Etymology
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pad (n.)

1550s, "bundle of straw to lie on," a word of obscure origin (perhaps a merger of several separate words), possibly from or related to Low German or obsolete Flemish pad "sole of the foot," which is perhaps from PIE *pent- "to tread, go" (see find (v.)), but see path (n.).

Sense of "soft cushion" is from 1560s, originally a soft saddle. Generalized sense of "something soft" is from c. 1700. Meaning "cushion-like part on the sole of an animal foot" in English is from 1790. The sense of "a number of sheets fastened or glued together at the edge" (in writing-pad, drawing-pad, etc.) is from 1865.

Sense of "takeoff or landing place for a helicopter or missile" is from 1949; the notion is of something to prevent friction or jarring. The word persisted in underworld slang from early 18c. in the sense "sleeping place," and this was popularized again c. 1959, originally in beatnik speech (later hippie slang) in its original English sense of "place to sleep temporarily," also "a room to use drugs."

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shoulder (n.)
Old English sculdor "shoulder," from West Germanic *skuldro (source also of Middle Dutch scouder, Dutch schouder, Old Frisian skoldere, Middle Low German scholder, Old High German scultra, German Schulter), of unknown origin, perhaps related to shield (n.). Meaning "edge of the road" is attested from 1933. Cold shoulder (Nehemiah ix.29) translates Latin humerum recedentum dare in Vulgate (but see cold shoulder). Shoulder-length, of hair, is from 1951.
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pad (v.2)

"to stuff with pads or padding, increase the amount of," 1827, from pad (n.). Of writing, "expand by insertion of extraneous matter," 1831; transferred to expense accounts, etc. by 1890. Related: Padded; padding. The idea of a padded cell in an asylum or prison (1862, padded room) is to prevent those inside from injuring themselves by dashing against the walls.

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pad (v.1)

"to walk, travel on foot, tramp slowly or wearily along," 1550s, probably from Middle Dutch paden "walk along a path, make a path," from pad, pat "path" (compare path). Originally a cant word among criminals and vagabonds, perhaps of imitative origin (sound of feet trudging on a dirt road). Related: Padded; padding. English also formerly had the noun pad meaning "path, foot path" (1560s), which might be from this verb, or from the Dutch noun, or a variant of path.

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shoulder (v.)
c. 1300, "to push with the shoulder," from shoulder (n.). Meaning "take a burden" first recorded 1580s. The military sense is from 1590s. Related: Shouldered; shouldering.
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lily-pad (n.)
"broad leaf of the water-lily," 1834, American English, from lily (n.) + pad (n.).
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cold shoulder (n.)

1816, in the figurative sense of "icy reception, studied neglect or indifference," first in Sir Walter Scott, probably originally a literal figure (see cold (adj.)), but commonly used with a punning reference to "cold shoulder of mutton," considered a poor man's dish and thus, perhaps, something one would set out for an unwanted guest with deliberate intention to convey displeasure.

How often have we admired the poor knight, who, to avoid the snares of bribery and dependence, was found making a second dinner from a cold shoulder of mutton, above the most affluent courtier, who had sold himself to others for a splendid pension! ["No Fiction," 1820]

Originally with to show, later to give. As a verb from 1845; related: cold-shouldered. Also compare cold roast, old slang for "something insignificant." Cold pig was a 19c. term for throwing cold water on a sleeping person to wake him or her.

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notepad (n.)

also note-pad, "pad of paper for writing notes," 1907, from note (n.) + pad (n.).

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keypad (n.)
"handheld pad of labelled buttons to work electronics," 1975, from key (n.1) + pad (n.).
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kneepad (n.)
also knee-pad, 1858, from knee (n.) + pad (n.).
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