1. Turma 1: favor enviar um email informando quem fará parte do seu trio. (Um email por trio, para evitar confusão!)

2. Turma 1: Se a soma do último dígito dos números USP dos integrantes do trio for par, trabalhem com o cilindro dourado, se for ímpar, com o cilindro prateado.

3. Detalhes sobre como entregar o relatório serão enviados em breve.



1. Medidas: diretas e indiretasMede-se diretamente com um instrumento de medida (régua, paquímetro, balança, cronômetro…)

Medidas indiretas são obtidas dos resultados das medidas diretas.

Exemplo: densidade de um objeto com formato de paralelepípedo.

⇢ =m

V=

m

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balança

Paquímetro

A densidade é medida indiretamente



2. Precisão dos instrumentos

Analógicos: metade da menor divisão (ou até menos!).

Digitais: menor divisão.

(2, 165± 0, 005)mm<latexit sha1_base64="p+I/9Mur7FfDh0JpQ2O8NC8cay8=">AAACBHicbVDLSgMxFM34rPU16rKbYBEqlJKpVl0W3bisYB/QGUomzbShycyQZIQydOHGX3HjQhG3foQ7/8a0nYW2HrhwOOde7r3HjzlTGqFva2V1bX1jM7eV397Z3du3Dw5bKkokoU0S8Uh2fKwoZyFtaqY57cSSYuFz2vZHN1O//UClYlF4r8cx9QQehCxgBGsj9exCqVp2LmpuLCAqI1SDp245daWAQkx6dhFV0AxwmTgZKYIMjZ795fYjkggaasKxUl0HxdpLsdSMcDrJu4miMSYjPKBdQ0MsqPLS2RMTeGKUPgwiaSrUcKb+nkixUGosfNMpsB6qRW8q/ud1Ex1ceSkL40TTkMwXBQmHOoLTRGCfSUo0HxuCiWTmVkiGWGKiTW55E4Kz+PIyaVUrzlmlenderF9nceRAARyDEnDAJaiDW9AATUDAI3gGr+DNerJerHfrY966YmUzR+APrM8f/ZuVJw==</latexit>

�micrometro = (0, 01)mm/2 = 0, 005mm<latexit sha1_base64="dEDOyDuPyDoVK+H6aweMsndZy+A=">AAACInicbVDLSgMxFM3UV62vqks3wSJUkJqpirooFN24rGAf0JYhk2ba0GQyJBmhDP0WN/6KGxeKuhL8GNOHoK0HLpyccy+59/gRZ9og9OmkFhaXllfSq5m19Y3Nrez2Tk3LWBFaJZJL1fCxppyFtGqY4bQRKYqFz2nd71+P/Po9VZrJ8M4MItoWuBuygBFsrORlL1uadQX2kpYSUDCipKBGySEswTw6Qu7hxBDD42LJvtEZ/BG8bA4V0BhwnrhTkgNTVLzse6sjSSxoaAjHWjddFJl2gpVhhNNhphVrGmHSx13atDTEgup2Mj5xCA+s0oGBVLZCA8fq74kEC60HwredApuenvVG4n9eMzbBRTthYRQbGpLJR0HMoZFwlBfsMEWJ4QNLMFHM7gpJDytMjE01Y0NwZ0+eJ7ViwT0pFG9Pc+WraRxpsAf2QR644ByUwQ2ogCog4AE8gRfw6jw6z86b8zFpTTnTmV3wB87XN7llod0=</latexit>



3. Erros e incertezasErro grosseiro.Erro sistemático: método não é o ideal, instrumento mal calibrado…Erro aleatório: dispersão na medida.

Exemplo: período de um pêndulo.

T = 2⇡

s`

g<latexit sha1_base64="ynQEOsAfPfJuT3SsU+j2YxY9IMI=">AAACBXicbVBNS8NAEJ34WetX1KMeFovgqSRV0ItQ9OKxQr+gKWWz3bRLN5u4uxFKyMWLf8WLB0W8+h+8+W/ctjlo64OBx3szzMzzY86Udpxva2l5ZXVtvbBR3Nza3tm19/abKkokoQ0S8Ui2fawoZ4I2NNOctmNJcehz2vJHNxO/9UClYpGo63FMuyEeCBYwgrWRevZR/arixQx56l7q1AskJqlHOc/SQZb17JJTdqZAi8TNSQly1Hr2l9ePSBJSoQnHSnVcJ9bdFEvNCKdZ0UsUjTEZ4QHtGCpwSFU3nX6RoROj9FEQSVNCo6n6eyLFoVLj0DedIdZDNe9NxP+8TqKDy27KRJxoKshsUZBwpCM0iQT1maRE87EhmEhmbkVkiE0S2gRXNCG48y8vkmal7J6VK3fnpep1HkcBDuEYTsGFC6jCLdSgAQQe4Rle4c16sl6sd+tj1rpk5TMH8AfW5w/3RZjl</latexit>

Valor verdadeiro: 10s.

T1 = 10, 15s, T2 = 9, 9s T3 = 9, 8s...<latexit sha1_base64="u+fsKZKM/a389csZo0xbcBO9EOI=">AAACFXicbVDLSgMxFM34rPU16tJNsAguhmGmVWwXhaIblxWmD2iHIZNm2tDMgyQjlKE/4cZfceNCEbeCO//GtJ2Ftp4QOPece0nu8RNGhbSsb21tfWNza7uwU9zd2z841I+O2yJOOSYtHLOYd30kCKMRaUkqGekmnKDQZ6Tjj29nfueBcEHjyJGThLghGkY0oBhJJXm64Xh23bYM+0oYsG/MjuOV6zWjJhYVdLyKKqvCNE1PL1mmNQdcJXZOSiBH09O/+oMYpyGJJGZIiJ5tJdLNEJcUMzIt9lNBEoTHaEh6ikYoJMLN5ltN4blSBjCIubqRhHP190SGQiEmoa86QyRHYtmbif95vVQGVTejUZJKEuHFQ0HKoIzhLCI4oJxgySaKIMyp+ivEI8QRlirIogrBXl55lbTLpl0xy/eXpcZNHkcBnIIzcAFscA0a4A40QQtg8AiewSt40560F+1d+1i0rmn5zAn4A+3zB6AdmXo=</latexit>

Flutuações intrínsecas à medida, erro aleatório: dispersão.

Tempo de reação do medidor.

Vento.

Distração.

Oscilação com muita amplitude.



T = 2⇡

s`

g<latexit sha1_base64="ynQEOsAfPfJuT3SsU+j2YxY9IMI=">AAACBXicbVBNS8NAEJ34WetX1KMeFovgqSRV0ItQ9OKxQr+gKWWz3bRLN5u4uxFKyMWLf8WLB0W8+h+8+W/ctjlo64OBx3szzMzzY86Udpxva2l5ZXVtvbBR3Nza3tm19/abKkokoQ0S8Ui2fawoZ4I2NNOctmNJcehz2vJHNxO/9UClYpGo63FMuyEeCBYwgrWRevZR/arixQx56l7q1AskJqlHOc/SQZb17JJTdqZAi8TNSQly1Hr2l9ePSBJSoQnHSnVcJ9bdFEvNCKdZ0UsUjTEZ4QHtGCpwSFU3nX6RoROj9FEQSVNCo6n6eyLFoVLj0DedIdZDNe9NxP+8TqKDy27KRJxoKshsUZBwpCM0iQT1maRE87EhmEhmbkVkiE0S2gRXNCG48y8vkmal7J6VK3fnpep1HkcBDuEYTsGFC6jCLdSgAQQe4Rle4c16sl6sd+tj1rpk5TMH8AfW5w/3RZjl</latexit>

Medidas de dispersão: desvio padrão da média

� =

vuutNX

i=1

(xi � x)2

N � 1<latexit sha1_base64="brlKLi8wKoSOzQqvgR80MAQna+o=">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</latexit>

Dispersão em torno do valor verdadeiro.

x =1

Ntot

NX

i=1

xi

<latexit sha1_base64="rRu3qjchfZ9wxXcYkbQflxKFYHk=">AAACF3icbVDLSsNAFJ3UV62vqEs3g0VwVZIq6KZQdOOqVLAPaGKYTCft0JkkzEykJeQv3Pgrblwo4lZ3/o3TNgttPTBwOOde7pzjx4xKZVnfRmFldW19o7hZ2tre2d0z9w/aMkoEJi0csUh0fSQJoyFpKaoY6caCIO4z0vFH11O/80CEpFF4pyYxcTkahDSgGCkteWbF8ZGAY1iDTiAQTu0sbXipIzhUkcoyRybcS2nNzu4bcOxRzyxbFWsGuEzsnJRBjqZnfjn9CCechAozJGXPtmLlpkgoihnJSk4iSYzwCA1IT9MQcSLddJYrgyda6cMgEvqFCs7U3xsp4lJOuK8nOVJDuehNxf+8XqKCSzelYZwoEuL5oSBhOjOclgT7VBCs2EQThAXVf4V4iHQ/SldZ0iXYi5GXSbtasc8q1dvzcv0qr6MIjsAxOAU2uAB1cAOaoAUweATP4BW8GU/Gi/FufMxHC0a+cwj+wPj8AW6Rn3w=</latexit>

Média aritmética: estimativa do valor verdadeiro

Valor verdadeiro: 10s.

T1 = 10, 15s, T2 = 9, 9s T3 = 9, 8s...<latexit sha1_base64="u+fsKZKM/a389csZo0xbcBO9EOI=">AAACFXicbVDLSgMxFM34rPU16tJNsAguhmGmVWwXhaIblxWmD2iHIZNm2tDMgyQjlKE/4cZfceNCEbeCO//GtJ2Ftp4QOPece0nu8RNGhbSsb21tfWNza7uwU9zd2z841I+O2yJOOSYtHLOYd30kCKMRaUkqGekmnKDQZ6Tjj29nfueBcEHjyJGThLghGkY0oBhJJXm64Xh23bYM+0oYsG/MjuOV6zWjJhYVdLyKKqvCNE1PL1mmNQdcJXZOSiBH09O/+oMYpyGJJGZIiJ5tJdLNEJcUMzIt9lNBEoTHaEh6ikYoJMLN5ltN4blSBjCIubqRhHP190SGQiEmoa86QyRHYtmbif95vVQGVTejUZJKEuHFQ0HKoIzhLCI4oJxgySaKIMyp+ivEI8QRlirIogrBXl55lbTLpl0xy/eXpcZNHkcBnIIzcAFscA0a4A40QQtg8AiewSt40560F+1d+1i0rmn5zAn4A+3zB6AdmXo=</latexit>

9.0 9.5 10.0 10.5 11.00

50

100

150

200

T (em s)<latexit sha1_base64="2YQE0Bvp6RwkGe+pXbAn4kBa/wI=">AAAB8XicbVA9TwJBEN3DL8Qv1NJmI5hgQ+6w0JJoY4kJXxEuZG+Zgw27e5fdPRNy4V/YWGiMrf/Gzn/jAlco+JJJXt6bycy8IOZMG9f9dnIbm1vbO/ndwt7+weFR8fikraNEUWjRiEeqGxANnEloGWY4dGMFRAQcOsHkbu53nkBpFsmmmcbgCzKSLGSUGCs9lptlXAGB9eWgWHKr7gJ4nXgZKaEMjUHxqz+MaCJAGsqJ1j3PjY2fEmUY5TAr9BMNMaETMoKepZII0H66uHiGL6wyxGGkbEmDF+rviZQIracisJ2CmLFe9ebif14vMeGNnzIZJwYkXS4KE45NhOfv4yFTQA2fWkKoYvZWTMdEEWpsSAUbgrf68jpp16reVbX2UCvVb7M48ugMnaMK8tA1qqN71EAtRJFEz+gVvTnaeXHenY9la87JZk7RHzifPzxcj1I=</latexit>

freq.<latexit sha1_base64="75wfMFwJU+tTAD1e+NQXv/ucJvw=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4Ckk96LHoxWMF0xbaUDbbSbt0s4m7G6GU/gYvHhTx6g/y5r9x2+agrQ8GHu/NMDMvygTXxvO+nbX1jc2t7dJOeXdv/+CwcnTc1GmuGAYsFalqR1Sj4BIDw43AdqaQJpHAVjS6nfmtJ1Sap/LBjDMMEzqQPOaMGisFscJHt1epeq43B1klfkGqUKDRq3x1+ynLE5SGCap1x/cyE06oMpwJnJa7ucaMshEdYMdSSRPU4WR+7JScW6VP4lTZkobM1d8TE5poPU4i25lQM9TL3kz8z+vkJr4OJ1xmuUHJFoviXBCTktnnpM8VMiPGllCmuL2VsCFVlBmbT9mG4C+/vEqaNde/dGv3tWr9poijBKdwBhfgwxXU4Q4aEAADDs/wCm+OdF6cd+dj0brmFDMn8AfO5w+aaY6K</latexit>

9.0 9.5 10.0 10.5 11.00

50

100

150

200

T (em s)<latexit sha1_base64="2YQE0Bvp6RwkGe+pXbAn4kBa/wI=">AAAB8XicbVA9TwJBEN3DL8Qv1NJmI5hgQ+6w0JJoY4kJXxEuZG+Zgw27e5fdPRNy4V/YWGiMrf/Gzn/jAlco+JJJXt6bycy8IOZMG9f9dnIbm1vbO/ndwt7+weFR8fikraNEUWjRiEeqGxANnEloGWY4dGMFRAQcOsHkbu53nkBpFsmmmcbgCzKSLGSUGCs9lptlXAGB9eWgWHKr7gJ4nXgZKaEMjUHxqz+MaCJAGsqJ1j3PjY2fEmUY5TAr9BMNMaETMoKepZII0H66uHiGL6wyxGGkbEmDF+rviZQIracisJ2CmLFe9ebif14vMeGNnzIZJwYkXS4KE45NhOfv4yFTQA2fWkKoYvZWTMdEEWpsSAUbgrf68jpp16reVbX2UCvVb7M48ugMnaMK8tA1qqN71EAtRJFEz+gVvTnaeXHenY9la87JZk7RHzifPzxcj1I=</latexit>

freq.<latexit sha1_base64="75wfMFwJU+tTAD1e+NQXv/ucJvw=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4Ckk96LHoxWMF0xbaUDbbSbt0s4m7G6GU/gYvHhTx6g/y5r9x2+agrQ8GHu/NMDMvygTXxvO+nbX1jc2t7dJOeXdv/+CwcnTc1GmuGAYsFalqR1Sj4BIDw43AdqaQJpHAVjS6nfmtJ1Sap/LBjDMMEzqQPOaMGisFscJHt1epeq43B1klfkGqUKDRq3x1+ynLE5SGCap1x/cyE06oMpwJnJa7ucaMshEdYMdSSRPU4WR+7JScW6VP4lTZkobM1d8TE5poPU4i25lQM9TL3kz8z+vkJr4OJ1xmuUHJFoviXBCTktnnpM8VMiPGllCmuL2VsCFVlBmbT9mG4C+/vEqaNde/dGv3tWr9poijBKdwBhfgwxXU4Q4aEAADDs/wCm+OdF6cd+dj0brmFDMn8AfO5w+aaY6K</latexit>



� =

vuutNX

i=1

(xi � x)2

N � 1<latexit sha1_base64="brlKLi8wKoSOzQqvgR80MAQna+o=">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</latexit>

x =1

Ntot

NX

i=1

xi

<latexit sha1_base64="rRu3qjchfZ9wxXcYkbQflxKFYHk=">AAACF3icbVDLSsNAFJ3UV62vqEs3g0VwVZIq6KZQdOOqVLAPaGKYTCft0JkkzEykJeQv3Pgrblwo4lZ3/o3TNgttPTBwOOde7pzjx4xKZVnfRmFldW19o7hZ2tre2d0z9w/aMkoEJi0csUh0fSQJoyFpKaoY6caCIO4z0vFH11O/80CEpFF4pyYxcTkahDSgGCkteWbF8ZGAY1iDTiAQTu0sbXipIzhUkcoyRybcS2nNzu4bcOxRzyxbFWsGuEzsnJRBjqZnfjn9CCechAozJGXPtmLlpkgoihnJSk4iSYzwCA1IT9MQcSLddJYrgyda6cMgEvqFCs7U3xsp4lJOuK8nOVJDuehNxf+8XqKCSzelYZwoEuL5oSBhOjOclgT7VBCs2EQThAXVf4V4iHQ/SldZ0iXYi5GXSbtasc8q1dvzcv0qr6MIjsAxOAU2uAB1cAOaoAUweATP4BW8GU/Gi/FufMxHC0a+cwj+wPj8AW6Rn3w=</latexit>

±<latexit sha1_base64="9fCtpFlmjA7RrbtInjcNN+axJn0=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvgqexWQY9FLx4r2g9ol5JNs21okl2SrFCW/gQvHhTx6i/y5r8xbfegrQ8GHu/NMDMvTAQ31vO+UWFtfWNzq7hd2tnd2z8oHx61TJxqypo0FrHuhMQwwRVrWm4F6ySaERkK1g7HtzO//cS04bF6tJOEBZIMFY84JdZJD71E9ssVr+rNgVeJn5MK5Gj0y1+9QUxTyZSlghjT9b3EBhnRllPBpqVealhC6JgMWddRRSQzQTY/dYrPnDLAUaxdKYvn6u+JjEhjJjJ0nZLYkVn2ZuJ/Xje10XWQcZWklim6WBSlAtsYz/7GA64ZtWLiCKGau1sxHRFNqHXplFwI/vLLq6RVq/oX1dr9ZaV+k8dRhBM4hXPw4QrqcAcNaAKFITzDK7whgV7QO/pYtBZQPnMMf4A+fwBXG43V</latexit>

x± �<latexit sha1_base64="aIeySjGvsk1k8lLZH/gKz8TgkbU=">AAAB+nicbVBNSwMxEJ31s9avrR69BIvgqexWQY9FLx4r2A/oLiWbZtvQJLskWbXU/hQvHhTx6i/x5r8xbfegrQ8GHu/NMDMvSjnTxvO+nZXVtfWNzcJWcXtnd2/fLR00dZIpQhsk4YlqR1hTziRtGGY4baeKYhFx2oqG11O/dU+VZom8M6OUhgL3JYsZwcZKXbcURFihRxSkAgWa9QXuumWv4s2AlomfkzLkqHfdr6CXkExQaQjHWnd8LzXhGCvDCKeTYpBpmmIyxH3asVRiQXU4np0+QSdW6aE4UbakQTP198QYC61HIrKdApuBXvSm4n9eJzPxZThmMs0MlWS+KM44Mgma5oB6TFFi+MgSTBSztyIywAoTY9Mq2hD8xZeXSbNa8c8q1dvzcu0qj6MAR3AMp+DDBdTgBurQAAIP8Ayv8OY8OS/Ou/Mxb11x8plD+APn8wdXPZNo</latexit>

: ~68% das vezes a medida estará neste intervalo

x± 2�<latexit sha1_base64="+6wUIu18+QCZUHR9of7kHG/yBw8=">AAAB+3icbVBNSwMxEJ2tX7V+rfXoJVgET2W3CnosevFYwX5AdynZNNuGJtklyUpL6V/x4kERr/4Rb/4b03YP2vpg4PHeDDPzopQzbTzv2ylsbG5t7xR3S3v7B4dH7nG5pZNMEdokCU9UJ8KaciZp0zDDaSdVFIuI03Y0upv77SeqNEvko5mkNBR4IFnMCDZW6rnlIMIKjVGQClQLNBsI3HMrXtVbAK0TPycVyNHouV9BPyGZoNIQjrXu+l5qwilWhhFOZ6Ug0zTFZIQHtGupxILqcLq4fYbOrdJHcaJsSYMW6u+JKRZaT0RkOwU2Q73qzcX/vG5m4ptwymSaGSrJclGccWQSNA8C9ZmixPCJJZgoZm9FZIgVJsbGVbIh+Ksvr5NWrepfVmsPV5X6bR5HEU7hDC7Ah2uowz00oAkExvAMr/DmzJwX5935WLYWnHzmBP7A+fwBzPaTpA==</latexit>

: ~95,5% das vezes a medida estará neste intervalo

x± 3�<latexit sha1_base64="nbwtFu/N/dxKRuHY9lxCjj7KHZ4=">AAAB+3icbVBNTwIxEO3iF+LXikcvjcTEE9kFEz0SvXjERMCE3ZDZ0oWGtrtpuwZC+CtePGiMV/+IN/+NBfag4EsmeXlvJjPzopQzbTzv2ylsbG5t7xR3S3v7B4dH7nG5rZNMEdoiCU/UYwSaciZpyzDD6WOqKIiI0040up37nSeqNEvkg5mkNBQwkCxmBIyVem45iEDhMQ5SgeuBZgMBPbfiVb0F8Drxc1JBOZo99yvoJyQTVBrCQeuu76UmnIIyjHA6KwWZpimQEQxo11IJgupwurh9hs+t0sdxomxJgxfq74kpCK0nIrKdAsxQr3pz8T+vm5n4OpwymWaGSrJcFGccmwTPg8B9pigxfGIJEMXsrZgMQQExNq6SDcFffXmdtGtVv16t3V9WGjd5HEV0is7QBfLRFWqgO9RELUTQGD2jV/TmzJwX5935WLYWnHzmBP2B8/kDzoCTpQ==</latexit>

: ~99,7% das vezes a medida estará neste intervalo



1p2⇡�

e�12

(x�µ)2

�2

<latexit sha1_base64="+xlUvLxuy0N3bodvBsc+u8Nl+Ao=">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</latexit>

Média

Desvio padrão

Densidade de probabilidade gaussiana:



Exemplo: mediu-se a densidade de um material e o valor encontrado foi:

⇢ = (18, 6± 0, 4)g/cm3<latexit sha1_base64="Kgcb7Sxjlx7W79Ec6p/REcJzqio=">AAACBHicbVDLSsNAFJ34rPUVddnNYBEqlJq0RbsRim5cVrAPaGKZTCft0JkkzEyEErpw46+4caGIWz/CnX/jtM1CWw9cOJxzL/fe40WMSmVZ38bK6tr6xmZmK7u9s7u3bx4ctmQYC0yaOGSh6HhIEkYD0lRUMdKJBEHcY6Ttja6nfvuBCEnD4E6NI+JyNAioTzFSWuqZOUcMQ3gJC3ateO5EHFrF6ikcnGF+X+mZeatkzQCXiZ2SPEjR6JlfTj/EMSeBwgxJ2bWtSLkJEopiRiZZJ5YkQniEBqSraYA4kW4ye2ICT7TSh34odAUKztTfEwniUo65pzs5UkO56E3F/7xurPyam9AgihUJ8HyRHzOoQjhNBPapIFixsSYIC6pvhXiIBMJK55bVIdiLLy+TVrlkV0rl22q+fpXGkQE5cAwKwAYXoA5uQAM0AQaP4Bm8gjfjyXgx3o2PeeuKkc4cgT8wPn8AxxqU+w==</latexit>

ouro: ⇢ = 19, 3g/cm3

platina: ⇢ = 21, 4g/cm3

uranio: ⇢ = 19, 1g/cm3<latexit sha1_base64="Wi7PAo1UIJA7wlB+/4RdQ036QMM=">AAACVXicbVHLagIxFM1MbWunD2277CZUC12IndFCH1CQdtOlhfoARyUTowYzyZBkCjL4k25K/6SbQuOjYNUbAodzzr1JToKIUaVd98uyd1K7e/vpA+fw6Pgkkz09qysRS0xqWDAhmwFShFFOappqRpqRJCgMGGkEo5eZ3vggUlHB3/U4Iu0QDTjtU4y0obpZZgaJR5j3C/Mlh+IJeg+FMhzc4LBTzkPfdyJmzBytuUpe4XbVFU u/gzjdMsz7s3WzObfozgtuAm8JcmBZ1W526vcEjkPCNWZIqZbnRrqdIKkpZmTi+LEiEcIjNCAtAzkKiWon81Qm8MowPdgX0myu4Zxd7UhQqNQ4DIwzRHqo1rUZuU1rxbp/304oj2JNOF4c1I8Z1ALOIoY9KgnWbGwAwpKau0I8RBJhbT7CMSF460/eBPVS0SsXS2+lXOV5GUcaXIBLcA08cAcq4BVUQQ1gMAXflmXZ1qf1Y6fsvYXVtpY95+Bf2ZlfTiqq8A==</latexit>

De que metal era feita a peça?

Prática 1: propagação de incertezas




⇢ =m

V<latexit sha1_base64="pyU2MrbzJYXZNXmkoGm8NpMpDtQ=">AAAB+nicbVDLSsNAFJ34rPWV6tLNYBFclaQKuhGKblxWsA9oQplMJ+3QeYSZiVJiPsWNC0Xc+iXu/BunbRbaeuDC4Zx7ufeeKGFUG8/7dlZW19Y3Nktb5e2d3b19t3LQ1jJVmLSwZFJ1I6QJo4K0DDWMdBNFEI8Y6UTjm6nfeSBKUynuzSQhIUdDQWOKkbFS360EaiThVRArhDOeZ+2871a9mjcDXCZ+QaqgQLPvfgUDiVNOhMEMad3zvcSEGVKGYkbycpBqkiA8RkPSs1QgTnSYzU7P4YlVBjCWypYwcKb+nsgQ13rCI9vJkRnpRW8q/uf1UhNfhhkVSWqIwPNFccqgkXCaAxxQRbBhE0sQVtTeCvEI2RSMTatsQ/AXX14m7XrNP6vV786rjesijhI4AsfgFPjgAjTALWiCFsDgETyDV/DmPDkvzrvzMW9dcYqZQ/AHzucPSjaUBg==</latexit>

m = m± �m<latexit sha1_base64="9KKOSo+Rxm+rCpEgonSWnoOg2DM=">AAAB/nicbVBNS8NAEN3Ur1q/ouLJy2IRPJWkCnoRil48VrAf0IQw2W7apbtJ2N0IJRT8K148KOLV3+HNf+O2zUFbHww83pthZl6Ycqa043xbpZXVtfWN8mZla3tnd8/eP2irJJOEtkjCE9kNQVHOYtrSTHPaTSUFEXLaCUe3U7/zSKViSfygxyn1BQxiFjEC2kiBfSSuvRAkFthLTSk2EBCIwK46NWcGvEzcglRRgWZgf3n9hGSCxppwUKrnOqn2c5CaEU4nFS9TNAUyggHtGRqDoMrPZ+dP8KlR+jhKpKlY45n6eyIHodRYhKZTgB6qRW8q/uf1Mh1d+TmL00zTmMwXRRnHOsHTLHCfSUo0HxsCRDJzKyZDkEC0SaxiQnAXX14m7XrNPa/V7y+qjZsijjI6RifoDLnoEjXQHWqiFiIoR8/oFb1ZT9aL9W59zFtLVjFziP7A+vwBKyeU+w==</latexit>

V = V ± �V<latexit sha1_base64="AwiWy+VZz4nxVgbLp4j75asoVHc=">AAAB/nicbVBNS8NAEJ3Ur1q/ouLJy2IRPJWkCnoRil48VrBpoQlhs922S3eTsLsRSij4V7x4UMSrv8Ob/8Ztm4O2Phh4vDfDzLwo5Uxpx/m2Siura+sb5c3K1vbO7p69f+CpJJOEtkjCE9mJsKKcxbSlmea0k0qKRcRpOxrdTv32I5WKJfGDHqc0EHgQsz4jWBsptI+8az/CEnnITwXyFRsIHHqhXXVqzgxombgFqUKBZmh/+b2EZILGmnCsVNd1Uh3kWGpGOJ1U/EzRFJMRHtCuoTEWVAX57PwJOjVKD/UTaSrWaKb+nsixUGosItMpsB6qRW8q/ud1M92/CnIWp5mmMZkv6mcc6QRNs0A9JinRfGwIJpKZWxEZYomJNolVTAju4svLxKvX3PNa/f6i2rgp4ijDMZzAGbhwCQ24gya0gEAOz/AKb9aT9WK9Wx/z1pJVzBzCH1ifP7+NlLY=</latexit>

�⇢ =?<latexit sha1_base64="hLnMZ45Wc1tXrurQPQ96OAIacgY=">AAAB9HicbVBNSwMxEJ31s9avqkcvwSJ4KrtV0ItY9OKxgv2A7lKyabYNzSZrki2Upb/DiwdFvPpjvPlvTNs9aOuDgcd7M8zMCxPOtHHdb2dldW19Y7OwVdze2d3bLx0cNrVMFaENIrlU7RBrypmgDcMMp+1EURyHnLbC4d3Ub42o0kyKRzNOaBDjvmARI9hYKfA168e466uBvL7plspuxZ0BLRMvJ2XIUe+WvvyeJGlMhSEca93x3MQEGVaGEU4nRT/VNMFkiPu0Y6nAMdVBNjt6gk6t0kORVLaEQTP190SGY63HcWg7Y2wGetGbiv95ndREV0HGRJIaKsh8UZRyZCSaJoB6TFFi+NgSTBSztyIywAoTY3Mq2hC8xZeXSbNa8c4r1YeLcu02j6MAx3ACZ+DBJdTgHurQAAJP8Ayv8OaMnBfn3fmYt644+cwR/IHz+QODwJHt</latexit>

Problema: qual a incerteza da densidade dadas as medidas de m e V?

⇢ =m

V± ?

<latexit sha1_base64="/bRyDBNsRB0oF1T6gjcHBpbzlqg=">AAACD3icbVDLSgMxFM3UV62vUZdugkVxUcpMFXQjFt24rGAf0BlKJs20oclkSDJCGeYP3Pgrblwo4tatO//GtJ2Fth643MM595LcE8SMKu0431ZhaXllda24XtrY3NresXf3WkokEpMmFkzIToAUYTQiTU01I51YEsQDRtrB6Gbitx+IVFRE93ocE5+jQURDipE2Us8+9uRQwEvohRLh1AuQhDyb9VYGvYoXc68Cr3p22ak6U8BF4uakDHI0evaX1xc44STSmCGluq4Taz9FUlPMSFbyEkVihEdoQLqGRogT5afTezJ4ZJQ+DIU0FWk4VX9vpIgrNeaBmeRID9W8NxH/87qJDi/8lEZxokmEZw+FCYNawEk4sE8lwZqNDUFYUvNXiIfIJKNNhCUTgjt/8iJp1aruabV2d1auX+dxFMEBOAQnwAXnoA5uQQM0AQaP4Bm8gjfryXqx3q2P2WjBynf2wR9Ynz+nU5si</latexit>

Casos mais simples:

g = ax<latexit sha1_base64="ECg9EANSGzdC5EgtFweWiA7jwgU=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9CIUvXisYNpCG8pku2mXbjZhdyOW0t/gxYMiXv1B3vw3btsctPXBwOO9GWbmhang2rjut7Oyura+sVnYKm7v7O7tlw4OGzrJFGU+TUSiWiFqJrhkvuFGsFaqGMahYM1weDv1m49MaZ7IBzNKWRBjX/KIUzRW8vvXBJ+6pbJbcWcgy8TLSRly1Lulr04voVnMpKECtW57bmqCMSrDqWCTYifTLEU6xD5rWyoxZjoYz46dkFOr9EiUKFvSkJn6e2KMsdajOLSdMZqBXvSm4n9eOzPRVTDmMs0Mk3S+KMoEMQmZfk56XDFqxMgSpIrbWwkdoEJqbD5FG4K3+PIyaVQr3nmlen9Rrt3kcRTgGE7gDDy4hBrcQR18oMDhGV7hzZHOi/PufMxbV5x85gj+wPn8ATpYjk0=</latexit>

A grandeza g e o produto de uma constante (que nao precisa ser medida)por um valor medido x± �x.

<latexit sha1_base64="tVchXgBaMwZTfW28hAxLx49TmUI=">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</latexit>

�g = a�x<latexit sha1_base64="vYJ4Se5Mepk3IoZW3XnaF8hOsXo=">AAAB/XicbZDLSsNAFIYn9VbrLV52bgaL4KokVdCNUHTjsoK9QBvCyXTSDp1JwsxErKH4Km5cKOLW93Dn2zhts9DWHwY+/nMO58wfJJwp7TjfVmFpeWV1rbhe2tjc2t6xd/eaKk4loQ0S81i2A1CUs4g2NNOcthNJQQSctoLh9aTeuqdSsTi606OEegL6EQsZAW0s3z7oKtYX4PfxJYacH3y77FScqfAiuDmUUa66b391ezFJBY004aBUx3US7WUgNSOcjkvdVNEEyBD6tGMwAkGVl02vH+Nj4/RwGEvzIo2n7u+JDIRSIxGYTgF6oOZrE/O/WifV4YWXsShJNY3IbFGYcqxjPIkC95ikRPORASCSmVsxGYAEok1gJROCO//lRWhWK+5ppXp7Vq5d5XEU0SE6QifIReeohm5QHTUQQY/oGb2iN+vJerHerY9Za8HKZ/bRH1mfP8q7lMw=</latexit>

1.



Casos mais simples: área do círculo dada medida do raio

2. z = r2<latexit sha1_base64="r9ltiu7R4QXsshG+Tq9oo66c9uQ=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9mtBb0IRS8eK9gPaNeSTbNtbDZZkqxQl/4HLx4U8er/8ea/MW33oK0PBh7vzTAzL4g508Z1v53cyura+kZ+s7C1vbO7V9w/aGqZKEIbRHKp2gHWlDNBG4YZTtuxojgKOG0Fo+up33qkSjMp7sw4pn6EB4KFjGBjpebTJVL3lV6x5JbdGdAy8TJSggz1XvGr25ckiagwhGOtO54bGz/FyjDC6aTQTTSNMRnhAe1YKnBEtZ/Orp2gE6v0USiVLWHQTP09keJI63EU2M4Im6Fe9Kbif14nMeGFnzIRJ4YKMl8UJhwZiaavoz5TlBg+tgQTxeytiAyxwsTYgAo2BG/x5WXSrJS9s3LltlqqXWVx5OEIjuEUPDiHGtxAHRpA4AGe4RXeHOm8OO/Ox7w152Qzh/AHzucPvDiOkw==</latexit>

r = (1, 1± 0, 1)cm<latexit sha1_base64="S6hUobN2vE11KuyGQVco4HLmXRU=">AAAB/3icbVDLSgMxFM34rPU1KrhxEyxChVImVdCNUHTjsoJ9QGcomTTThiaZIckIZezCX3HjQhG3/oY7/8b0sdDWAxcO59zLvfeECWfaeN63s7S8srq2ntvIb25t7+y6e/sNHaeK0DqJeaxaIdaUM0nrhhlOW4miWIScNsPBzdhvPlClWSzvzTChgcA9ySJGsLFSxz1UV0VUQn4ioFdCp5mvBCRi1HELXtmbAC4SNCMFMEOt43753ZikgkpDONa6jbzEBBlWhhFOR3k/1TTBZIB7tG2pxILqIJvcP4InVunCKFa2pIET9fdEhoXWQxHaToFNX897Y/E/r52a6DLImExSQyWZLopSDk0Mx2HALlOUGD60BBPF7K2Q9LHCxNjI8jYENP/yImlUyuisXLk7L1SvZ3HkwBE4BkWAwAWogltQA3VAwCN4Bq/gzXlyXpx352PauuTMZg7AHzifP+8MlCI=</latexit>

�z = �r2 = 2r�r = 2(1, 1cm)⇥ (0, 1cm) = 0, 22cm2<latexit sha1_base64="4wB8TDOJmtAEoqSjwRdv8jBuYjM=">AAACPnicbVDLSgMxFM3UV62vqks3wSJUKGVmFHRTKLpxWcE+oDMdMmnahiYzQ5IR6tAvc+M3uHPpxoUibl2atiPU1gOBc8+5l9x7/IhRqUzzxcisrK6tb2Q3c1vbO7t7+f2DhgxjgUkdhywULR9JwmhA6ooqRlqRIIj7jDT94fXEb94TIWkY3KlRRFyO+gHtUYyUlrx83ZG0z5H3UElJIjr2uGKLtBSwAu2iVbJg4ggOMR+fOopyIovmvFYxS7b9W3VsL18wy+YUcJlYKSmAFDUv/+x0QxxzEijMkJRty4yUmyChKGZknHNiSSKEh6hP2poGSG/gJtPzx/BEK13YC4V+gYJTdX4iQVzKEfd1J0dqIBe9ifif145V79JNaBDFigR49lEvZlCFcJIl7FJBsGIjTRAWVO8K8QAJhJVOPKdDsBZPXiYNu2ydle3b80L1Ko0jC47AMSgCC1yAKrgBNVAHGDyCV/AOPown4834NL5mrRkjnTkEf2B8/wAIYayJ</latexit>

Dois Algarismos

Significativos

r2 = (1, 21± 0, 22)cm2<latexit sha1_base64="R8JgfcsorCe+oAvHZfwWSHsEEe8=">AAACB3icbVDLSgMxFM34rPU16lKQYBEqlDIzCroRim5cVrAP6ExLJk3b0CQzJBmhDN258VfcuFDErb/gzr8xbWehrQcuHM65l3vvCWNGlXacb2tpeWV1bT23kd/c2t7Ztff26ypKJCY1HLFINkOkCKOC1DTVjDRjSRAPGWmEw5uJ33ggUtFI3OtRTAKO+oL2KEbaSB37SLY9eAWLbslz/ZhDp+R5p6kvOcR83PY6dsEpO1PAReJmpAAyVDv2l9+NcMKJ0JghpVquE+sgRVJTzMg47yeKxAgPUZ+0DBWIExWk0z/G8MQoXdiLpCmh4VT9PZEirtSIh6aTIz1Q895E/M9rJbp3GaRUxIkmAs8W9RIGdQQnocAulQRrNjIEYUnNrRAPkERYm+jyJgR3/uVFUvfK7lnZuzsvVK6zOHLgEByDInDBBaiAW1AFNYDBI3gGr+DNerJerHfrY9a6ZGUzB+APrM8f93SWNw==</latexit>

3. z = ⇡r2<latexit sha1_base64="TfXxybTkbWkD2AsRK0uO+lbOVbY=">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRbBU9mtBb0IRS8eK9gPadeSTbNtaJJdkqxQl/4KLx4U8erP8ea/MW33oK0PBh7vzTAzL4g508Z1v53cyura+kZ+s7C1vbO7V9w/aOooUYQ2SMQj1Q6wppxJ2jDMcNqOFcUi4LQVjK6nfuuRKs0ieWfGMfUFHkgWMoKNle6fLrsxQ+qh0iuW3LI7A1omXkZKkKHeK351+xFJBJWGcKx1x3Nj46dYGUY4nRS6iaYxJiM8oB1LJRZU++ns4Ak6sUofhZGyJQ2aqb8nUiy0HovAdgpshnrRm4r/eZ3EhBd+ymScGCrJfFGYcGQiNP0e9ZmixPCxJZgoZm9FZIgVJsZmVLAheIsvL5NmpeydlSu31VLtKosjD0dwDKfgwTnU4Abq0AACAp7hFd4c5bw4787HvDXnZDOH8AfO5w8Hqo/m</latexit>

�z = ⇡�r = ⇡ ⇥ (0, 22)cm2 = 0.69115 ⇡ 0.69<latexit sha1_base64="RXSFHZvHa2IPtbh45cIK6vL5jWA=">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</latexit>

2<latexit sha1_base64="jk/1fpohXujb3eq/tOFNvjxoFrw=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZq1frrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJu1b1Lqq15mWlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8ffW+Mug==</latexit>

z = ⇡r2 = ⇡ ⇥ (1, 21)cm2 = 3, 80133cm2 ⇡ 3, 80 cm2<latexit sha1_base64="Amp/Ggx9ay0kpimBljxDegT5CWo=">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</latexit>

A = (3, 80± 0, 69) cm2<latexit sha1_base64="YjcCO3n4+HVigZ9qXYsFihMHmtc=">AAACBnicbVDLSsNAFJ3UV62vqEsRBotQoZSkFa0LoerGZQX7gCaWyXTSDp1JwsxEKKErN/6KGxeKuPUb3Pk3TtsstHrgwuGce7n3Hi9iVCrL+jIyC4tLyyvZ1dza+sbmlrm905RhLDBp4JCFou0hSRgNSENRxUg7EgRxj5GWN7ya+K17IiQNg1s1iojLUT+gPsVIaalr7l+cFyrFqgWdiEOreHJ25BQTR3CI+fiu3DXzVsmaAv4ldkryIEW9a346vRDHnAQKMyRlx7Yi5SZIKIoZGeecWJII4SHqk46mAeJEusn0jTE81EoP+qHQFSg4VX9OJIhLOeKe7uRIDeS8NxH/8zqx8qtuQoMoViTAs0V+zKAK4SQT2KOCYMVGmiAsqL4V4gESCCudXE6HYM+//Jc0yyW7UirfHOdrl2kcWbAHDkAB2OAU1MA1qIMGwOABPIEX8Go8Gs/Gm/E+a80Y6cwu+AXj4xtcPJXm</latexit>

cm2<latexit sha1_base64="TvaF+tPkXGmj0unNBbrb+XwKseY=">AAAB8XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME88BkDbOT2WTIPJaZWSEs+QsvHhTx6t9482+cJHvQxIKGoqqb7q4o4cxY3//2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCKK92OsKGcSdqwzHLaTjTFIuK0FY1upn7riWrDlLy344SGAg8kixnB1kkPWVcLRMTksdorlf2KPwNaJkFOypCj3it9dfuKpIJKSzg2phP4iQ0zrC0jnE6K3dTQBJMRHtCOoxILasJsdvEEnTqlj2KlXUmLZurviQwLY8Yicp0C26FZ9Kbif14ntfFVmDGZpJZKMl8UpxxZhabvoz7TlFg+dgQTzdytiAyxxsS6kIouhGDx5WXSrFaC80r17qJcu87jKMAxnMAZBHAJNbiFOjSAgIRneIU3z3gv3rv3MW9d8fKZI/gD7/MHK1KQlQ==</latexit>



⇢ =m

V<latexit sha1_base64="pyU2MrbzJYXZNXmkoGm8NpMpDtQ=">AAAB+nicbVDLSsNAFJ34rPWV6tLNYBFclaQKuhGKblxWsA9oQplMJ+3QeYSZiVJiPsWNC0Xc+iXu/BunbRbaeuDC4Zx7ufeeKGFUG8/7dlZW19Y3Nktb5e2d3b19t3LQ1jJVmLSwZFJ1I6QJo4K0DDWMdBNFEI8Y6UTjm6nfeSBKUynuzSQhIUdDQWOKkbFS360EaiThVRArhDOeZ+2871a9mjcDXCZ+QaqgQLPvfgUDiVNOhMEMad3zvcSEGVKGYkbycpBqkiA8RkPSs1QgTnSYzU7P4YlVBjCWypYwcKb+nsgQ13rCI9vJkRnpRW8q/uf1UhNfhhkVSWqIwPNFccqgkXCaAxxQRbBhE0sQVtTeCvEI2RSMTatsQ/AXX14m7XrNP6vV786rjesijhI4AsfgFPjgAjTALWiCFsDgETyDV/DmPDkvzrvzMW9dcYqZQ/AHzucPSjaUBg==</latexit>

m = m± �m<latexit sha1_base64="9KKOSo+Rxm+rCpEgonSWnoOg2DM=">AAAB/nicbVBNS8NAEN3Ur1q/ouLJy2IRPJWkCnoRil48VrAf0IQw2W7apbtJ2N0IJRT8K148KOLV3+HNf+O2zUFbHww83pthZl6Ycqa043xbpZXVtfWN8mZla3tnd8/eP2irJJOEtkjCE9kNQVHOYtrSTHPaTSUFEXLaCUe3U7/zSKViSfygxyn1BQxiFjEC2kiBfSSuvRAkFthLTSk2EBCIwK46NWcGvEzcglRRgWZgf3n9hGSCxppwUKrnOqn2c5CaEU4nFS9TNAUyggHtGRqDoMrPZ+dP8KlR+jhKpKlY45n6eyIHodRYhKZTgB6qRW8q/uf1Mh1d+TmL00zTmMwXRRnHOsHTLHCfSUo0HxsCRDJzKyZDkEC0SaxiQnAXX14m7XrNPa/V7y+qjZsijjI6RifoDLnoEjXQHWqiFiIoR8/oFb1ZT9aL9W59zFtLVjFziP7A+vwBKyeU+w==</latexit>

V = V ± �V<latexit sha1_base64="AwiWy+VZz4nxVgbLp4j75asoVHc=">AAAB/nicbVBNS8NAEJ3Ur1q/ouLJy2IRPJWkCnoRil48VrBpoQlhs922S3eTsLsRSij4V7x4UMSrv8Ob/8Ztm4O2Phh4vDfDzLwo5Uxpx/m2Siura+sb5c3K1vbO7p69f+CpJJOEtkjCE9mJsKKcxbSlmea0k0qKRcRpOxrdTv32I5WKJfGDHqc0EHgQsz4jWBsptI+8az/CEnnITwXyFRsIHHqhXXVqzgxombgFqUKBZmh/+b2EZILGmnCsVNd1Uh3kWGpGOJ1U/EzRFJMRHtCuoTEWVAX57PwJOjVKD/UTaSrWaKb+nsixUGosItMpsB6qRW8q/ud1M92/CnIWp5mmMZkv6mcc6QRNs0A9JinRfGwIJpKZWxEZYomJNolVTAju4svLxKvX3PNa/f6i2rgp4ijDMZzAGbhwCQ24gya0gEAOz/AKb9aT9WK9Wx/z1pJVzBzCH1ifP7+NlLY=</latexit>

�⇢ =?<latexit sha1_base64="hLnMZ45Wc1tXrurQPQ96OAIacgY=">AAAB9HicbVBNSwMxEJ31s9avqkcvwSJ4KrtV0ItY9OKxgv2A7lKyabYNzSZrki2Upb/DiwdFvPpjvPlvTNs9aOuDgcd7M8zMCxPOtHHdb2dldW19Y7OwVdze2d3bLx0cNrVMFaENIrlU7RBrypmgDcMMp+1EURyHnLbC4d3Ub42o0kyKRzNOaBDjvmARI9hYKfA168e466uBvL7plspuxZ0BLRMvJ2XIUe+WvvyeJGlMhSEca93x3MQEGVaGEU4nRT/VNMFkiPu0Y6nAMdVBNjt6gk6t0kORVLaEQTP190SGY63HcWg7Y2wGetGbiv95ndREV0HGRJIaKsh8UZRyZCSaJoB6TFFi+NgSTBSztyIywAoTY3Mq2hC8xZeXSbNa8c4r1YeLcu02j6MAx3ACZ+DBJdTgHurQAAJP8Ayv8OaMnBfn3fmYt644+cwR/IHz+QODwJHt</latexit>

Problema: qual a incerteza da densidade dadas as medidas de m e V?

⇢ =m

V± ?

<latexit sha1_base64="/bRyDBNsRB0oF1T6gjcHBpbzlqg=">AAACD3icbVDLSgMxFM3UV62vUZdugkVxUcpMFXQjFt24rGAf0BlKJs20oclkSDJCGeYP3Pgrblwo4tatO//GtJ2Fth643MM595LcE8SMKu0431ZhaXllda24XtrY3NresXf3WkokEpMmFkzIToAUYTQiTU01I51YEsQDRtrB6Gbitx+IVFRE93ocE5+jQURDipE2Us8+9uRQwEvohRLh1AuQhDyb9VYGvYoXc68Cr3p22ak6U8BF4uakDHI0evaX1xc44STSmCGluq4Taz9FUlPMSFbyEkVihEdoQLqGRogT5afTezJ4ZJQ+DIU0FWk4VX9vpIgrNeaBmeRID9W8NxH/87qJDi/8lEZxokmEZw+FCYNawEk4sE8lwZqNDUFYUvNXiIfIJKNNhCUTgjt/8iJp1aruabV2d1auX+dxFMEBOAQnwAXnoA5uQQM0AQaP4Bm8gjfryXqx3q2P2WjBynf2wR9Ynz+nU5si</latexit>

Resolva por partes e aplique as fórmulas a cada passo.

V = L1L2L3 = (L1L2)L3 = AL3<latexit sha1_base64="bZ/qXWOmz6FEJnXB/xI0FMucoeo=">AAACBnicbZBNS8MwGMfT+TbnW9WjCMEhzMtoN0Evg6kXDztMcC+wlZJm6RaWpiVJhVF28uJX8eJBEa9+Bm9+G9OtB918IOHH//88JM/fixiVyrK+jdzK6tr6Rn6zsLW9s7tn7h+0ZRgLTFo4ZKHoekgSRjlpKaoY6UaCoMBjpOONb1K/80CEpCG/V5OIOAEacupTjJSWXPO4XYMN1264lYZbrZXmeJbylb5cs2iVrVnBZbAzKIKsmq751R+EOA4IV5ghKXu2FSknQUJRzMi00I8liRAeoyHpaeQoINJJZmtM4alWBtAPhT5cwZn6eyJBgZSTwNOdAVIjueil4n9eL1b+pZNQHsWKcDx/yI8ZVCFMM4EDKghWbKIBYUH1XyEeIYGw0skVdAj24srL0K6U7Wq5cnderF9nceTBETgBJWCDC1AHt6AJWgCDR/AMXsGb8WS8GO/Gx7w1Z2Qzh+BPGZ8/9L6VnA==</latexit>

�A =?<latexit sha1_base64="nqKAKDhWfmQLOMfcddTDRwGB/s0=">AAAB8XicbVBNSwMxEJ2tX7V+VT16CRbBU9mtBb2IVS8eK9gPbJeSTbNtaJJdkqxQlv4LLx4U8eq/8ea/MW33oK0PBh7vzTAzL4g508Z1v53cyura+kZ+s7C1vbO7V9w/aOooUYQ2SMQj1Q6wppxJ2jDMcNqOFcUi4LQVjG6nfuuJKs0i+WDGMfUFHkgWMoKNlR67mg0E7l1fXvWKJbfszoCWiZeREmSo94pf3X5EEkGlIRxr3fHc2PgpVoYRTieFbqJpjMkID2jHUokF1X46u3iCTqzSR2GkbEmDZurviRQLrccisJ0Cm6Fe9Kbif14nMeGFnzIZJ4ZKMl8UJhyZCE3fR32mKDF8bAkmitlbERlihYmxIRVsCN7iy8ukWSl7Z+XKfbVUu8niyMMRHMMpeHAONbiDOjSAgIRneIU3RzsvzrvzMW/NOdnMIfyB8/kD64+Qaw==</latexit>

�⇢ =m�V + V �m

V 2<latexit sha1_base64="N/TGIHKi/SuytwHC+f6c/wjwyGM=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiCUJIq2I1QdOOygk0LTQyT6aQdOpOEmYlQQn7Fjb/ixoUi3Yk/4zTNQlsPXDhzzr3MvSdIGJXKsr6MldW19Y3NylZ1e2d3b988OHRknApMOjhmsegFSBJGI9JRVDHSSwRBPGCkG4xvZ373iQhJ4+hBTRLicTSMaEgxUlryzaYr6ZAj3xWjGF67oUA4cwMkIC8NB54XbweWAs8z57GR+2bNqlsF4DKxS1IDJdq+OXUHMU45iRRmSMq+bSXKy5BQFDOSV91UkgThMRqSvqYR4kR6WXFhDk+1MoBhLHRFChbq74kMcSknPNCdHKmRXPRm4n9eP1Vh08tolKSKRHj+UZgyqGI4iwsOqCBYsYkmCAuqd4V4hHRKSoda1SHYiycvE6dRty/qjfvLWuumjKMCjsEJOAM2uAItcAfaoAMweAav4B18GC/Gm/FpTOetK0Y5cwT+wPj+Af9Yo2Y=</latexit>

Depois de encontrar a incerteza de V aplique a fórmula final

Prática 1: compatibilidade entre medidas





Exemplo: dois grupos mediram a densidade de duas peças de metal e encontraram

⇢2 = (19, 2± 0, 3)g/cm3<latexit sha1_base64="Q+5YSYaJrnAn0XYi+ryv2KGDEHE=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBahQqlJKqgLoejGZQX7gCaGyXTaDp1JwsxEKKEbN/6KGxeKuPUf3Pk3TtsstHrgwuGce7n3niBmVCrL+jJyC4tLyyv51cLa+sbmlrm905RRIjBp4IhFoh0gSRgNSUNRxUg7FgTxgJFWMLya+K17IiSNwls1ionHUT+kPYqR0pJv7rtiEPnORck+LztuzKFVrh6l/WPMx3dV3yxaFWsK+JfYGSmCDHXf/HS7EU44CRVmSMqObcXKS5FQFDMyLriJJDHCQ9QnHU1DxIn00ukXY3iolS7sRUJXqOBU/TmRIi7liAe6kyM1kPPeRPzP6ySqd+alNIwTRUI8W9RLGFQRnEQCu1QQrNhIE4QF1bdCPEACYaWDK+gQ7PmX/5KmU7GrFefmpFi7zOLIgz1wAErABqegBq5BHTQABg/gCbyAV+PReDbejPdZa87IZnbBLxgf37mIlio=</latexit>

As peças eram feitas do mesmo material? Sim? Não? Talvez?Teste de compatibilidade: Duas medidas são compatíveis (podem ser medidas da mesma coisa) se a diferença entre as duas pode ser zero (no sentido estatístico).

x1 ± �1<latexit sha1_base64="GtjchniNDjrDdFNlZv3DqUrFqaI=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSyCp5JUQY9FLx4r2A9oQthsN+3S3STsbqQl9K948aCIV/+IN/+N2zYHbX0w8Hhvhpl5YcqZ0o7zbZU2Nre2d8q7lb39g8Mj+7jaUUkmCW2ThCeyF2JFOYtpWzPNaS+VFIuQ0244vpv73ScqFUviRz1NqS/wMGYRI1gbKbCrk8BFXiqQp9hQ4CB3Z4Fdc+rOAmiduAWpQYFWYH95g4RkgsaacKxU33VS7edYakY4nVW8TNEUkzEe0r6hMRZU+fni9hk6N8oARYk0FWu0UH9P5FgoNRWh6RRYj9SqNxf/8/qZjm78nMVppmlMlouijCOdoHkQaMAkJZpPDcFEMnMrIiMsMdEmrooJwV19eZ10GnX3st54uKo1b4s4ynAKZ3ABLlxDE+6hBW0gMIFneIU3a2a9WO/Wx7K1ZBUzJ/AH1ucPHfCT2Q==</latexit>

x2 ± �2<latexit sha1_base64="Xfpm5owgIDrH4DSchCpFdMNyVRg=">AAAB+3icbVDLSsNAFJ3UV62vWJduBovgqiRR0GXRjcsK9gFNCJPppB06MwkzE2kJ+RU3LhRx64+482+ctllo64ELh3Pu5d57opRRpR3n26psbG5t71R3a3v7B4dH9nG9q5JMYtLBCUtkP0KKMCpIR1PNSD+VBPGIkV40uZv7vSciFU3Eo56lJOBoJGhMMdJGCu36NPSgn3LoKzriKMy9IrQbTtNZAK4TtyQNUKId2l/+MMEZJ0JjhpQauE6qgxxJTTEjRc3PFEkRnqARGRgqECcqyBe3F/DcKEMYJ9KU0HCh/p7IEVdqxiPTyZEeq1VvLv7nDTId3wQ5FWmmicDLRXHGoE7gPAg4pJJgzWaGICypuRXiMZIIaxNXzYTgrr68Trpe071seg9XjdZtGUcVnIIzcAFccA1a4B60QQdgMAXP4BW8WYX1Yr1bH8vWilXOnIA/sD5/ACEIk9s=</latexit>

{|x1 � x2|� 2�(x1�x2), |x1 � x2|+ 2�(x1�x2)}<latexit sha1_base64="n7xFLBLohEcAD/PZS1A8edrQhUs=">AAACK3icbVDLSsNAFJ3UV62vqEs3g0WoaEsSBV2WunFZwT6gCWEynbZDJw9mJtKS9n/c+CsudOEDt/6H0zaItj1w4XDOvdx7jxcxKqRhfGiZldW19Y3sZm5re2d3T98/qIsw5pjUcMhC3vSQIIwGpCapZKQZcYJ8j5GG17+Z+I0HwgUNg3s5jIjjo25AOxQjqSRXr9gJHA1cszhwrVHRsgXt+shNCql0Oobnv/bZMtseu3reKBlTwEVipiQPUlRd/cVuhzj2SSAxQ0K0TCOSToK4pJiRcc6OBYkQ7qMuaSkaIJ8IJ5n+OoYnSmnDTshVBRJO1b8TCfKFGPqe6vSR7Il5byIu81qx7Fw7CQ2iWJIAzxZ1YgZlCCfBwTblBEs2VARhTtWtEPcQR1iqeHMqBHP+5UVSt0rmRcm6u8yXK2kcWXAEjkEBmOAKlMEtqIIawOARPIM38K49aa/ap/Y1a81o6cwh+Aft+wdZxaVN</latexit>

O zero deve estar contido no intervalo de 2 desvio padrões da diferença

(2sigma é uma convenção [95,5% de confiança], 3 sigma daria resultado parecido).




Exemplo: dois grupos mediram a densidade de duas peças de metal e encontraram

⇢2 = (19, 2± 0, 3)g/cm3<latexit sha1_base64="Q+5YSYaJrnAn0XYi+ryv2KGDEHE=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBahQqlJKqgLoejGZQX7gCaGyXTaDp1JwsxEKKEbN/6KGxeKuPUf3Pk3TtsstHrgwuGce7n3niBmVCrL+jJyC4tLyyv51cLa+sbmlrm905RRIjBp4IhFoh0gSRgNSUNRxUg7FgTxgJFWMLya+K17IiSNwls1ionHUT+kPYqR0pJv7rtiEPnORck+LztuzKFVrh6l/WPMx3dV3yxaFWsK+JfYGSmCDHXf/HS7EU44CRVmSMqObcXKS5FQFDMyLriJJDHCQ9QnHU1DxIn00ukXY3iolS7sRUJXqOBU/TmRIi7liAe6kyM1kPPeRPzP6ySqd+alNIwTRUI8W9RLGFQRnEQCu1QQrNhIE4QF1bdCPEACYaWDK+gQ7PmX/5KmU7GrFefmpFi7zOLIgz1wAErABqegBq5BHTQABg/gCbyAV+PReDbejPdZa87IZnbBLxgf37mIlio=</latexit>

As peças eram feitas do mesmo material? Sim? Não? Talvez?

�⇢1�⇢2 =q

�21 + �2

2 = 0.5g/cm3

<latexit sha1_base64="Uv3AL2LL+Xri95fWMpDwPxgI7JU=">AAACKHicbZDLSgMxFIYz9VbrrerSTbAIglhnpopuikU3LivYC3TaIZOmbWgyMyYZoQzzOG58FTciinTrk5i2s9DWAyEf/38Oyfm9kFGpTHNsZJaWV1bXsuu5jc2t7Z387l5dBpHApIYDFoimhyRh1Cc1RRUjzVAQxD1GGt7wduI3noiQNPAf1CgkbY76Pu1RjJSW3Py1I2mfIzd2xCBwrdPpZSewDB35KFSc2lbHPknR7thJ2S xe9M8w75TcfMEsmtOCi2ClUABpVd38u9MNcMSJrzBDUrYsM1TtGAlFMSNJzokkCREeoj5pafQRJ7IdTxdN4JFWurAXCH18Bafq74kYcSlH3NOdHKmBnPcm4n9eK1K9q3ZM/TBSxMezh3oRgyqAk9RglwqCFRtpQFhQ/VeIB0ggrHS2OR2CNb/yItTtolUq2vfnhcpNGkcWHIBDcAwscAkq4A5UQQ1g8AxewQf4NF6MN+PLGM9aM0Y6sw/+lPH9A9iBpSk=</latexit>

|⇢1 � ⇢2| = 0.6g/cm3<latexit sha1_base64="gpIGh4jBh7n5jg6n56J/ut+MAlY=">AAACAnicbVDLSsNAFJ3UV62vqCtxEyyCG2PSiroRim5cVrAPaGOYTKft0JlMmJkIJS1u/BU3LhRx61e482+cpllo64HLPZxzLzP3BBElUjnOt5FbWFxaXsmvFtbWNza3zO2duuSxQLiGOOWiGUCJKQlxTRFFcTMSGLKA4kYwuJ74jQcsJOHhnRpG2GOwF5IuQVBpyTf3Rm3R5757nLbS6NKxz3oniN2XfbPo2E4Ka564GSmCDFXf/Gp3OIoZDhWiUMqW60TKS6BQBFE8LrRjiSOIBrCHW5qGkGHpJekJY+tQKx2ry4WuUFmp+nsjgUzKIQv0JIOqL2e9ifif14pV98JLSBjFCodo+lA3ppbi1iQPq0MERooONYFIEP1XC/WhgEjp1Ao6BHf25HlSL9lu2S7dnhYrV1kcebAPDsARcME5qIAbUAU1gMAjeAav4M14Ml6Md+NjOpozsp1d8AfG5w8zV5YB</latexit>

|⇢1 � ⇢2| = (0, 6± 0, 5)g/cm3<latexit sha1_base64="WOTczUEdsY8gf5RIzmBWWMsPU6o=">AAACC3icbVDLTgIxFO34RHyNunTTQEwwQZwBXxsTohuXmMgjYUbSKQUa2umk7ZgQYO/GX3HjQmPc+gPu/BvLY6HgSW7uyTn3pr0niBhV2nG+rYXFpeWV1cRacn1jc2vb3tmtKBFLTMpYMCFrAVKE0ZCUNdWM1CJJEA8YqQbd65FffSBSURHe6V5EfI7aIW1RjLSRGnZq4MmOaLhH45YfXGac7JkXcehkTw/bx5jfFxp22sk5Y8B54k5JGkxRathfXlPgmJNQY4aUqrtOpP0+kppiRoZJL1YkQriL2qRuaIg4UX5/fMsQHhilCVtCmgo1HKu/N/qIK9XjgZnkSHfUrDcS//PqsW5d+H0aRrEmIZ481IoZ1AKOgoFNKgnWrGcIwpKav0LcQRJhbeJLmhDc2ZPnSSWfcwu5/O1Jung1jSMB9kEKZIALzkER3IASKAMMHsEzeAVv1pP1Yr1bH5PRBWu6swf+wPr8ATHMmJQ=</latexit>

Resposta: São compatíveis, é possível que as peças sejam do mesmo material. (É até bastante provável, a diferença é zero com pouco mais de um desvio padrão.)

Prática 1: Resumo das atividades


P1: Resumo das atividades (ver apostila)


1. Meça todas as peças com paquímetro (para cálculo do volume)2. Faça desenho esquemático das peças e suas medidas.3. Calcule o volume de cada peça com a incerteza associada.4. Relatório: Faça uma tabela com os resultados.

1. Meça a massa das peças com balança2. Usando o volume medido indiretamente obtenha as densidades (com

incerteza).3. Relatório: De acordo com suas medidas (com incerteza) proponha uma

identificação para o material da peça (usar tabelas de ref).

Medidas sem dispersão (TURMA 1)



Medidas com dispersão (TURMA 1)

1. Você vai receber os dados de 10 medidas por email.2. Calcule o diâmetro médio.3. Calcule o desvio absoluto médio.4. Calcule o desvio padrão da média!5. Compare o desvio padrão da média com a incerteza do

micrômetro. Qual é maior? Qual a incerteza final da medida?6. Relatório: Faça a discussão do item e) da apostila.



1. Meça todas as peças com paquímetro e micrômetro (para cálculo do volume)

2. Faça desenho esquemático das peças e suas medidas.3. Calcule o volume de cada peça com a incerteza associada.4. Meça o volume (com incerteza) diretamente com a proveta.5. Relatório: Faça uma tabela com os resultados, compare as medidas

(levando em conta as incertezas!).

1. Meça a massa das peças com balança (cuidado com erros sistemáticos)2. Usando o volume medido indiretamente obtenha as densidades (com

incerteza).3. Relatório: De acordo com suas medidas (com incerteza) proponha uma

identificação para o material da peça (usar tabelas de ref).

Medidas sem dispersão (TURMA 2)



Medida direta do volume com a proveta (TURMA 2)

A água exerce força de empuxo sobre a peça, que exerce a mesma força em direção contrária sobre a água: balança indica mais massa.

Empuxo é o peso (mg) da massa de água deslocada.

Fe = ⇢H2OVobg<latexit sha1_base64="1xzdbVX+YU1cLkxR7ryqmwPeUJM=">AAACBnicbVBNS8NAEN3Ur1q/oh5FWCwFTyWpgl6EoiC9WcG2QhPCZrttl252w+5GKCEnL/4VLx4U8epv8Oa/cdvmoK0PBh7vzTAzL4wZVdpxvq3C0vLK6lpxvbSxubW9Y+/utZVIJCYtLJiQ9yFShFFOWppqRu5jSVAUMtIJR1cTv/NApKKC3+lxTPwIDTjtU4y0kQL78DogF54ciiBtBLWbDLaD1JMRFGEGBzCwy07VmQIuEjcnZZCjGdhfXk/gJCJcY4aU6rpOrP0USU0xI1nJSxSJER6hAekaylFElJ9O38hgxSg92BfSFNdwqv6eSFGk1Dgyl1UipIdq3puI/3ndRPfP/ZTyONGE49mifsKgFnCSCexRSbBmY0MQltTcCvEQSYS1Sa5kQnDnX14k7VrVPanWbk/L9cs8jiI4AEfgGLjgDNRBAzRBC2DwCJ7BK3iznqwX6936mLUWrHxmH/yB9fkDgP2X3A==</latexit>

Se a balança estiver zerada, o valor indicado é o volume da peça em ml:

⇢H2O Vob = X g

Vob =X g

⇢H2O=

Xg

1g/ml= Xml.

<latexit sha1_base64="yFtPFcDEJCHyexYcc7Cp6thzaMw=">AAACcHicbVHLSgMxFM2M7/qquhFUjBZFpNSZKuhGEN24U8HWQlOGTJppQ5PJkGSEMsza/3PnR7jxC0wfSn1cCJx7zr03uSdhwpk2nvfmuFPTM7Nz8wuFxaXlldXi2npdy1QRWiOSS9UIsaacxbRmmOG0kSiKRcjpU9i7GehPz1RpJuNH009oS+BOzCJGsLFUUHw5RKorg+w2qN7lqAzrQYaUgDLM4SVsQFRG5SHRyREqHP5QUa QwyRrfBXk2MWqiAH7r/hc8ETy/tJ2j3CaVoFjyKt4w4F/gj0EJjOM+KL6itiSpoLEhHGvd9L3EtDKsDCOc5gWUappg0sMd2rQwxoLqVjY0LIcHlmnDSCp7YgOH7GRHhoXWfWHXPBDYdPVvbUD+pzVTE120MhYnqaExGV0UpRwaCQfuwzZTlBjetwATxexbIelia5Oxf1SwJvi/V/4L6tWKf1qpPpyVrq7HdsyDLbAPjoAPzsEVuAX3oAYIeHc2nG1nx/lwN91dd29U6jrjng3wI9zjTzAOuas=</latexit>

Aceleração da gravidade

Medida da balança em gramas (g!)

⇢ =0, 997g

ml<latexit sha1_base64="wCNhsZxGKipc5aPrs776LRXw+ys=">AAACCXicbVDLSgMxFM3UV62vUZdugkVwIWWmCrULoejGZQX7gE4pmTTThiaZIckIZZitG3/FjQtF3PoH7vwb0+kstPXA5R7OuZfkHj9iVGnH+bYKK6tr6xvFzdLW9s7unr1/0FZhLDFp4ZCFsusjRRgVpKWpZqQbSYK4z0jHn9zM/M4DkYqG4l5PI9LnaCRoQDHSRhrY0JPj8MoLJMKJc1av1xJPcjhK06xzlg7sslNxMsBl4uakDHI0B/aXNwxxzInQmCGleq4T6X6CpKaYkbTkxYpECE/QiPQMFYgT1U+yS1J4YpQhDEJpSmiYqb83EsSVmnLfTHKkx2rRm4n/eb1YB5f9hIoo1kTg+UNBzKAO4SwWOKSSYM2mhiAsqfkrxGNkUtEmvJIJwV08eZm0qxX3vFK9uyg3rvM4iuAIHINT4IIaaIBb0AQtgMEjeAav4M16sl6sd+tjPlqw8p1D8AfW5w+kzZms</latexit>

A 25ºC



Medidas com dispersão (TURMA 2)

1. Meça 10 vezes o diâmetro do fio com micrômetro, em diferentes pontos, para que tenhamos dispersão nos dados.

2. Calcule o diâmetro médio3. Calcule o desvio absoluto médio.4. Calcule o desvio padrão da média!5. Compare o desvio padrão da média com a incerteza do

micrômetro. Qual é maior? Qual a incerteza final da medida?6. Relatório: Faça a discussão do item e) da apostila.

prática 1: medidas e incertezas

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