Abstract The Foucault pendulum is a physical apparatus that shows the Earth’s rotation through the variation of the plane of oscillation over time. The main physical agent responsible for this phenomenon is the non-inertial Coriolis force, which couples the coordinates in the equation of motion. Although the Foucault pendulum is more than 170 years old, its theoretical description still lacks completeness, as there is no exact vector solution to the problem or even a decoupled equation of motion. Furthermore, the main prediction about the Foucault pendulum, the dependence of the pendulum’s plane of oscillation rotation on the latitude, needs to be revised as such dependence is more complex than is shown in the literature, once it depends essentially on the initial conditions. Here all these points raised are successfully answered.

Throughout the undergraduate course in Physics, the student frequently encounters problems involving oscillations. The primary model used to investigate oscillatory phenomena is the simple harmonic oscillator. As students advance in their academic education, they also learn various methods for solving differential equations. It is standard to employ such novel techniques in problems whose solution is previously known, to test the method’s quality and for didactic purposes. Placing these two lines together, I present twelve different ways to solve the well-known simple harmonic oscillator problem. They fall into three categories: basic, intermediate, and advanced. This division is based on the level of knowledge necessary to solve the problem, progressively acquired throughout the student’s academic education. Therefore, this material serves as a reference for students at distinct stages of their education.

Ao longo do curso de graduação em Física, o(a) estudante frequentemente se depara com problemas envolvendo oscilações. O principal modelo usado para investigar fenômenos oscilatórios é o oscilador harmônico simples. À medida que os(as) estudantes avançam em sua formação acadêmica, eles(as) também aprendem vários métodos para resolver equações diferenciais. É padrão empregar essas novas técnicas em problemas cuja solução é previamente conhecida, para testar a qualidade do método e para fins didáticos. Juntando essas duas linhas, apresento doze maneiras diferentes de resolver o conhecido problema do oscilador harmônico simples. Elas se enquadram em três categorias: básicas, intermediárias e avançadas. Essa divisão é baseada no nível de conhecimento necessário para resolver o problema, progressivamente adquirido ao longo da formação acadêmica do(a) aluno(a). Portanto, este material serve de referência para estudantes em diferentes etapas de sua formação.

The motion of a particle evaluated from the noninertial frame of the rotating Earth with constant angular velocity ω is solved analytically without carrying out a priori approximations, that is, considering both the Coriolis and the centrifugal terms. Position vector solutions written in terms of generic initial conditions were obtained from a decoupled sixth-order equation of motion, obtained using a fully vectorial approach, without specifically resorting to coordinates. By comparing the exact solutions for the complete and approximate (ignoring the centrifugal term from the beginning) problems, great differences can be seen in the order of the decoupled differential equation as well as in the temporal dependencies of the noninertial terms of the solutions. Noninertial effects for the solution were classified according to the order of magnitude in ω, where first-order effects are oriented east-west and the second-order ones are oriented north-south.

ABSTRACT OBJECTIVE To estimate the 2020 all-cause and COVID-19 excess mortality according to sex, age, race/color, and state, and to compare mortality rates by selected causes with that of the five previous years in Brazil. METHODS Data from the Mortality Information System were used. Expected deaths for 2020 were estimated from 2015 to 2019 data using a negative binomial log-linear model. RESULTS Excess deaths in Brazil in 2020 amounted to 13.7%, and the ratio of excess deaths to COVID-19 deaths was 0.90. Reductions in deaths from cardiovascular diseases (CVD), respiratory diseases, and external causes, and an increase in ill-defined causes were all noted. Excess deaths were also found to be heterogeneous, being higher in the Northern, Center-Western, and Northeastern states. In some states, the number of COVID-19 deaths was lower than that of excess deaths, whereas the opposite occurred in others. Moreover, excess deaths were higher in men aged 20 to 59, and in black, yellow, or indigenous individuals. Meanwhile, excess mortality was lower in women, in individuals aged 80 years or older, and in whites. Additionally, deaths among those aged 0 to 19 were 7.2% lower than expected, with reduction in mortality from respiratory diseases and external causes. There was also a drop in mortality due to external causes in men and in those aged 20 to 39 years. Moreover, reductions in deaths from CVD and neoplasms were noted in some states and groups. CONCLUSION There is evidence of underreporting of COVID-19 deaths and of the possible impact of restrictive measures in the reduction of deaths from external causes and respiratory diseases. The impacts of COVID-19 on mortality were heterogeneous among the states and groups, revealing that regional, demographic, socioeconomic, and racial differences expose individuals in distinct ways to the risk of death from both COVID-19 and other causes.

ABSTRACT OBJECTIVE To estimate the 2020 all-cause and COVID-19 excess mortality according to sex, age, race/color, and state, and to compare mortality rates by selected causes with that of the five previous years in Brazil. METHODS Data from the Mortality Information System were used. Expected deaths for 2020 were estimated from 2015 to 2019 data using a negative binomial log-linear model. RESULTS Excess deaths in Brazil in 2020 amounted to 13.7%, and the ratio of excess deaths to COVID-19 deaths was 0.90. Reductions in deaths from cardiovascular diseases (CVD), respiratory diseases, and external causes, and an increase in ill-defined causes were all noted. Excess deaths were also found to be heterogeneous, being higher in the Northern, Center-Western, and Northeastern states. In some states, the number of COVID-19 deaths was lower than that of excess deaths, whereas the opposite occurred in others. Moreover, excess deaths were higher in men aged 20 to 59, and in black, yellow, or indigenous individuals. Meanwhile, excess mortality was lower in women, in individuals aged 80 years or older, and in whites. Additionally, deaths among those aged 0 to 19 were 7.2% lower than expected, with reduction in mortality from respiratory diseases and external causes. There was also a drop in mortality due to external causes in men and in those aged 20 to 39 years. Moreover, reductions in deaths from CVD and neoplasms were noted in some states and groups. CONCLUSION There is evidence of underreporting of COVID-19 deaths and of the possible impact of restrictive measures in the reduction of deaths from external causes and respiratory diseases. The impacts of COVID-19 on mortality were heterogeneous among the states and groups, revealing that regional, demographic, socioeconomic, and racial differences expose individuals in distinct ways to the risk of death from both COVID-19 and other causes.

O estudo teve como objetivos, descrever o atraso na notificação de óbitos por COVID-19 na cidade de São Luís, Maranhão, Brasil, e demonstrar o impacto sobre o monitoramento oportuno e modelagem da pandemia. O estudo teve como objetivo secundário determinar a medida em que a nowcasting é capaz de diminuir a defasagem na notificação de óbitos. Analisamos os dados de mortalidade por COVID-19 registrados diariamente no Boletim Epidemiológico da Secretaria de Estado da Saúde do Maranhão e calculamos o atraso na notificação entre 23 de março e 29 de agosto de 2020. Para ilustrar o impacto do atraso na notificação de óbitos e testar a efetividade de uma nowcasting bayesiana para melhorar a qualidade dos dados, ajustamos um modelo hierárquico bayesiano semi-mecanístico. Apenas 17,8% dos óbitos foram notificados sem atraso ou no dia seguinte, enquanto 40,5% foram atrasados em mais de 30 dias. Devido ao atraso na notificação, houve uma subestimação inicial nos óbitos. Entre 7 de junho e 29 de agosto, 644 óbitos foram notificados, mas apenas 116 mortes ocorreram nesse período. O uso da técnica de nowcasting bayesiana melhorou parcialmente a qualidade dos dados de mortalidade no pico da epidemia, apresentando estimativas mais ajustadas ao cenário observado na cidade, mas não se mostrou útil quase dois meses depois do pico. O atraso na notificação de óbitos pode interferir diretamente nas decisões assertivas e oportunas sobre o combate à pandemia da COVID-19. Portanto, o sistema brasileiro de vigilância epidemiológica deve ser revisto urgentemente, e o registro da data do óbito deve ser obrigatório. A técnica de nowcasting mostrou ser parcialmente eficaz na melhoria dos dados de mortalidade no auge da pandemia, mas não depois.

This study describes the COVID-19 death reporting delay in the city of São Luís, Maranhão State, Brazil, and shows its impact on timely monitoring and modeling of the COVID-19 pandemic, while seeking to ascertain how nowcasting can improve death reporting delay. We analyzed COVID-19 death data reported daily in the Epidemiological Bulletin of the State Health Secretariat of Maranhão and calculated the reporting delay from March 23 to August 29, 2020. A semi-mechanistic Bayesian hierarchical model was fitted to illustrate the impact of death reporting delay and test the effectiveness of a Bayesian Nowcasting in improving data quality. Only 17.8% of deaths were reported without delay or the day after, while 40.5% were reported more than 30 days late. Following an initial underestimation due to reporting delay, 644 deaths were reported from June 7 to August 29, although only 116 deaths occurred during this period. Using the Bayesian nowcasting technique partially improved the quality of mortality data during the peak of the pandemic, providing estimates that better matched the observed scenario in the city, becoming unusable nearly two months after the peak. As delay in death reporting can directly interfere with assertive and timely decision-making regarding the COVID-19 pandemic, the Brazilian epidemiological surveillance system must be urgently revised and notifying the date of death must be mandatory. Nowcasting has proven somewhat effective in improving the quality of mortality data, but only at the peak of the pandemic.

La propuesta de este estudio es describir la demora en la notificación de muertes por COVID-19, en la ciudad São Luís, Maranhão, Brasil, y demostrar su impacto en el seguimiento puntual, así como en el modelaje de la pandemia de COVID-19. Un objetivo secundario fue confirmar el alcance, donde la previsión inmediata es capaz de mejorar el retraso en la notificación de las muertes. Analizamos los datos de muertes por COVID-19 diariamente en el Boletín Epidemiológico de la Secretaría de Estado de la Salud de Maranhão y calculamos los atrasos notificados desde el 23 de marzo al 29 de agosto, 2020. Con el fin de ilustrar el impacto del retraso en la notificación de muertes, y para probar la efectividad de la predicción inmediata bayesiana en la mejora de los datos de calidad, ajustamos un modelo jerárquico bayesiano semi-mecanicista. Solo un 17.8% de las muertes se notificaron sin atrasos o el día después, mientras que un 40.5% se vieron retrasadas durante más de 30 días. Debido a la demora informada, se produjo una subestimación inicial de muertes. No obstante, desde el 7 de junio al 29 de agosto, se informó de 644 muertes, pero solamente 116 muertes se produjeron durante este periodo. El uso de la técnica de predicción inmediata bayesiana mejoró parcialmente la calidad de la información de mortalidad durante el pico de la epidemia, presentando estimaciones que se ajustan mejor al escenario observado en la ciudad, pero no fue útil casi 2 meses después del pico. El retraso en la notificación de muertes podría interferir directamente en la toma de decisiones asertivas y puntuales, respecto a la pandemia de COVID-19. Por consiguiente, se debe revisar urgentemente el sistema brasileño de vigilancia epidemiológica y la notificación de la fecha de muerte debería ser obligatoria. La técnica de predicción inmediata ha demostrado ser bastante efectiva para mejorar la calidad de los datos de mortalidad solamente en el pico pandémico, pero no después.

O estudo teve como objetivos, descrever o atraso na notificação de óbitos por COVID-19 na cidade de São Luís, Maranhão, Brasil, e demonstrar o impacto sobre o monitoramento oportuno e modelagem da pandemia. O estudo teve como objetivo secundário determinar a medida em que a nowcasting é capaz de diminuir a defasagem na notificação de óbitos. Analisamos os dados de mortalidade por COVID-19 registrados diariamente no Boletim Epidemiológico da Secretaria de Estado da Saúde do Maranhão e calculamos o atraso na notificação entre 23 de março e 29 de agosto de 2020. Para ilustrar o impacto do atraso na notificação de óbitos e testar a efetividade de uma nowcasting bayesiana para melhorar a qualidade dos dados, ajustamos um modelo hierárquico bayesiano semi-mecanístico. Apenas 17,8% dos óbitos foram notificados sem atraso ou no dia seguinte, enquanto 40,5% foram atrasados em mais de 30 dias. Devido ao atraso na notificação, houve uma subestimação inicial nos óbitos. Entre 7 de junho e 29 de agosto, 644 óbitos foram notificados, mas apenas 116 mortes ocorreram nesse período. O uso da técnica de nowcasting bayesiana melhorou parcialmente a qualidade dos dados de mortalidade no pico da epidemia, apresentando estimativas mais ajustadas ao cenário observado na cidade, mas não se mostrou útil quase dois meses depois do pico. O atraso na notificação de óbitos pode interferir diretamente nas decisões assertivas e oportunas sobre o combate à pandemia da COVID-19. Portanto, o sistema brasileiro de vigilância epidemiológica deve ser revisto urgentemente, e o registro da data do óbito deve ser obrigatório. A técnica de nowcasting mostrou ser parcialmente eficaz na melhoria dos dados de mortalidade no auge da pandemia, mas não depois.

This study describes the COVID-19 death reporting delay in the city of São Luís, Maranhão State, Brazil, and shows its impact on timely monitoring and modeling of the COVID-19 pandemic, while seeking to ascertain how nowcasting can improve death reporting delay. We analyzed COVID-19 death data reported daily in the Epidemiological Bulletin of the State Health Secretariat of Maranhão and calculated the reporting delay from March 23 to August 29, 2020. A semi-mechanistic Bayesian hierarchical model was fitted to illustrate the impact of death reporting delay and test the effectiveness of a Bayesian Nowcasting in improving data quality. Only 17.8% of deaths were reported without delay or the day after, while 40.5% were reported more than 30 days late. Following an initial underestimation due to reporting delay, 644 deaths were reported from June 7 to August 29, although only 116 deaths occurred during this period. Using the Bayesian nowcasting technique partially improved the quality of mortality data during the peak of the pandemic, providing estimates that better matched the observed scenario in the city, becoming unusable nearly two months after the peak. As delay in death reporting can directly interfere with assertive and timely decision-making regarding the COVID-19 pandemic, the Brazilian epidemiological surveillance system must be urgently revised and notifying the date of death must be mandatory. Nowcasting has proven somewhat effective in improving the quality of mortality data, but only at the peak of the pandemic.

La propuesta de este estudio es describir la demora en la notificación de muertes por COVID-19, en la ciudad São Luís, Maranhão, Brasil, y demostrar su impacto en el seguimiento puntual, así como en el modelaje de la pandemia de COVID-19. Un objetivo secundario fue confirmar el alcance, donde la previsión inmediata es capaz de mejorar el retraso en la notificación de las muertes. Analizamos los datos de muertes por COVID-19 diariamente en el Boletín Epidemiológico de la Secretaría de Estado de la Salud de Maranhão y calculamos los atrasos notificados desde el 23 de marzo al 29 de agosto, 2020. Con el fin de ilustrar el impacto del retraso en la notificación de muertes, y para probar la efectividad de la predicción inmediata bayesiana en la mejora de los datos de calidad, ajustamos un modelo jerárquico bayesiano semi-mecanicista. Solo un 17.8% de las muertes se notificaron sin atrasos o el día después, mientras que un 40.5% se vieron retrasadas durante más de 30 días. Debido a la demora informada, se produjo una subestimación inicial de muertes. No obstante, desde el 7 de junio al 29 de agosto, se informó de 644 muertes, pero solamente 116 muertes se produjeron durante este periodo. El uso de la técnica de predicción inmediata bayesiana mejoró parcialmente la calidad de la información de mortalidad durante el pico de la epidemia, presentando estimaciones que se ajustan mejor al escenario observado en la ciudad, pero no fue útil casi 2 meses después del pico. El retraso en la notificación de muertes podría interferir directamente en la toma de decisiones asertivas y puntuales, respecto a la pandemia de COVID-19. Por consiguiente, se debe revisar urgentemente el sistema brasileño de vigilancia epidemiológica y la notificación de la fecha de muerte debería ser obligatoria. La técnica de predicción inmediata ha demostrado ser bastante efectiva para mejorar la calidad de los datos de mortalidad solamente en el pico pandémico, pero no después.

Resumo O oscilador harmônico amortecido por atrito de contato superficial é revisitado com uma descrição física e matemática mais robustas e abrangentes. Várias propriedades do sistema são apresentadas e discutidas, como a solução para o caso geral de condições iniciais, envoltórias do movimento, número exato de oscilações, dissipação instantânea e média de energia, fator de mérito, diagrama de fase dentre outras. Este texto serve como fundamentação teórica para simulações computacionais e experimentos didáticos que explorem sistemas oscilantes amortecidos por forças de atrito superficial.

Abstract The harmonic oscillator damped by frictional forces is revisited with a more robust and comprehensive physical and mathematical descriptions. Several system properties are presented and discussed, such as the solution to the general case of initial conditions, displacement envelopes, exact number of oscillations, instantaneous dissipation and average energy, quality factor, phase diagram and others. This text serves as a theoretical foundation for computer simulations and didactic experiments that explore oscillating systems damped by frictional forces.