Mechanics- forces in two dimensions ?
In a competition of strength, four people pull a mass with ropes at diff angles. The direction in which the mass moves determines the winner. Arjun wants the mass to go north, Bob wants it to go east, Chen wants it to go south and David wants it to go west. The men pull with the forces in the directions shown in the diagram. Find the direction of the resultant motion and determine who wins.
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On the face of it, Arjun will win, but it could be Bob because he gets a little help from both Arjun and Chen. Well, let's see...
A: 330 @ 80 deg,
B: 250 @ 5 deg,
C: 300 @ 280 deg,
D: 280 @ 190 deg.
330 cos(80) i + 330 sin(80) j;
250 cos(5) i + 250 sin(5) j;
300 cos(280) i + 300 sin(280) j;
280 cos(190) i + 280 sin(190) j.
Use calculator:
57.30 i + 324.99 j;
249.05 i + 21.79 j;
52.09 i - 295.44 j;
-275.75 i - 48.62 j.
Add up:
In my head, it's ABOUT 83i + 3j.
So Bob wins!!
Use calculator for a good sum,
then take arctan(y/x) to get exact direction.
Resulting force F :
Fy = cos 10°(330-300)+250*sin 15°-280*sin 10° = 45.6 N
Fx = sin 10°(330+300)+250*cos 15°-280*cos 10° = 75.1 N
F is located in the 1st quadrant with heading angle = arctan 45.6/75.1 = 31.3° NoE
off hand, I'd say east is closest. But technically none of them win, as I doubt it goes exactly in a cardinal direction.
get the x and y components of each and add
starting with bob and moving CCW
F1x = 250cos5 = 249
F1y = 250sin5 = 22
F2x = 330cos80 = 57
F2y = 330sin80 = 325
F3x = –280cos10 = –276
F3y = –280sin10 = –40
F4x = 300cos80 = 52
F3y = –300sin80 = –295
add them up
Fx = 82
Fy = 12
F = √(82² + 12²) = 83
angle = arctan(12/82) = 8.3º (with + x axis)
so nobody wins, as I predicted.