Introduction:
A simple pendulum consists of a mass called the bob, which is attached to a string or rod of length x. When the pendulum is displaced from its equilibrium position and released, it oscillates back and forth. In this scenario, the pendulum is held at two different angles, theta1 and theta2, with respect to the vertical. The speeds at which it passes the lowest point, denoted as v1 and v2, are given. We need to find the ratio v1/v2.

Solution:
To solve this problem, we will use the conservation of mechanical energy and the principle of conservation of angular momentum.

Conservation of Mechanical Energy:
The mechanical energy of the pendulum is conserved throughout its motion. At the highest point, when the bob is released, it has only potential energy. As it swings down, this potential energy is converted into kinetic energy. At the lowest point, the bob has only kinetic energy.

The potential energy at the highest point is given by:
PE1 = m * g * x * (1 - cos(theta1))

The kinetic energy at the lowest point is given by:
KE2 = (1/2) * m * v2^2

Since the mechanical energy is conserved, we can equate the potential energy at the highest point with the kinetic energy at the lowest point:
PE1 = KE2

Simplifying the equation, we get:
m * g * x * (1 - cos(theta1)) = (1/2) * m * v2^2

Principle of Conservation of Angular Momentum:
The angular momentum of the pendulum is conserved throughout its motion. At the highest point, the angular momentum is given by:
L1 = m * x * v1 * sin(theta1)

At the lowest point, the angular momentum is given by:
L2 = m * x * v2 * sin(theta2)

Since the angular momentum is conserved, we can equate L1 and L2:
L1 = L2

Simplifying the equation, we get:
m * x * v1 * sin(theta1) = m * x * v2 * sin(theta2)

Calculating v1/v2:
Dividing the equation derived from the conservation of mechanical energy by the equation derived from the conservation of angular momentum, we get:
(m * g * x * (1 - cos(theta1))) / ((1/2) * m * v2^2) = (m * x * v1 * sin(theta1)) / (m * x * v2 * sin(theta2))

Simplifying the equation and canceling out common terms, we get:
2 * g * (1 - cos(theta1)) = v1^2 / v2^2 * (sin(theta1) / sin(theta2))

Taking the square root of both sides, we get:
sqrt(2 * g * (1 - cos(theta1))) = v1 / v2 * sqrt(sin(theta1) / sin(theta2))

Finally, rearranging the equation, we get:
v1 / v2 = sqrt(2 * g * (1 - cos(theta1))) / sqrt(sin(theta1) / sin(theta2))

Conclusion:
The ratio v1/v2 can be calculated using the equation v1 /

(1-costheta1/1-costheta2)^1/2