Problem 45 - University Physics with Modern Physics

Question

You pull a simple pendulum 0.240 m long to the side through an angle of \(3.50^{\circ}\) and release it. (a) How much time does it take the pendulum bob to reach its highest speed? (b) How much time does it take if the pendulum is released at an angle of \(1.75^{\circ}\) instead of \(3.50^{\circ} ?\)

Step-by-Step Solution

Answer

(a) Time to highest speed: 0.245 s; (b) Time is the same: 0.245 s.

Step 1: Understand the Pendulum Motion

A simple pendulum exhibits simple harmonic motion when displaced at small angles. The time it takes to reach the highest speed from rest is a quarter of its period. The period (T) of a simple pendulum is given by:\[ T = 2\pi \sqrt{\frac{L}{g}} \]where L is the length of the pendulum and g is the acceleration due to gravity (approximately \( 9.81 \text{ m/s}^2 \)).

Step 2: Calculate the Period of the Pendulum

Using the formula for the period, substitute the known values:\[ T = 2\pi \sqrt{\frac{0.240}{9.81}} \]Calculate the result to find the period of the pendulum.

Step 3: Determine Time to Highest Speed for Initial Angle 3.50°

The time to reach the highest speed is a quarter of the period because the pendulum swings from rest to maximum speed within a quarter cycle. Thus, the time is:\[ t = \frac{T}{4} \]

Step 4: Calculate Time for Initial Angle 1.75°

The period of a simple pendulum only depends on its length and gravity, not the initial angle. Therefore, for an initial angle of \(1.75^{\circ}\), repeat steps 2 and 3 with the same values for length and gravity.

Key Concepts

Understanding Harmonic Motion

A simple pendulum moves in what is called simple harmonic motion, especially when it's released from small angles. This type of motion refers to the back-and-forth oscillation that happens in a repetitive and periodic manner. For a pendulum, this occurs between its highest points on either side and its lowest point in the middle.

• At the highest points, the pendulum comes to a temporary stop before swinging back.
• At the lowest point, the pendulum achieves its highest speed.

Simple harmonic motion is predictable. The pendulum's motion can be described using a sine or cosine function in physics. This is what makes pendulums helpful not only in clocks but also in various experiments where precise timing is required.

Influence of Pendulum Length

The length of a pendulum significantly affects how it swings and the period of its swing. The length is the distance from the pivot point to the center of mass of the pendulum bob. Longer pendulums have longer periods, meaning they take more time to complete a full swing.

• A short pendulum swings faster and has a shorter period.
• A long pendulum swings slower and has a longer period.

It's important to note that this length should remain constant for the pendulum to maintain uniform motion. In a classroom or lab, ensuring precise measurement of the pendulum's length is vital to achieving accurate results.

Role of Acceleration due to Gravity

The acceleration due to gravity, denoted as 'g', is crucial in determining the period of a pendulum. On Earth, this value is approximately 9.81 m/s², but it can vary slightly depending on location. This factor is part of the formula used to calculate the pendulum's period: \[ T = 2\pi \sqrt{\frac{L}{g}} \]Here, 'L' represents the pendulum's length, and 'g' is the local gravitational acceleration. On planets with higher gravity, a pendulum would swing faster, reducing the period. Conversely, on planets with lower gravity, the pendulum's oscillations would slow down, increasing the period. Thus, understanding 'g' is key in predicting the pendulum's behavior in different environments.

Calculating the Pendulum Period

The period of a pendulum is the time it takes to complete one full back-and-forth swing. It's crucial for determining how fast or slow a pendulum moves. The period is influenced by both the length of the pendulum and the acceleration due to gravity.
For a simple pendulum, the period can be calculated using the formula: \[ T = 2\pi \sqrt{\frac{L}{g}} \]From this formula, you can see that the period is independent of the initial angle of release, as long as the swings remain small. Understanding how to calculate the pendulum period is important for solving problems related to timekeeping and in physics experiments where timing accuracy is crucial.