Answer:
Step-by-step explanation:
To completely balance the system, the net radial force and the net tangential force acting on the rotating shaft should be zero. Let's analyze the forces in the system and solve for the magnitude and angular position of mass B.
First, let's calculate the radial forces:
Radial force due to mass A (FA) = mass A * radial distance from the axis of rotation
FA = 7.5 kg * 0.03 m = 0.225 N
Radial force due to mass C (FC) = mass C * radial distance from the axis of rotation
FC = 5 kg * 0.04 m = 0.2 N
Radial force due to mass D (FD) = mass D * radial distance from the axis of rotation
FD = 4 kg * 0.035 m = 0.14 N
Now, let's calculate the tangential forces:
Tangential force due to mass A (TA) = FA * sin(angle between mass A and B)
TA = 0.225 N * sin(47.5°) = 0.149 N
Tangential force due to mass C (TC) = FC * sin(angle between mass C and B)
TC = 0.2 N * sin(90°) = 0.2 N (since mass C and B are at right angles)
Tangential force due to mass D (TD) = FD * sin(angle between mass D and B)
TD = 0.14 N * sin(180° - 47.5°) = 0.091 N
The net radial force is given by the sum of the individual radial forces:
Net radial force = FA + FC + FD
Net radial force = 0.225 N + 0.2 N + 0.14 N = 0.565 N
The net tangential force is given by the sum of the individual tangential forces:
Net tangential force = TA + TC + TD
Net tangential force = 0.149 N + 0.2 N + 0.091 N = 0.44 N
For the system to be completely balanced, both the net radial force and the net tangential force should be zero. Since the net tangential force is already non-zero, we need to adjust the magnitude and angular position of mass B to make it zero.
Let's assume the magnitude of mass B as mB kg and the angular position as θ degrees. The radial distance from the axis of rotation for mass B is 0.038 m.
The radial force due to mass B (FB) = mB * 0.038 N
The tangential force due to mass B (TB) = FB * sin(angle between mass B and B)
TB = (mB * 0.038 N) * sin(θ - 90°)
To balance the system, the following condition should be satisfied:
Net radial force = 0
FB = -(FA + FC + FD)
mB * 0.038 N = -(0.225 N + 0.2 N + 0.14 N)
mB = -(0.565 N) / 0.038 N
Net tangential force = 0
TB = -(TA + TC + TD)
(mB * 0.038 N) * sin(θ - 90°) = -(0.149 N + 0.2 N + 0.091 N)
Solve these two equations to find the magnitude (mB) and angular position (θ) of mass B that will completely balance the system.