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 = 0.157 kg located at θ1 = 22.5° and a second mass m2 = 0.219 kg located at θ2 = 283°. Calculate the mass m3, and location (in degrees), θ3, which will balance the system and the ring will remain stationary.

Answer 1

Given,

m1 = 0.157 kg ; theta1 = 22.5 deg ; m2 = 0.219 kg ; theta2 = 283

We need to find m3 and theta3

m2 will rotate by:

283 - 22.5 = 260.5 deg

theta2' = 260.5 - 180 = 80.5 deg

W2 = m2 g cos(theta2')

W2(y) = 0.219 x 9.81 x cos80.5 = 0.355 N

W2(x) = 0.219 x 9.81 x sin80.5 = 2.12 N

Wnet-y = m1g - W2x

Wnet-y = 0.157 x 9.81 - 0.355 = 1.185 N

W3 = sqrt (1.185^2 + 2.12^2) = 2.429 N

m3 = 2.429/9.81 = 0.248 kg

theta = tan^-1(1.185/2.12) = 29.2 deg

theta3 = 22.5 + 29.2 + 90 = 141.7 deg

Hence, m3 = 0.248 kg ; theta3 =141.7 deg