smc-3577-20-Force

Vectors, EXT1 V1 SM-Bank 32

A beached whale is being rescued by two jet-skis with the forces measured in kilonewtons (kN).

The angle of the forces are measured against an east-west axis line.

What is the resultant force of the two jet-skis in magnitude and direction? (3 marks)

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`text{120 kN on a bearing of 096°.}`

Show Worked Solution

`text{Split each force into component forces:`

`underset~u = (55 cos 20, 55 sin 20)`

`underset~v = (75 cos 25, -75 sin 25)`

`underset~F_{text{net}}`	`= underset~u + underset~v = ((55 cos 20 + 75 cos 25),(55 sin 20 – 75 sin 25))`
`\| underset~F_{text{net}}\|`	`= sqrt{(55 cos 20 + 75 cos 25)^2 + (55 sin 20 – 75 sin 25)^2}`
	`= 120. 347 …`

`tan theta`	`= {\| 55 sin 20 – 75 sin 25 \|}/{ 55 cos 20 + 75 cos 25}`
	`= 0.10768`
`theta`	`= 6^@`

`:. \ text{The resultant force is 120 kN on a bearing of 096°.}`

Vectors, EXT1 V1 EQ-Bank 3

A force described by the vector `underset~F = ((3),(6))` newtons is applied to a line `l` which is parallel to the vector `((4),(3))`.

Find the component of the force `underset~F` in the direction of `l`. (2 marks)

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What is the component of the force `underset~F` in the direction perpendicular to the line? (1 mark)

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`((4.8),(3.6))`
`((−1.8),(2.4))`

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i. `underset~F = ((3),(6)), \ underset~v = ((4),(3))`

`underset~overset^v = underset~v/(|underset~v|) = underset~v/sqrt(4^2 + 3^2) = 1/5 underset~v=((0.8),(0.6))`

`underset~ F · underset~overset^v`	`= ((3),(6))((0.8),(0.6))`
	`= 3 xx 0.8 + 6 xx 0.6`
	`= 6`

`text(proj)_(underset~v) underset~F`	`= (underset~F · underset~overset^v) underset~overset^v`
	`= 6((0.8),(0.6))`
	`= ((4.8),(3.6))`

ii. `text(Component of)\ underset~F ⊥ l`

`= ((3),(6)) – ((4.8),(3.6))`

`= ((−1.8),(2.4))`

Vectors, EXT1 V1 2015 SPEC2 15 MC

The projection of the force `underset~F = aunderset~i + bunderset~j`, where `a` and `b` are non-zero real constants, in the direction of the vector `underset~w = underset~i + underset~j`, is

A. `((a + b)/2)underset~w`

B. `underset~F/(a + b)`

C. `((a + b)/(a^2 + b^2))underset~F`

D. `((a + b)/sqrt2)underset~w`

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`A`

Show Worked Solution

`hatw`	`= tildew/sqrt(1+1)`
	`= (tildei + tildej)/sqrt2`

`tildeF*hat w = (a + b)/sqrt2`

♦ Mean mark 49%.

`(tildeF*hat w)hatw`	`= ((a + b)/sqrt2) tildew/sqrt2`
	`= ((a + b)/2) tildew`

`=> A`