To better understand the concept of static equilibrium a laboratory procedure as

Question

To better understand the concept of static equilibrium a laboratory procedure asks the student to make a calculation before performing the experiment. The apparatus consists of a round level table in the center of which is a massless ring. There are three strings tied to different spots on the ring. Each string passes parallel over the table and is individually strung over a frictionless pulley (there are three pulleys) where a mass is hung. The table is degree marked to indicate the position or angle of each string. There is a mass m1 = 0.153 kg located at 1 = 22.5° and a second mass m2 = 0.227 kg located at 2 = 279°. Calculate the mass m3, and location (in degrees), 3, which will balance the system and the ring will remain stationary.

Explanation / Answer

m1 = 153g = 0.153kg.

Weight = (.153 x 9.81) = 1.5N.
m2 = 0.227kg., = 2.226N. weight.
Let's "rotate" the table 22.5 degrees anticlockwise,

so m1 is at 0. That puts m2 at (279- 22.5) = 256.5 degrees.
(256.5 - 180) = 76.5 degrees "west of south"

South component of m2 = (cos 76.5) x 2.226, = 0.5196 N.
West component = (sin 76.5) x 2.226, = 2.1644N.
Subtract South component from weight m1, = 0.9804N., acting North.

Weight of M3 = sqrt. (2.1644^2 + 0.9804^2), = 2.376N., divided by g = mass of 0.2422kg., or 242g.
Direction = arctan (0.9804/2.1644) = 24.36 degrees S of E.,
Now "rotate" the table back the 22.5 degrees clockwise, (22.5 + 24.36+ 90) = 136.86 degrees, is where to place the m3 of 242g. (The "90" is because east is 90 deg. from North, and the 22.5 was from E).

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