A 200 g ball is tied to a string. It is pulled to an angle...

A 200 \mathrm{g} ball is tied to a string. It is pulled to an angle of $8.0^{\circ}$ and released to swing as a pendulum. A student with a stopwatch finds that 10 oscillations take 12 s. How long is the string?

Key Concepts

Simple Pendulum

A simple pendulum consists of a small mass suspended by a string or rod that swings under the influence of gravity. Its motion, when displaced slightly from its equilibrium position, approximates simple harmonic motion, making it an important system for studying oscillatory behavior in classical mechanics.

Period of a Pendulum

The period is the time required for one complete cycle of the pendulum’s swing. For a pendulum, the period is given by the formula T = 2??(L/g), which shows that the period depends on the length of the string and the acceleration due to gravity, but not on the mass of the bob.

Small Angle Approximation

This approximation assumes that for small angles (typically less than about 15°), the sine of the angle is approximately equal to the angle in radians. It simplifies the equations of motion for the pendulum, allowing the motion to be treated as simple harmonic motion.

Experimental Determination of the Period

Measuring the time taken for multiple oscillations and then dividing by the number of oscillations helps to reduce measurement errors in timing and yields an average period. This method is commonly used in experiments to more accurately determine the period of an oscillatory system.

Gravitational Acceleration

Gravitational acceleration, denoted by g (approximately 9.8 m/s² near the Earth's surface), is the acceleration experienced by an object due to Earth's gravity. In the context of a pendulum, g is a crucial factor in determining the period through the relationship T = 2??(L/g).

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