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 0.73 m. With the string hanging vertically, the object is given an initial velocity of 2.0 m/s parallel to the ground and swings upward in 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.

Question

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 0.73 m. With the string hanging vertically, the object is given an initial velocity of 2.0 m/s parallel to the ground and swings upward in 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.

Answer 1

To find the angle θ, we can use the principle of conservation of mechanical energy. At the initial point, the pendulum has both kinetic energy and potential energy. As it swings upward, the potential energy is converted into kinetic energy until it comes to a stop momentarily.

Step 1: Find the total mechanical energy (E) of the pendulum at the initial point.
Since the object has an initial velocity of 2.0 m/s parallel to the ground, its kinetic energy (K) can be calculated using the formula:
K = (1/2) * m * v^2
where m is the mass of the object and v is the velocity.

Step 2: Find the potential energy (PE) of the pendulum at the initial point.
At the initial point, the potential energy is zero, as the object is at ground level. When the object reaches the highest point in its swing, all of its initial kinetic energy has been converted into potential energy:
PE = m * g * h
where m is the mass of the object, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the object from the ground.

Step 3: Set the total mechanical energy at the initial point (E_initial) equal to the potential energy at the highest point (PE_highest):
E_initial = PE_highest
Therefore, K + PE = m * g * h

Step 4: Calculate the height (h) at the highest point using the law of conservation of energy:
h = (K + PE) / (m * g)

Step 5: Find the angle (θ) by using the relationship between the height (h) and the length of the pendulum (L):
h = L * (1 - cos(θ))
where L is the length of the string of the pendulum.

Step 6: Rearrange the equation to solve for θ:
θ = cos^(-1)((h / L) - 1)

By substituting the values of m, v, g, and L into the equations, you can calculate the angle θ.

78

To find the angle θ, we can use the principle of conservation of mechanical energy. At the initial point, the pendulum has both kinetic energy and potential energy. As it swings upward, the potential energy is converted into kinetic energy until it comes to a stop momentarily.