Question

Explain any three applications of a couple.​

Answer 1

Explanation:

Sup­pose we want to com­pute all of the devi­a­tions of a list of num­bers. A devi­a­tion is the dis­tance from the num­ber to the mean of the list, in other words x−μ. Here is naïve approach to the prob­lem.

len :: Frac­tional b => [a] -> b

len [] = 0

len (_ : xs) = 1 + (len xs)

aver­age :: Frac­tional a => [a] -> a

aver­age xs = (sum xs) / (len xs)

devi­a­tions :: Frac­tional a => [a] -> [a]

devi­a­tions xs = map (sub­tract (aver­age xs)) xs

This is fine, but it tra­verses the list three times. Once for sum, once for len, and once for map. We can improve this sit­u­a­tion by com­put­ing the sum and len simul­ta­ne­ously.

aver­age' :: Frac­tional a => [a] -> (a, a)

aver­age' [] = (0, 0)

aver­age' (x : xs) = (x + sum, 1 + len)

where (sum, len) = aver­age' xs

aver­age :: Frac­tional a => [a] -> a

aver­age xs = sum / len

where (sum, len) = aver­age' xs

devi­a­tions :: Frac­tional a => [a] -> [a]

devi­a­tions xs = map (sub­tract (aver­age xs)) xs

Here we only have to trav­ese the list twice. Once for aver­age’ and once for map.

Can we do even bet­ter? Can we com­pute this with just a sin­gle tra­ver­sal? If you think it’s pos­si­ble, try to write it. If you think it’s impos­si­ble, why?

Read this short post and under­stand how trace com­putes rep­Min. I’ve also posted my own notes on cir­cu­lar pro­gram­ming.

Come up with an appro­pri­ate type for devi­a­tions’. We’ve put in a, b, c, d type para­me­ters as a place­holder. You should replace these with the appro­pri­ate types.

Write devi­a­tions’ so that devi­a­tions only tra­verses the list once. You should only write devi­a­tions’ and no other recur­sive helper func­tions. Our def­i­n­i­tion of trace is slightly dif­fer­ent than the arti­cle; we don’t use the unnec­es­sary tuple in the input. This is a super­fi­cial dif­fer­ence. (Hint: You may have to use a lazy pat­tern match.)

Answer 2

Answer:

In mechanics, a couple is a system of forces with a resultant (a.k.a. net or sum) moment but no resultant force.[1]

A better term is force couple or pure moment. Its effect is to create rotation without translation, or more generally without any acceleration of the centre of mass. In rigid body mechanics, force couples are free vectors, meaning their effects on a body are independent of the point of application.

The resultant moment of a couple is called a torque. This is not to be confused with the term torque as it is used in physics, where it is merely a synonym of moment.[2] Instead, torque is a special case of moment. Torque has special properties that moment does not have, in particular the property of being independent of reference point, as described below.

Explanation:

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