Four people are pulling in different directions on a shopping cart. The magnitude of their pulling forces and the directions relative to straight east are indicated below. Jane: 80 pounds, straight east (0 degrees relative to east) Brittany: 100 pounds, north (90 degrees relative to east) Robyn: 80 pounds, southwest (225 degrees relative to east) Scott: 100 pounds, northwest (135 degrees relative to east) Which way with the shopping cart go?

17⁰ north of west

Explanation:

We shall represent forces by vector notations . i will represent unit vector towards east and j will represent unit vector towards north .

Jane: 80 pounds, straight east

F₁ = 80 i

Brittany: 100 pounds, north (90 degrees relative to east)

F₂ = 100 j

Robyn: 80 pounds, southwest (225 degrees relative to east ) or 45⁰ south of west

F₃ = - 80cos 45  i - 80 sin45 j = - 80/√2 x ( i + j )

= - 40√2 ( i + j )

Scott: 100 pounds, northwest (135 degrees relative to east) or 45⁰ west of north .

F₄ = - 100 cos 45 i + 100 sin45 j =  50√2 ( - i + j )

Resultant force

= F₁ + F₂ + F₃ + F₄

F =  80 i + 100 j + - 40√2 ( i + j ) + 50√2 ( - i + j )

F = 80 i - 90√2 i + 110√2 j

80 i - 127.26 i + 114.14 j

F = - 47.26 i + 114.14 j

Direction of force

TanФ = 114.14 / 47.26 = 2.4

Ф = 67⁰

67⁰ north of west .

0.863333

86%

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