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11 days ago

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23. While sliding a couch across a floor, Andrea and Jennifer exert forces F → A and F → J on the couch. Andrea’s force is due n

orth with a magnitude of 130.0 N and Jennifer’s force is 32° east of north with a magnitude of 180.0 N. (a) Find the net force in component form. (b) Find the magnitude and direction of the net force. (c) If Andrea and Jennifer’s housemates, David and Stephanie, disagree with the move and want to prevent its relocation, with what combined force F → DS should they push so that the couch does not move?

1 answer:

Softa [913]11 days ago

4 0

Response:

a) (95.4 i^ + 282.6 j^) N, b) 298.27 N 71.3º and c) F' = 298.27 N θ = 251.4º

Justification:

a) Using trigonometric functions, we can resolve Jennifer's force

sin θ = Fjy / Fj

cos θ = Fjx / Fj

Calculating the angle, which is 32º to the east of north, taken from the positive x-axis:

T = 90 -32 = 58º

Fjy = Fj sin 58

Fjx = FJ cos 58

Fjx = 180 cos 58 = 95.4 N

Fjy = 180 sin 58 = 152.6 N

Andrea's exerted force is

Fa = 130.0 j ^

Summing forces across both axes:

X axis

Fx = Fjx

Fx = 95.4 N

Y axis

Fy = Fjy + Fa

Fy = 152.6 + 130

Fy = 282.6 N

F = (95.4 i ^ + 282.6 j ^) N

b) Now let's employ the Pythagorean theorem and trigonometry

F² = Fx² + Fy²

F = √ (95.4² + 282.6²)

F = √ (88963)

F = 298.27 N

tan θ = Fy / Fx

θ = tan-1 (282.6 / 95.4)

θ = tan-1 (2.962)

θ = 71.3º

c) To counteract the motion, they must apply an opposing force of equal intensity:

F' = 298.27 N

θ' = 180 + 71.3

θ = 251.4º

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