To solve this problem, we need to understand the dynamics of a simple pendulum and how tension in the string changes during its motion. The tension in the string when the pendulum passes through the mean position (lowest point) involves both the gravitational force and the centripetal force required to keep the bob moving in a circular path.
Understanding the forces: When the pendulum passes through its mean position, the forces acting on it are:
Condition at the mean position: At the lowest point, the tension in the string must provide the centripetal force required for circular motion:
\(T - mg = \frac{mv^2}{l}\),
where \(v\) is the speed of the pendulum bob at the mean position and \(l\) is the length of the pendulum.
Given that the tension in the string is 3 times the weight of the pendulum bob:
\(T = 3mg\)
Substituting the value of \(T\) in the previous equation, we get:
\(3mg - mg = \frac{mv^2}{l}\)
\(2mg = \frac{mv^2}{l}\)
Canceling mass \((m)\) from both sides, we obtain:
\(v^2 = 2gl\)
Finding the maximum angle of displacement: The maximum displacement of the pendulum corresponds to the maximum height it reaches. Using energy conservation between the highest and lowest points:
\(\frac{1}{2}mv^2 = mgh\)
Substitute \(v^2 = 2gl\) and simplify:
\(\frac{1}{2}m(2gl) = mgh\)
\(gl = gh\)
\(l = h\)
Thus, the bob reaches a height equal to the length \(l\) of the pendulum. This indicates that the pendulum reaches a position where it is horizontal, implying a displacement of:
\(\theta = 90^\circ\)
The correct answer is that the maximum displacement of the pendulum with respect to the vertical is 90°.