Question

What is the period of a simple pendulum that is \(1.00 \mathrm{~m}\) long in each situation? a) in the physics lab b) in an clevator accelerating at \(2.10 \mathrm{~m} / \mathrm{s}^{2}\) upward c) in an elevator accelerating \(2.10 \mathrm{~m} / \mathrm{s}^{2}\) downward d) in an elevator that is in free fall

Short answer

Answer: The period of the simple pendulum is: a) Approximately 2.00 s in the physics lab b) Approximately 1.82 s in an elevator accelerating upward c) Approximately 2.26 s in an elevator accelerating downward d) Undefined in an elevator in free fall

Step-by-step solution

Calculate the period in normal conditions (physics lab)

To find the period of the pendulum in the physics lab, we'll use the formula \(T=2\pi\sqrt{\frac{l}{g}}\), where \(l\) is the length of the pendulum and \(g\) is the standard gravitational acceleration (\(9.81 \mathrm{~m} / \mathrm{s}^{2}\)). Given that the length of the pendulum is \(1.00 \mathrm{~m}\), we can plug the values into the formula to find the period. T = 2π × √(1.00 m / 9.81 m/s²) ≈ 2.00 s

Calculate the period in an elevator accelerating upward

When the elevator is accelerating upward, the effective gravitational acceleration is the sum of the normal gravitational acceleration and the acceleration of the elevator. In this case, \(g'=g+a\), where \(a = 2.10 \mathrm{~m} / \mathrm{s}^{2}\). We can then use the formula again to find the period. g' = 9.81 m/s² + 2.10 m/s² = 11.91 m/s² T = 2π × √(1.00 m / 11.91 m/s²) ≈ 1.82 s

Calculate the period in an elevator accelerating downward

When the elevator is accelerating downward, the effective gravitational acceleration is the difference between the normal gravitational acceleration and the acceleration of the elevator. In this case, \(g'=g-a\), where \(a = 2.10 \mathrm{~m} / \mathrm{s}^{2}\). We can then use the formula again to find the period. g' = 9.81 m/s² - 2.10 m/s² = 7.71 m/s² T = 2π × √(1.00 m / 7.71 m/s²) ≈ 2.26 s

Calculate the period in an elevator in free fall

When the elevator is in free fall, the effective gravitational acceleration is 0 m/s² (since the acceleration due to gravity is canceled by the acceleration of the elevator). Since the denominator of the formula would be 0, the period of the pendulum would be undefined (meaning the pendulum would not oscillate back and forth). In summary, the period of the simple pendulum is: a) \( T \approx 2.00 \mathrm{~s}\) in the physics lab b) \( T \approx 1.82 \mathrm{~s}\) in an elevator accelerating upward c) \( T \approx 2.26 \mathrm{~s}\) in an elevator accelerating downward d) Undefined in an elevator in free fall

Key concepts

Gravitational Acceleration

Understanding gravitational acceleration is essential when studying the motion of a simple pendulum. Gravitational acceleration, denoted as \( g \), is the acceleration imparted to objects due to the force of gravity exerted by a massive body, such as Earth. On the surface of the Earth, this value is approximately \( 9.81 \text{m/s}^2 \), though it can vary slightly depending on location.

When dealing with pendulum motion, gravitational acceleration is a crucial part of the formula to calculate the period, which is the duration of one complete cycle of movement. The formula for the period \( T \) of a simple pendulum is \( T = 2\text{π} \times \text{√}\frac{l}{g} \), where \( l \) represents the length of the pendulum. As gravitational acceleration increases, the period decreases, meaning the pendulum swings more quickly. If \( g \) decreases, the period increases and the pendulum swings more slowly.

Thus, gravitational acceleration sets the pace at which the pendulum oscillates when other factors remain constant. Having a strong grasp on how \( g \) influences the pendulum's period is fundamental to understanding pendulum dynamics in different scenarios.

Effective Gravitational Acceleration

In scenarios where the gravitational conditions are altered, such as inside an accelerating elevator, we introduce the concept of effective gravitational acceleration, denoted by \( g' \). It expresses the combined effects of Earth's gravity and other accelerations acting upon a system.

For example, when a simple pendulum is in an elevator accelerating upward, the acceleration of the elevator \( a \) adds to Earth’s gravitational pull, yielding \( g' = g + a \). Conversely, if the elevator accelerates downward, the effective gravitational acceleration becomes \( g' = g - a \), since the elevator's acceleration works against Earth's gravity. It's important to note that these conditions alter how the pendulum swings, affecting the period accordingly.

Interestingly, if an elevator were in free fall, the acceleration would negate gravity’s effect completely, leading to \( g' = 0 \), and the pendulum would experience weightlessness, resulting in no oscillation. This effectively demonstrates the significance of effective gravitational acceleration in altered environments and is a critical concept for students to understand when considering the motion of pendulums in non-standard conditions.

Pendulum Oscillation

Pendulum oscillation refers to the regular, repetitive movement of a pendulum swinging back and forth under the influence of gravity. Each swing from its starting position to the opposite side and back is considered one complete oscillation. A key feature of pendulum motion is that it follows a regular time period, if not acted upon by external forces.

The time it takes to complete one full oscillation is referred to as the period, which we’ve seen can be calculated using the length of the pendulum and the gravitational acceleration acting upon it. Factors that affect pendulum oscillation include the length of the pendulum and the strength of gravity, but not the mass of the pendulum or the amplitude of the swing, as long as the amplitudes are small. This assumption is recognized in the context of the simple harmonic motion that pendulums exhibit, where the restoring force is directly proportional to the displacement.

Understanding the intricacies of pendulum oscillation not only deepens knowledge in physics regarding periodic motion but also paves the way for comprehending more complex systems influenced by a range of forces and accelerations.