Rotações: momento de inércia e conservação do momento angular

Rotações: momento de inércia e conservação do momento angular

Lab. de Física A - Eng. Amb.Diogo Boito, Junho de 2020

Diogo Boito, Junho de 2020

Rotações: torque e momento de inérciaInércia: resistência a mudar o estado de movimento

(velocidade)

F = ma<latexit sha1_base64="IhRE8B0Kz7CL433STMkcIreWgOY=">AAAB7nicbVBNSwMxEJ34WetX1aOXYBE8ld0q6EUoCuKxgv2AdinZNNuGJtklyQpl6Y/w4kERr/4eb/4b03YP2vpg4PHeDDPzwkRwYz3vG62srq1vbBa2its7u3v7pYPDpolTTVmDxiLW7ZAYJrhiDcutYO1EMyJDwVrh6Hbqt56YNjxWj3acsECSgeIRp8Q6qXWHr7HEpFcqexVvBrxM/JyUIUe9V/rq9mOaSqYsFcSYju8lNsiItpwKNil2U8MSQkdkwDqOKiKZCbLZuRN86pQ+jmLtSlk8U39PZEQaM5ah65TEDs2iNxX/8zqpja6CjKsktUzR+aIoFdjGePo77nPNqBVjRwjV3N2K6ZBoQq1LqOhC8BdfXibNasU/r1QfLsq1mzyOAhzDCZyBD5dQg3uoQwMojOAZXuENJegFvaOPeesKymeO4A/Q5w+kII51</latexit>

Inércia da translação é a massa

Quanto maior a massa, mais difícil colocar o corpo em movimento, ou alterar a velocidade.


Qual o análogo em rotações? O que é a “força”? O que faz o papel de massa das rotações?

Torque é a “força" das rotações.

Rotações: torque e momento de inércia


Torque é a “força" das rotações.

Rotações: torque e momento de inércia

⌧ = Fd sin(✓)<latexit sha1_base64="hTzXhBrbzHASVUXGSXwLRFr8bog=">AAACAXicbVBNS8NAEN3Ur1q/ol4EL4tFqJeSVEEvQlEQjxXsBzShbDbbdulmE3YnQin14l/x4kERr/4Lb/4bt20O2vpg4PHeDDPzgkRwDY7zbeWWlldW1/LrhY3Nre0de3evoeNUUVansYhVKyCaCS5ZHTgI1koUI1EgWDMYXE/85gNTmsfyHoYJ8yPSk7zLKQEjdewDD0iKL/ENDrGnuSx50GdATnDHLjplZwq8SNyMFFGGWsf+8sKYphGTQAXRuu06CfgjooBTwcYFL9UsIXRAeqxtqCQR0/5o+sEYHxslxN1YmZKAp+rviRGJtB5GgemMCPT1vDcR//PaKXQv/BGXSQpM0tmibiowxHgSBw65YhTE0BBCFTe3YtonilAwoRVMCO78y4ukUSm7p+XK3VmxepXFkUeH6AiVkIvOURXdohqqI4oe0TN6RW/Wk/VivVsfs9aclc3soz+wPn8A1DGVNQ==</latexit>

[N ·m]<latexit sha1_base64="on4a4FxMps1n8cMwtVjEuvVdTRA=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04kkq2A9MQ9lsNu3SzW7Y3Qgl9F948aCIV/+NN/+N2zYHbX0w8Hhvhpl5YcqZNq777aysrq1vbJa2yts7u3v7lYPDtpaZIrRFJJeqG2JNORO0ZZjhtJsqipOQ0044upn6nSeqNJPiwYxTGiR4IFjMCDZWevTveiSSBiVBv1J1a+4MaJl4BalCgWa/8tWLJMkSKgzhWGvfc1MT5FgZRjidlHuZpikmIzygvqUCJ1QH+eziCTq1SoRiqWwJg2bq74kcJ1qPk9B2JtgM9aI3Ff/z/MzEV0HORJoZKsh8UZxxZCSavo8ipigxfGwJJorZWxEZYoWJsSGVbQje4svLpF2veee1+v1FtXFdxFGCYziBM/DgEhpwC01oAQEBz/AKb452Xpx352PeuuIUM0fwB87nDwJykHs=</latexit>

~⌧ = ~r ⇥ ~F<latexit sha1_base64="ewGBQlZshy22T6tzHE9Tbju9MLg=">AAACB3icbZBNS8NAEIY39avWr6hHQRaL4KkkVdCLUBTEYwXbCk0om+2mXbrZhN1JoZTevPhXvHhQxKt/wZv/xm2ag7YOLDzzzgyz8waJ4Boc59sqLC2vrK4V10sbm1vbO/buXlPHqaKsQWMRq4eAaCa4ZA3gINhDohiJAsFaweB6Wm8NmdI8lvcwSpgfkZ7kIacEjNSxD70ho9gDkl5mpAzziGmcZTcdu+xUnCzwIrg5lFEe9Y795XVjmkZMAhVE67brJOCPiQJOBZuUvFSzhNAB6bG2QUnMLn+c3THBx0bp4jBW5knAmfp7YkwirUdRYDojAn09X5uK/9XaKYQX/pjLJAUm6WxRmAoMMZ6agrtcMQpiZIBQxc1fMe0TRSgY60rGBHf+5EVoVivuaaV6d1auXeV2FNEBOkInyEXnqIZuUR01EEWP6Bm9ojfryXqx3q2PWWvBymf20Z+wPn8Abq6YYg==</latexit>


Rotações: torque e momento de inérciaInércia: proporcional à massa (intuitivo), Depende da geometria também! (dependência quadrática)

Quanto mais longe a massa estiver do eixo de rotação, mais difícil rodar.

É 2,5 vezes "mais difícil” rodar o anel do que a esfera sólida.

É 6 vezes "mais difícil” rodar um disco sólido que uma barra de (com L=R)

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Momento de inércia: [kgm2]<latexit sha1_base64="DeugSXe487RxWla0EIvufYdavVw=">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRbBg5TdKuix6MVjBfsh27Vk02wbmmSXJCuUpb/CiwdFvPpzvPlvTNs9aOuDgcd7M8zMCxPOtHHdb6ewsrq2vlHcLG1t7+zulfcPWjpOFaFNEvNYdUKsKWeSNg0znHYSRbEIOW2Ho5up336iSrNY3ptxQgOBB5JFjGBjpQd/NOieicda0CtX3Ko7A1omXk4qkKPRK391+zFJBZWGcKy177mJCTKsDCOcTkrdVNMEkxEeUN9SiQXVQTY7eIJOrNJHUaxsSYNm6u+JDAutxyK0nQKboV70puJ/np+a6CrImExSQyWZL4pSjkyMpt+jPlOUGD62BBPF7K2IDLHCxNiMSjYEb/HlZdKqVb3zau3uolK/zuMowhEcwyl4cAl1uIUGNIGAgGd4hTdHOS/Ou/Mxby04+cwh/IHz+QMI9Y/n</latexit>

k e um numero que depende da geometria<latexit sha1_base64="QSr3bLkJxvD8zNxE9gQDOYpyyok=">AAACOXicbVA7TwMxDPbxprwKjCwRBalTdVcGGBEsjCBRQGqrKpe6bdRcciQ5pKrib7HwL9iQWBhAwMrGhHt04GXJ9ufPn5XYcaqk82F4H0xMTk3PzM7NFxYWl5ZXiqtrZ85kVmBNGGXsRcwdKqmx5qVXeJFa5Ems8DzuH47651donTT61A9SbCa8q2VHCu6JMsUybEGfnMEreQYJRQ0flBEsGKouiUXKbYopuc7RqOYUu4RNrvakl8BbxVJYCXNjf0E0BiUY23GreNdoG5ElqL1Q3Ll6FKa+OeTWS6HwutDIHKZc9HkX6wQ1T9A1h/nm12ybmDbrGEuuPcvZ7xNDnjg3SGJSJtz33O/eiPyvV898Z685lDrNPGrx9VAnU8wbNjoja0uLwqsBAS6spL8y0eOWC0/HLtARot8r/wVn1Uq0U6meVEv7B+NzzMEGbEIZItiFfTiCY6iBgBt4gCd4Dm6Dx+AlePuSTgTjmXX4YcH7Jyw8ntA=</latexit>


Rotações: torque e momento de inércia(Link para um vídeo com demonstração no e-disciplinas USP)


Translação vs rotação

torque

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a<latexit sha1_base64="wqZLPcGml9Og5FDdUdFUNWEF4FE=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlJu2XK27VnYOsEi8nFcjR6Je/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSrlW9i2qteVmp3+RxFOEETuEcPLiCOtxBA1rAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MHxKuM6Q==</latexit>

↵[rad/s2]<latexit sha1_base64="izD1XTBhQp2cRe0L8Rjvq0uVTl8=">AAACAHicbVDLSsNAFJ34rPUVdeHCzWARXNWkCrosunFZwT4gieVmMmmHTh7MTIQSsvFX3LhQxK2f4c6/cdpmoa0HLhzOuZd77/FTzqSyrG9jaXlldW29slHd3Nre2TX39jsyyQShbZLwRPR8kJSzmLYVU5z2UkEh8jnt+qObid99pEKyJL5X45R6EQxiFjICSkt989AFng4BYyd3RYQFBMWZfGh4fbNm1a0p8CKxS1JDJVp988sNEpJFNFaEg5SObaXKy0EoRjgtqm4maQpkBAPqaBpDRKWXTx8o8IlWAhwmQles8FT9PZFDJOU48nVnBGoo572J+J/nZCq88nIWp5miMZktCjOOVYInaeCACUoUH2sCRDB9KyZDEECUzqyqQ7DnX14knUbdPq837i5qzesyjgo6QsfoFNnoEjXRLWqhNiKoQM/oFb0ZT8aL8W58zFqXjHLmAP2B8fkDdBSVqw==</latexit>

aceleração angularaceleração

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F<latexit sha1_base64="oAfPRSTQ5uqd9KYOzOVm3vcLsJY=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9FgUxGMV+wFtKJvtpF262YTdjVBC/4EXD4p49R9589+4bXPQ1gcDj/dmmJkXJIJr47rfzsrq2vrGZmGruL2zu7dfOjhs6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0M/VbT6g0j+WjGSfoR3QgecgZNVZ6uCW9UtmtuDOQZeLlpAw56r3SV7cfszRCaZigWnc8NzF+RpXhTOCk2E01JpSN6AA7lkoaofaz2aUTcmqVPgljZUsaMlN/T2Q00nocBbYzomaoF72p+J/XSU145WdcJqlByeaLwlQQE5Pp26TPFTIjxpZQpri9lbAhVZQZG07RhuAtvrxMmtWKd16p3l+Ua9d5HAU4hhM4Aw8uoQZ3UIcGMAjhGV7hzRk5L8678zFvXXHymSP4A+fzB/FpjPg=</latexit>

⌧<latexit sha1_base64="Wa8bCywmcQCIXorlDsXU45DPVf0=">AAAB63icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGC/YA2lM120y7d3YTdiVBC/4IXD4p49Q9589+YtDlo64OBx3szzMwLYiksuu63s7a+sbm1Xdop7+7tHxxWjo7bNkoM4y0Wych0A2q5FJq3UKDk3dhwqgLJO8HkLvc7T9xYEelHnMbcV3SkRSgYxVzqI00Glapbc+cgq8QrSBUKNAeVr/4wYoniGpmk1vY8N0Y/pQYFk3xW7ieWx5RN6Ij3Mqqp4tZP57fOyHmmDEkYmaw0krn6eyKlytqpCrJORXFsl71c/M/rJRje+KnQcYJcs8WiMJEEI5I/TobCcIZymhHKjMhuJWxMDWWYxVPOQvCWX14l7XrNu6zVH66qjdsijhKcwhlcgAfX0IB7aEILGIzhGV7hzVHOi/PufCxa15xi5gT+wPn8ASJ7jkw=</latexit>

I<latexit sha1_base64="2YL8+LeSuf2tkZkxzVvgMLFabyk=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BL3pLwDwgWcLspDcZMzu7zMwKIeQLvHhQxKuf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mHGCfkQHkoecUWOl+n2vWHLL7hxklXgZKUGGWq/41e3HLI1QGiao1h3PTYw/ocpwJnBa6KYaE8pGdIAdSyWNUPuT+aFTcmaVPgljZUsaMld/T0xopPU4CmxnRM1QL3sz8T+vk5rw2p9wmaQGJVssClNBTExmX5M+V8iMGFtCmeL2VsKGVFFmbDYFG4K3/PIqaVbK3kW5Ur8sVW+yOPJwAqdwDh5cQRXuoAYNYIDwDK/w5jw6L86787FozTnZzDH8gfP5A6BLjNE=</latexit>

!<latexit sha1_base64="tlCosEe7acB15WG/y1yAvFPNxVY=">AAAB7XicbVDLSgNBEJyNrxhfUY9eBoPgKexGQY9BLx4jmAckS5id9CZj5rHMzAphyT948aCIV//Hm3/jJNmDJhY0FFXddHdFCWfG+v63V1hb39jcKm6Xdnb39g/Kh0cto1JNoUkVV7oTEQOcSWhaZjl0Eg1ERBza0fh25refQBum5IOdJBAKMpQsZpRYJ7V6SsCQ9MsVv+rPgVdJkJMKytHol796A0VTAdJSTozpBn5iw4xoyyiHaamXGkgIHZMhdB2VRIAJs/m1U3zmlAGOlXYlLZ6rvycyIoyZiMh1CmJHZtmbif953dTG12HGZJJakHSxKE45tgrPXscDpoFaPnGEUM3crZiOiCbUuoBKLoRg+eVV0qpVg4tq7f6yUr/J4yiiE3SKzlGArlAd3aEGaiKKHtEzekVvnvJevHfvY9Fa8PKZY/QH3ucPkX+PHw==</latexit>

Massa

Força

Velocidade Velocidade angular

Momento de inércia

Translação Rotação

! = 2⇡/(60s) = 0.10472rad/s<latexit sha1_base64="/OesdfmOil2n6z+TeGSg2alkhVg=">AAACCHicbVDLSsNAFJ34rPUVdenCwSLUTZrEYt0Uim5cVrAPaEKZTCbt0MmDmYlQQpdu/BU3LhRx6ye482+ctllo64ELh3Pu5d57vIRRIU3zW1tZXVvf2CxsFbd3dvf29YPDtohTjkkLxyzmXQ8JwmhEWpJKRroJJyj0GOl4o5up33kgXNA4upfjhLghGkQ0oBhJJfX1EycOyQDVbSehlfKlKc7rpmGZ1ZoNOfIroq+XTMOcAS4TKyclkKPZ178cP8ZpSCKJGRKiZ5mJdDPEJcWMTIpOKkiC8AgNSE/RCIVEuNnskQk8U4oPg5iriiScqb8nMhQKMQ491RkiORSL3lT8z+ulMrhyMxolqSQRni8KUgZlDKepQJ9ygiUbK4Iwp+pWiIeIIyxVdkUVgrX48jJp24Z1Ydh31VLjOo+jAI7BKSgDC9RAA9yCJmgBDB7BM3gFb9qT9qK9ax/z1hUtnzkCf6B9/gAjUJbZ</latexit>

Exemplo: vel. angular do ponteiro dos segundos de um relógio


Translação vs rotação

F = ma<latexit sha1_base64="IhRE8B0Kz7CL433STMkcIreWgOY=">AAAB7nicbVBNSwMxEJ34WetX1aOXYBE8ld0q6EUoCuKxgv2AdinZNNuGJtklyQpl6Y/w4kERr/4eb/4b03YP2vpg4PHeDDPzwkRwYz3vG62srq1vbBa2its7u3v7pYPDpolTTVmDxiLW7ZAYJrhiDcutYO1EMyJDwVrh6Hbqt56YNjxWj3acsECSgeIRp8Q6qXWHr7HEpFcqexVvBrxM/JyUIUe9V/rq9mOaSqYsFcSYju8lNsiItpwKNil2U8MSQkdkwDqOKiKZCbLZuRN86pQ+jmLtSlk8U39PZEQaM5ah65TEDs2iNxX/8zqpja6CjKsktUzR+aIoFdjGePo77nPNqBVjRwjV3N2K6ZBoQq1LqOhC8BdfXibNasU/r1QfLsq1mzyOAhzDCZyBD5dQg3uoQwMojOAZXuENJegFvaOPeesKymeO4A/Q5w+kII51</latexit>

Translação Rotação

⌧ = I↵<latexit sha1_base64="GmZ2fFYeeFLobdCCOWjqKStALRU=">AAAB+HicbVDLSgNBEJz1GeMjqx69DAbBU9iNgl6EoBe9RTAPyC6hdzKbDJmdXeYhxJAv8eJBEa9+ijf/xkmyB00saCiquunuijLOlPa8b2dldW19Y7OwVdze2d0rufsHTZUaSWiDpDyV7QgU5UzQhmaa03YmKSQRp61oeDP1W49UKpaKBz3KaJhAX7CYEdBW6rqlQIPBV/gOB8CzAXTdslfxZsDLxM9JGeWod92voJcSk1ChCQelOr6X6XAMUjPC6aQYGEUzIEPo046lAhKqwvHs8Ak+sUoPx6m0JTSeqb8nxpAoNUoi25mAHqhFbyr+53WMji/DMROZ0VSQ+aLYcKxTPE0B95ikRPORJUAks7diMgAJRNusijYEf/HlZdKsVvyzSvX+vFy7zuMooCN0jE6Rjy5QDd2iOmogggx6Rq/ozXlyXpx352PeuuLkM4foD5zPH0lXkjM=</latexit>

Torque é igual a momento de inércia vezes aceleração angular

torque

v<latexit sha1_base64="gfGd9lilLkuqmwOX+dBCP3vq/yw=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELx4hkUcCGzI79MLI7OxmZpaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWj+7nfGqPSPJaPZpKgH9GB5CFn1FipPu4VS27ZXYCsEy8jJchQ6xW/uv2YpRFKwwTVuuO5ifGnVBnOBM4K3VRjQtmIDrBjqaQRan+6OHRGLqzSJ2GsbElDFurviSmNtJ5Ege2MqBnqVW8u/ud1UhPe+lMuk9SgZMtFYSqIicn8a9LnCpkRE0soU9zeStiQKsqMzaZgQ/BWX14nzUrZuypX6tel6l0WRx7O4BwuwYMbqMID1KABDBCe4RXenCfnxXl3PpatOSebOYU/cD5/AOR/jP4=</latexit>

a<latexit sha1_base64="wqZLPcGml9Og5FDdUdFUNWEF4FE=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlJu2XK27VnYOsEi8nFcjR6Je/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSrlW9i2qteVmp3+RxFOEETuEcPLiCOtxBA1rAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MHxKuM6Q==</latexit>

↵[rad/s2]<latexit sha1_base64="izD1XTBhQp2cRe0L8Rjvq0uVTl8=">AAACAHicbVDLSsNAFJ34rPUVdeHCzWARXNWkCrosunFZwT4gieVmMmmHTh7MTIQSsvFX3LhQxK2f4c6/cdpmoa0HLhzOuZd77/FTzqSyrG9jaXlldW29slHd3Nre2TX39jsyyQShbZLwRPR8kJSzmLYVU5z2UkEh8jnt+qObid99pEKyJL5X45R6EQxiFjICSkt989AFng4BYyd3RYQFBMWZfGh4fbNm1a0p8CKxS1JDJVp988sNEpJFNFaEg5SObaXKy0EoRjgtqm4maQpkBAPqaBpDRKWXTx8o8IlWAhwmQles8FT9PZFDJOU48nVnBGoo572J+J/nZCq88nIWp5miMZktCjOOVYInaeCACUoUH2sCRDB9KyZDEECUzqyqQ7DnX14knUbdPq837i5qzesyjgo6QsfoFNnoEjXRLWqhNiKoQM/oFb0ZT8aL8W58zFqXjHLmAP2B8fkDdBSVqw==</latexit>

aceleração angularaceleração

m<latexit sha1_base64="hgrbGyfUs1b/Mr5+M32Ck4D9zmA=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3a3STsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgkfYMtwI7CQKqQwEPgTj25n/8IRK8zi6N5MEfUmHEQ85o8ZKTdkvV9yqOwdZJV5OKpCj0S9/9QYxSyVGhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFraUQlaj+bHzolZ1YZkDBWtiJD5urviYxKrScysJ2SmpFe9mbif143NeG1n/EoSQ1GbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucP1tuM9Q==</latexit>

F<latexit sha1_base64="oAfPRSTQ5uqd9KYOzOVm3vcLsJY=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9FgUxGMV+wFtKJvtpF262YTdjVBC/4EXD4p49R9589+4bXPQ1gcDj/dmmJkXJIJr47rfzsrq2vrGZmGruL2zu7dfOjhs6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0M/VbT6g0j+WjGSfoR3QgecgZNVZ6uCW9UtmtuDOQZeLlpAw56r3SV7cfszRCaZigWnc8NzF+RpXhTOCk2E01JpSN6AA7lkoaofaz2aUTcmqVPgljZUsaMlN/T2Q00nocBbYzomaoF72p+J/XSU145WdcJqlByeaLwlQQE5Pp26TPFTIjxpZQpri9lbAhVZQZG07RhuAtvrxMmtWKd16p3l+Ua9d5HAU4hhM4Aw8uoQZ3UIcGMAjhGV7hzRk5L8678zFvXXHymSP4A+fzB/FpjPg=</latexit>

⌧<latexit sha1_base64="Wa8bCywmcQCIXorlDsXU45DPVf0=">AAAB63icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGC/YA2lM120y7d3YTdiVBC/4IXD4p49Q9589+YtDlo64OBx3szzMwLYiksuu63s7a+sbm1Xdop7+7tHxxWjo7bNkoM4y0Wych0A2q5FJq3UKDk3dhwqgLJO8HkLvc7T9xYEelHnMbcV3SkRSgYxVzqI00Glapbc+cgq8QrSBUKNAeVr/4wYoniGpmk1vY8N0Y/pQYFk3xW7ieWx5RN6Ij3Mqqp4tZP57fOyHmmDEkYmaw0krn6eyKlytqpCrJORXFsl71c/M/rJRje+KnQcYJcs8WiMJEEI5I/TobCcIZymhHKjMhuJWxMDWWYxVPOQvCWX14l7XrNu6zVH66qjdsijhKcwhlcgAfX0IB7aEILGIzhGV7hzVHOi/PufCxa15xi5gT+wPn8ASJ7jkw=</latexit>

I<latexit sha1_base64="2YL8+LeSuf2tkZkxzVvgMLFabyk=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BL3pLwDwgWcLspDcZMzu7zMwKIeQLvHhQxKuf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mHGCfkQHkoecUWOl+n2vWHLL7hxklXgZKUGGWq/41e3HLI1QGiao1h3PTYw/ocpwJnBa6KYaE8pGdIAdSyWNUPuT+aFTcmaVPgljZUsaMld/T0xopPU4CmxnRM1QL3sz8T+vk5rw2p9wmaQGJVssClNBTExmX5M+V8iMGFtCmeL2VsKGVFFmbDYFG4K3/PIqaVbK3kW5Ur8sVW+yOPJwAqdwDh5cQRXuoAYNYIDwDK/w5jw6L86787FozTnZzDH8gfP5A6BLjNE=</latexit>

!<latexit sha1_base64="tlCosEe7acB15WG/y1yAvFPNxVY=">AAAB7XicbVDLSgNBEJyNrxhfUY9eBoPgKexGQY9BLx4jmAckS5id9CZj5rHMzAphyT948aCIV//Hm3/jJNmDJhY0FFXddHdFCWfG+v63V1hb39jcKm6Xdnb39g/Kh0cto1JNoUkVV7oTEQOcSWhaZjl0Eg1ERBza0fh25refQBum5IOdJBAKMpQsZpRYJ7V6SsCQ9MsVv+rPgVdJkJMKytHol796A0VTAdJSTozpBn5iw4xoyyiHaamXGkgIHZMhdB2VRIAJs/m1U3zmlAGOlXYlLZ6rvycyIoyZiMh1CmJHZtmbif953dTG12HGZJJakHSxKE45tgrPXscDpoFaPnGEUM3crZiOiCbUuoBKLoRg+eVV0qpVg4tq7f6yUr/J4yiiE3SKzlGArlAd3aEGaiKKHtEzekVvnvJevHfvY9Fa8PKZY/QH3ucPkX+PHw==</latexit>

Massa

Força

VelocidadeVelocidade angular

Momento de inércia

p = mv<latexit sha1_base64="0B/0U7mWYz3Eh2TineB1TZg/Jz0=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRbBU9mtBb0IRS8eK9gPaJeSTbNtaJJdkmyhLP0LXjwo4tU/5M1/Y7bdg7Y+GHi8N8PMvCDmTBvX/XYKG5tb2zvF3dLe/sHhUfn4pK2jRBHaIhGPVDfAmnImacsww2k3VhSLgNNOMLnP/M6UKs0i+WRmMfUFHkkWMoJNJsW3YjooV9yquwBaJ15OKpCjOSh/9YcRSQSVhnCsdc9zY+OnWBlGOJ2X+ommMSYTPKI9SyUWVPvp4tY5urDKEIWRsiUNWqi/J1IstJ6JwHYKbMZ61cvE/7xeYsIbP2UyTgyVZLkoTDgyEcoeR0OmKDF8ZgkmitlbERljhYmx8ZRsCN7qy+ukXat6V9XaY73SuMvjKMIZnMMleHANDXiAJrSAwBie4RXeHOG8OO/Ox7K14OQzp/AHzucPAP2ONg==</latexit>

Momento L = I!<latexit sha1_base64="UHaK849VABs2N9uDlOKYJIgoU8c=">AAAB8HicbVDLSgNBEJz1GeMr6tHLYBA8hd0o6EUIelHwEME8JFnC7KSTDJnHMjMrhCVf4cWDIl79HG/+jZNkD5pY0FBUddPdFcWcGev7397S8srq2npuI7+5tb2zW9jbrxuVaAo1qrjSzYgY4ExCzTLLoRlrICLi0IiG1xO/8QTaMCUf7CiGUJC+ZD1GiXXS493lbVsJ6JNOoeiX/CnwIgkyUkQZqp3CV7uraCJAWsqJMa3Aj22YEm0Z5TDOtxMDMaFD0oeWo5IIMGE6PXiMj53SxT2lXUmLp+rviZQIY0Yicp2C2IGZ9ybif14rsb2LMGUyTixIOlvUSzi2Ck++x12mgVo+coRQzdytmA6IJtS6jPIuhGD+5UVSL5eC01L5/qxYucriyKFDdIROUIDOUQXdoCqqIYoEekav6M3T3ov37n3MWpe8bOYA/YH3+QNEdpAP</latexit>

Momento angular


Momento angular

L = I!<latexit sha1_base64="UHaK849VABs2N9uDlOKYJIgoU8c=">AAAB8HicbVDLSgNBEJz1GeMr6tHLYBA8hd0o6EUIelHwEME8JFnC7KSTDJnHMjMrhCVf4cWDIl79HG/+jZNkD5pY0FBUddPdFcWcGev7397S8srq2npuI7+5tb2zW9jbrxuVaAo1qrjSzYgY4ExCzTLLoRlrICLi0IiG1xO/8QTaMCUf7CiGUJC+ZD1GiXXS493lbVsJ6JNOoeiX/CnwIgkyUkQZqp3CV7uraCJAWsqJMa3Aj22YEm0Z5TDOtxMDMaFD0oeWo5IIMGE6PXiMj53SxT2lXUmLp+rviZQIY0Yicp2C2IGZ9ybif14rsb2LMGUyTixIOlvUSzi2Ck++x12mgVo+coRQzdytmA6IJtS6jPIuhGD+5UVSL5eC01L5/qxYucriyKFDdIROUIDOUQXdoCqqIYoEekav6M3T3ov37n3MWpe8bOYA/YH3+QNEdpAP</latexit>

Momento angular

Análogo ao momento linear: grandeza conservada

Para L fixo, maior I tem menor omega, e inversamente.

(Link para um vídeo com demonstração no e-disciplinas USP)

kg m2/s<latexit sha1_base64="pbWfNK8QfZlKL/44OunYEsviLoY=">AAAB8XicbVA9T8MwEL3wWcpXgZHFokViKkkYYKxgYSwS/RBtqBzXaa3aTmQ7SFXUf8HCAEKs/Bs2/g1umwFannTS03t3ursXJpxp47rfzsrq2vrGZmGruL2zu7dfOjhs6jhVhDZIzGPVDrGmnEnaMMxw2k4UxSLktBWObqZ+64kqzWJ5b8YJDQQeSBYxgo2VHkYDJCqPfuVc90plt+rOgJaJl5My5Kj3Sl/dfkxSQaUhHGvd8dzEBBlWhhFOJ8VuqmmCyQgPaMdSiQXVQTa7eIJOrdJHUaxsSYNm6u+JDAutxyK0nQKboV70puJ/Xic10VWQMZmkhkoyXxSlHJkYTd9HfaYoMXxsCSaK2VsRGWKFibEhFW0I3uLLy6TpV72Lqn/nl2vXeRwFOIYTOAMPLqEGt1CHBhCQ8Ayv8OZo58V5dz7mrStOPnMEf+B8/gDcJI+5</latexit>


Prática com o simulador PhET




Trabalhar em radianos

Sem freio

A joaninha tem massa zero



1. Escolha uma massa um raio externo e um raio interno para o disco.

2. Aplique um torque qq com o mouse e registre os valores de L e omega.

Parte 1: verificação de L = I!<latexit sha1_base64="n3sGbOnWhnbpcUUVgU2FumMskx0=">AAAB8nicdVDLSsNAFJ3UV62vqks3g0VwFZLWR10IRTcKLirYB6ShTKaTduhkJsxMhBL6GW5cKOLWr3Hn3zhNI6jogQuHc+7l3nuCmFGlHefDKiwsLi2vFFdLa+sbm1vl7Z22EonEpIUFE7IbIEUY5aSlqWakG0uCooCRTjC+nPmdeyIVFfxOT2LiR2jIaUgx0kbybuA5vO6JiAxRv1xx7GPHPTtxoGM7GTJSd2sudHOlAnI0++X33kDgJCJcY4aU8lwn1n6KpKaYkWmplygSIzxGQ+IZylFElJ9mJ0/hgVEGMBTSFNcwU79PpChSahIFpjNCeqR+ezPxL89LdFj3U8rjRBOO54vChEEt4Ox/OKCSYM0mhiAsqbkV4hGSCGuTUsmE8PUp/J+0q7Zbs6u3R5XGRR5HEeyBfXAIXHAKGuAKNEELYCDAA3gCz5a2Hq0X63XeWrDymV3wA9bbJyO9kIU=</latexit>

3. Repita para mais 4 valores. (Cinco pares (L, omega) no total)

4. Faça o gráfico e verifique se é aproximadamente linear.

5. Use o MMQ para encontrar I. Compare com o valor esperado para o seu disco.

Não esqueça de registrar as variáveis

fixas!



1. Escolha uma massa para disco e comece com r=4m. Deixe o raio interno em zero!

2. Aplique um torque qq com o mouse e registre L. Registre R e omega.

Parte 2: verificação de

3. Agora mantenha um valor de L fixo! Altere o raio do disco pouco a pouco e registre ao menos cinco pares (R,omega) variando bastante o raio!.

5. Faça o gráfico de Log(omega) (no eixo y) por Log(R). Dá uma reta?

6. Use o MMQ para e encontrar o coeficiente angular deste último gráfico. É o esperado?

I = kMR2<latexit sha1_base64="CkdvzZK7Y6PLb1cWL2ZAczB8c8E=">AAAB8nicdVDLSgMxFM3UV62vqks3wSK4GjKtj7oQim50IVSxD5iOJZOmbWgmMyQZoQz9DDcuFHHr17jzb0ynI6jogQuHc+7l3nv8iDOlEfqwcnPzC4tL+eXCyura+kZxc6upwlgS2iAhD2Xbx4pyJmhDM81pO5IUBz6nLX90PvVb91QqFopbPY6oF+CBYH1GsDaSewlP4QhewZu7crdYQvYhck6OEEQ2SpGSqlNxoJMpJZCh3i2+d3ohiQMqNOFYKddBkfYSLDUjnE4KnVjRCJMRHlDXUIEDqrwkPXkC94zSg/1QmhIapur3iQQHSo0D33QGWA/Vb28q/uW5se5XvYSJKNZUkNmifsyhDuH0f9hjkhLNx4ZgIpm5FZIhlphok1LBhPD1KfyfNMu2U7HL1wel2lkWRx7sgF2wDxxwDGrgAtRBAxAQggfwBJ4tbT1aL9brrDVnZTPb4Aest0/bp4+u</latexit>

4. Faça o gráfico de 1/omega (no eixo y) por R. Tem mais ou menos a cara que vc esperaria?

7. Com L fixo, varie a massa do disco (mantendo R fixo). Registre pares de (M,omega)

8. Faça o gráfico de 1/omega (eixo y) por m. Dá uma reta? Era o esperado?

Não esqueça de registrar as variáveis

fixas!

L = (kMR2)!<latexit sha1_base64="UChDn92dYI/uSwUSUBUR+s3tol4=">AAAB+HicdVDLSsNAFJ3UV62PRl26GSxC3ZSk9VEXQtGNC4Uq9gFtLJPppB06yYSZiVBDv8SNC0Xc+inu/BunaQQVPXDhcM693HuPGzIqlWV9GJm5+YXFpexybmV1bT1vbmw2JY8EJg3MGRdtF0nCaEAaiipG2qEgyHcZabmjs6nfuiNCUh7cqHFIHB8NAupRjJSWemb+4qQ4ury+Le91uU8GqGcWrNKBZR8fWtAqWQkSUrUrNrRTpQBS1Hvme7fPceSTQGGGpOzYVqicGAlFMSOTXDeSJER4hAako2mAfCKdODl8Ane10oceF7oCBRP1+0SMfCnHvqs7faSG8rc3Ff/yOpHyqk5MgzBSJMCzRV7EoOJwmgLsU0GwYmNNEBZU3wrxEAmElc4qp0P4+hT+T5rlkl0pla/2C7XTNI4s2AY7oAhscARq4BzUQQNgEIEH8ASejXvj0XgxXmetGSOd2QI/YLx9AlhfkkA=</latexit>



! Ao calcular os valores verdadeiros (de I por exemplo) considere Incertezas na massa e nos raios (+/- 1 na última casa decimal

que o programa dá).


Prática com o simulador PhETCada grupo deve enviar três arquivos de dados até as 18:00.

1. Cinco pares de (L, Omega) com massa e raios fixos.

2. Cinco pares de (R, omega) com L e m fixos.

3. Cinco pares de (M, Omega) com L e R fixos.

Os arquivos só devem conter números!

GrupoX_P4_LxOmega.dat

GrupoX_P4_RxOmega.dat

GrupoX_P4_MxOmega.dat

Os arquivos pode ser .txt também

Use ponto e não vírgula para separar os decimais (obrigado!).Variáveis fixas devem aparecer somente no relatório.

Rotações: momento de inércia e conservação do momento angular

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