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\(^\circ\)?

Short answer

(a) 0.245 s (b) 0.245 s

Step-by-step solution

Understanding the Problem

We need to find the time it takes for a pendulum to reach its highest speed when released from two different initial angles: 3.50° and 1.75°, with a pendulum length of 0.240 m. As the pendulum is released from a small angle, we can assume simple harmonic motion.

Recall the Formula for Period of a Pendulum

The period T of a simple pendulum, which is the time for a complete back and forth swing, is given by the formula:\[ T = 2\pi \sqrt{\frac{L}{g}} \]where \( L = 0.240 \, \text{m} \) is the length of the pendulum and \( g \approx 9.81 \, \text{m/s}^2 \) is the acceleration due to gravity.

Calculate the Period of the Pendulum

Substitute the given length into the formula to find the period:\[ T = 2\pi \sqrt{\frac{0.240}{9.81}} \]Calculate this to find the period for the pendulum.

Simplify the Period Calculation

Calculate the inside of the square root:\[ \sqrt{\frac{0.240}{9.81}} \approx 0.156 \]Therefore, the period \( T \) becomes:\[ T = 2\pi \times 0.156 \approx 0.980 \, \text{s} \]

Determine Time to Reach Highest Speed

The pendulum reaches its highest speed at the lowest point of the swing, which is one-quarter of the period. Therefore, the time \( t \) to reach maximum speed is:\[ t = \frac{T}{4} = \frac{0.980}{4} \approx 0.245 \, \text{s} \]

Consider the Second Angle

For small angles, the pendulum period depends only on length and gravity, not the angle. Thus the time to reach the highest speed is the same even for an initial angle of 1.75°.

Key concepts

Harmonic Motion

Simple harmonic motion is a type of periodic motion where an object moves back and forth around an equilibrium position. In the context of a pendulum, this occurs when the pendulum swings from side to side. The motion is governed by the restoring force, which is proportional to the displacement. This means that the further the pendulum is pulled from its resting position, the stronger the force trying to bring it back. This force causes the pendulum to accelerate towards the center position, reaching its maximum speed as it passes through this point. The simplicity of this motion allows us to model and predict the pendulum's behavior under small angle assumptions, where the motion closely approximates simple harmonic motion.

Pendulum Period

The period of a pendulum is the time it takes to complete one full swing, from one extreme to the other and back again. For a simple pendulum, this period is determined by the length of the string and the acceleration due to gravity, not the mass of the pendulum or the angle of release (as long as the angle is small). The formula for the period \( T \) is given by \( T = 2\pi \sqrt{\frac{L}{g}} \). Here, \( L \) represents the pendulum's length, and \( g \), the acceleration due to gravity, is approximately \(9.81\,\text{m/s}^2\). Through this formula, we see that as the length of the pendulum increases, so too does the period. This means a longer pendulum takes more time to complete a swing, highlighting the independence of the period from the angle of release.

Pendulum Length

Pendulum length is a crucial factor in determining the period of the pendulum. In our scenario, the length is given as 0.240 meters. This length, alongside the constant acceleration due to gravity, allows us to calculate the period directly using the formula discussed. A longer pendulum would result in a longer period, while a shorter one would swing back and forth more quickly. It's important to understand that the length affects only the period and not the amplitude (how far it swings) of the pendulum's motion.

Acceleration Due to Gravity

Acceleration due to gravity, often symbolized as \( g \), is a constant that affects the motion of pendulums and other falling objects. On Earth, \( g \) is around \( 9.81 \, \text{m/s}^2 \). This value is crucial for calculating the pendulum's period using the formula \( T = 2\pi \sqrt{\frac{L}{g}} \). Because gravity is a constant factor in this equation, the location (on Earth) remains a constant influence on the pendulum's timing. Thus, a pendulum will swing with the same period regardless of where it is on Earth, provided the length remains constant and the assumption of a simple pendulum holds true. Understanding gravity's role helps clarify why only pendulum length alters the period, underlining the distinct role each parameter plays in harmonic motion.