Considere um corpo esférico de raio !$ r !$ totalmente envolvido por um fluido de viscosidade !$ η !$ com velocidade média !$ v !$. De acordo com a lei de Stokes, para baixas velocidades, esse corpo sofrerá a ação de uma força de arrasto viscoso dada por !$ F=-6\pi rηv !$. A dimensão de !$ η !$ é dada por

A figura mostra uma placa fina de peso !$ P !$ dobrada em Ângulo reto e disposta sobre uma esfera fixa de raio !$ a !$. O coeficiente de atrito mínimo entre estes objetos para que a placa não escorregue é

Considere as afirmações a seguir:

I. Se !$ z !$ e !$ w !$ são números complexos tais que !$ z-iw=1-2i !$ e !$ w-z=2+3i !$, então !$ z^2+w^2=-3+6i !$.

II. A soma de todos os números complexos z que satisfazem !$ 2 \left\vert z \right\vert^2+z^2=4+2i !$ é igual a zero.

III. Se !$ z=1-i !$, então !$ z^{59}=2^{29}(-1+i) !$.

É (são) verdadeira(s)

Your Facial Bone Structure Has a Big Influence on How People See You

(…) Selfies, headshots, mug shots — photos of oneself convey more these days than snapshots ever did back in the Kodak era. Most digitally minded people continually post and update pictures of themselves at professional, social media and dating sites such as LinkedIn, Facebook, Match.com and Tinder. For better or worse, viewers then tend to make snap judgments about someone’s personality or character from a single shot. As such, it can be a stressful task to select the photo that conveys the best impression of ourselves. For those of us seeking to appear friendly and trustworthy to others, a new study underscores an old, chipper piece of advice: Put on a happy face.

A newly published series of experiments by cognitive neuroscientists at New York University is reinforcing the relevance of facial expressions to perceptions of characteristics such as trustworthiness and friendliness. More importantly, the research also revealed the unexpected finding that perceptions of abilities such as physical strength are not dependent on facial expressions but rather on facial bone structure.

The team’s first experiment featured photographs of 10 different people presenting five different facial expressions each. Study subjects rated how friendly, trustworthy or strong the person in each photo appeared. A separate group of subjects scored each face on an emotional scale from “very angry” to “very happy.” And three experts not involved in either of the previous two ratings to avoid confounding results calculated the facial width-to-height ratio for each face. An analysis revealed that participants generally ranked people with a happy expression as friendly and trustworthy but not those with angry expressions. Surprisingly, participants did not rank faces as indicative of physical strength based on facial expression but graded faces that were very broad as that of a strong individual.

In a second survey facial expression and facial structure were manipulated in computer-generated faces. Participants rated each face for the same traits as in the first survey, with the addition of a rating for warmth. Again, people thought a happy expression, but not an angry one, indicated friendliness, trustworthiness — and in this case, warmth. The researchers then showed two additional sets of participants the same faces, this time either with areas relevant to facial expressions obscured or the width cropped. In the first variation, for faces lacking emotional cues, people could no longer perceive personality traits but could still perceive strength based on width. Similarly, for those faces lacking structural cues, people could no longer perceive strength but could still perceive personality traits based on facial expressions.

In a third iteration of the survey participants had to pick four faces out of a lineup of eight faces varied for expression and width that they might select either as their financial advisor or as the winner of a power-lifting competition. As might be expected, participants picked faces with happier expressions as financial advisors and selected broader faces as belonging to power-lifting champs.

In a final survey the researchers generated more than 100 variations of one individual “base face” by varying facial features. Participants saw two faces at a time, and then picked one as either trustworthy or high in ability or as a good financial advisor or power-lifting winner. Using these results, a computer then created an average face for each of these four categories, which were shown to a separate set of participants who had to pick which face appeared either more trustworthy or stronger. Most of the participants found the computer-generated averages to be good representations of trustworthiness or strength — and generally saw the average “financial advisor” face as more trustworthy and the “powerlifter” face as stronger. The findings from all four surveys were published in the Personality and Social Psychology Bulletin on June 18.

Adaptado de www.scientific.american.com/article/your-facial-bone-strecture-has-a-big-influence-on-how-people-see-you.(acesso em 20/8/2015)

Uma corda de cobre, com seção de raio !$ r_C !$, está submetida a uma tensão !$ T !$. Uma corda de ferro, com seção de raio !$ r_F !$ , de mesmo comprimento e emitindo ondas de mesma frequência que a do cobre, está submetida a uma tensão !$ T/3 !$. Sendo de !$ 1,15 !$ a razão entre as densidades do cobre e do ferro, e sabendo que ambas oscilam no modo fundamental, a razão !$ r_C/r_F !$ é igual a

Pintam-se !$ N !$ cubos iguais utilizando-se 6 cores diferentes, uma para cada face. Considerando que cada cubo pode ser perfeitamente distinguido dos demais, o maior valor possível de !$ N !$ é igual a

Seja !$ (a_1, a_2, a_3,...) !$ a sequência definida da seguinte forma: !$ a_1 = 1000 !$ e !$ a_n = \log_{10}(1 + a_{n-1}) !$ para !$ n \ge 2 !$. Considere as afirmações a seguir:

I. A sequência !$ (a_n) !$ é decrescente.

II. !$ a_n > 0 !$ para todo !$ n \ge 1 !$.

III. !$ a_n < 1 !$ para todo !$ n \ge 3 !$.

É (são) verdadeira(s)