A pendulum is made by tying a 75 g ball to a 130-cm-long string. The ball is pulled 5.0° to the side and released. How many times does the ball pass through the lowest point of its arc in 7.5 s?

Verified step by step guidance

1. Convert the given mass of the ball (75 g) into kilograms by dividing by 1000, as SI units require mass in kilograms.
2. Determine the length of the pendulum in meters by converting the given length of 130 cm into meters (divide by 100).
3. Use the formula for the period of a simple pendulum: $T=2\pi \sqrt{\frac{L}{g}}$, where $L$ is the length of the pendulum and $g$ is the acceleration due to gravity (approximately 9.8 m/s²). Substitute the value of $L$ into the formula to calculate the period $T$.
4. Determine the number of oscillations the pendulum completes in 7.5 seconds by dividing the total time (7.5 s) by the period $T$.
5. Since the ball passes through the lowest point of its arc twice per oscillation (once in each direction), multiply the number of oscillations by 2 to find the total number of times the ball passes through the lowest point in 7.5 seconds.

Key Concepts

Pendulum Motion

A pendulum exhibits periodic motion, swinging back and forth around a pivot point. The time it takes to complete one full cycle is called the period, which depends on the length of the string and the acceleration due to gravity, but not on the mass of the pendulum bob. For small angles, the motion can be approximated as simple harmonic motion.

Period of a Pendulum

The period of a simple pendulum can be calculated using the formula T = 2π√(L/g), where T is the period, L is the length of the string, and g is the acceleration due to gravity (approximately 9.81 m/s²). This formula shows that the period increases with the length of the pendulum and is independent of the mass of the bob, making it crucial for determining how often the pendulum swings.

Frequency of Motion

Frequency is the number of complete cycles of motion that occur in a unit of time, typically measured in hertz (Hz). It is the reciprocal of the period (f = 1/T). In the context of the pendulum, knowing the frequency allows us to calculate how many times the pendulum passes through the lowest point in a given time frame, such as 7.5 seconds.