1) Suppose one force vector is 100 N at theta= 45 degrees, and a second force ve

Question

1) Suppose one force vector is 100 N at theta= 45 degrees, and a second force vector is 75 N at theta =123 degrees both with respect to the positive x-axis. What is a third force that would cause zero net force if all forces are acting on the same mass particle?

2) Suppose one force vector is 150 N at theta = 45 degrees, and a second force vector is 172 N at theta =
300 degrees both with respect to the positive x-axis. What is a third force that would cause zero net force if all forces are acting on the same mass particle?

Explanation / Answer

1) F vector = Fcos@ i + Fsin@ j

in (i and j form)

F1 = 100cos45i + 100sin45j N

F2 = 75cos123i + 75sin123 j N

F3 = Fcos@i + Fsin@j

Fnet = F1 + F2 + F3 = 0

100cos45i + 100sin45j N + 75cos123i + 75sin123 j + Fcos@i + Fsin@j = 0

(100cos45 + 75cos123i + Fcos@ ) i + (100sin45 + 75sin123 +Fsin@) j = 0

100cos45 + 75cos123i + Fcos@ = 0

Fcos@ = -29.86 ..(i)

100sin45 + 75sin123 +Fsin@ = 0

Fsin@ = -133.61 ...(ii)

(ii) / (i)

tan@ = 133.61 / 29.86 =4.47

@ = 77.40 degree

both sin and cos are -ve
so its in 3rd quad.

so from +ve x-axis @ = 180 + 77.40 = 257.40 degrees

F = -29.86 / cos257.40 = 136.91 N

so 136.91 N at 2257.40 degere

2) in similar way (cut in short)

150cos45 +172cos300 + Fcos@ = 0

Fcos@ = -192.07

150sin45 + 172sin300 + Fsin@ = 0

Fsin@ = 62.92

Fsin@ / Fcos@ = tan@ = 62.92 / 192.07 = 18.14 degree

cos is -ve and sin is +ve -> that means 2nd quadrant.

angle = 180 - 18.14 =161.86 degrees

F = -192.07/ cos161.86 = 202.11 N

202.11N at 161.86 degrees

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