A pendulum consists of a small object hanging from the ceiling at the end of a string of negligible mass. The string has a length of 0.80 m. With the string hanging vertically, the object is given an initial velocity of 2.6 m/s parallel to the ground and swings upward on a circular arc. Eventually, the object comes to a momentary halt at a point where the string makes an angle θ with its initial vertical orientation and then swings back downward. Find the angle θ.

To find the angle θ, we can apply the conservation of energy principle and use the equation for the conservation of mechanical energy of the pendulum.

The conservation of mechanical energy states that the total mechanical energy of a system is constant, assuming there are no external forces that do work on the system.

In this case, the initial mechanical energy of the pendulum is given by the initial kinetic energy, and the final mechanical energy is given by the final potential energy. At the highest point of the swing, the object comes to a momentary halt, meaning its velocity is zero. This implies that all the initial kinetic energy gets converted into potential energy.

Let's break down the problem step by step:

1. Determine the initial kinetic energy (KEi):
- Given that the object has an initial velocity of 2.6 m/s parallel to the ground, we can calculate its initial kinetic energy using the equation KE = 1/2 * m * v^2, where m is the mass of the object (which cancels out in this case).
- KEi = 1/2 * (2.6 m/s)^2

2. Determine the final potential energy (PEf):
- At the highest point, the object comes to a momentary halt, meaning its velocity is zero. At this point, all the initial kinetic energy is converted into potential energy.
- To calculate the potential energy, we need to consider the height difference from the initial vertical position to the highest point of the swing, which is given by the length of the string (0.80 m) multiplied by the difference in height (which is sin(θ)).
- PEf = m * g * h, where g is the acceleration due to gravity, and h is the height difference.
- PEf = m * g * (0.80 m * sin(θ))

3. Equate the initial kinetic energy to the final potential energy:
- Set KEi = PEf and solve for θ.
- 1/2 * (2.6 m/s)^2 = m * g * (0.80 m * sin(θ))

Now, to actually find the angle θ, we need to know the mass of the object and the acceleration due to gravity. Once you have that information, you can substitute the values into the equation and solve for θ.