A pendulum on a 75-cm-long string has a maximum speed of 0.25 m/s. What is the pendulum's maximum angle in degrees?

Verified step by step guidance

1. Step 1: Recognize that the pendulum's motion is governed by the principles of conservation of energy. The total mechanical energy (sum of potential energy and kinetic energy) remains constant throughout the motion.
2. Step 2: Write the expression for the total mechanical energy. At the lowest point of the pendulum's swing, the energy is purely kinetic: \( E = \frac{1}{2}mv^2 \), where \( m \) is the mass of the pendulum and \( v \) is its maximum speed (0.25 m/s).
3. Step 3: At the maximum angle, the pendulum's energy is purely potential: \( E = mgh \), where \( h \) is the height the pendulum rises relative to its lowest point, and \( g \) is the acceleration due to gravity (9.8 m/s²). Equate the kinetic energy at the lowest point to the potential energy at the maximum height: \( \frac{1}{2}mv^2 = mgh \).
4. Step 4: Solve for \( h \), the height: \( h = \frac{v^2}{2g} \). Substitute \( v = 0.25 \, \text{m/s} \) and \( g = 9.8 \, \text{m/s}^2 \) into the equation to find \( h \).
5. Step 5: Relate the height \( h \) to the maximum angle \( \theta \) using trigonometry. The pendulum's string length \( L \) is 0.75 m. The height \( h \) corresponds to the vertical displacement, so \( \cos(\theta) = \frac{L - h}{L} \). Solve for \( \theta \) in degrees: \( \theta = \cos^{-1}\left(\frac{L - h}{L}\right) \).

Key Concepts

Pendulum Motion

A pendulum exhibits simple harmonic motion, where it swings back and forth around a pivot point. The motion is periodic, and the restoring force acting on the pendulum is gravity, which pulls it back toward its equilibrium position. The length of the string and the maximum speed are crucial in determining the pendulum's behavior.

Maximum Speed and Energy Conservation

The maximum speed of a pendulum occurs at its lowest point in the swing, where potential energy is converted into kinetic energy. The principle of conservation of energy states that the total mechanical energy (potential + kinetic) remains constant if we neglect air resistance and friction. This relationship allows us to relate speed to height and angle.

Angle of Displacement

The angle of displacement refers to the maximum angle the pendulum makes with the vertical at its highest point. This angle can be calculated using trigonometric relationships, considering the height reached by the pendulum and the length of the string. The relationship between the maximum speed and the angle can be derived from energy conservation principles.