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 elevator 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

Question: Calculate the period of a simple pendulum with a length of 1 meter in the following situations: (a) in a physics lab, (b) in an elevator accelerating upward at 2.10 m/s², (c) in an elevator accelerating downward at 2.10 m/s², and (d) in an elevator in free fall. Answer: (a) The period of the pendulum in the physics lab is 2.006 s. (b) The period of the pendulum in an elevator accelerating upward is 1.795 s. (c) The period of the pendulum in an elevator accelerating downward is 2.282 s. (d) The period of the pendulum in an elevator in free fall cannot be calculated as it approaches infinity.

Step-by-step solution

Define the knowns and given values

Length of the pendulum, \(l = 1.00 \mathrm{~m}\). Acceleration due to gravity (only considering Earth), \(g = 9.81 \mathrm{~m/s^2}\) Period formula, \(T= 2\pi\sqrt{\frac{l}{g'}}\).

Calculate the period of the pendulum in the physics lab

In the physics lab, no extra forces act on the pendulum other than Earth's gravitational pull, so \(g'=g\). Therefore, the period \(T\) can be calculated using the formula, \(T = 2\pi\sqrt{\frac{l}{g}}\) Plug in the known values and solve for \(T\). \(T = 2\pi\sqrt{\frac{1.00}{9.81}}\) \(T = 2.006 \mathrm{~s}\) (rounded to three decimal places).

Calculate the period of the pendulum in the elevator accelerating upward

When the elevator is accelerating upward, the effective gravitational acceleration experienced by the pendulum is \(g' = g + a\), where \(a\) is the acceleration of the elevator. In this case, \(a = 2.10 \mathrm{~m/s^2}\) Now, we can find the period using the formula, \(T = 2\pi\sqrt{\frac{l}{g'}}\) \(T = 2\pi\sqrt{\frac{1.00}{9.81+2.10}}\) \(T = 1.795 \mathrm{~s}\) (rounded to three decimal places).

Calculate the period of the pendulum in the elevator accelerating downward

When the elevator is accelerating downward, the effective gravitational acceleration experienced by the pendulum is \(g' = g - a\), where \(a\) is the acceleration of the elevator. In this case, \(a = 2.10 \mathrm{~m/s^2}\) Now, we can find the period using the formula, \(T = 2\pi\sqrt{\frac{l}{g'}}\) \(T = 2\pi\sqrt{\frac{1.00}{9.81-2.10}}\) \(T = 2.282 \mathrm{~s}\) (rounded to three decimal places).

Calculate the period of the pendulum in an elevator in free fall

In free fall, the elevator and pendulum are both experiencing the same downward acceleration due to gravity, and the effective gravitational acceleration experienced by the pendulum becomes \(g'=0\). In this situation, the period of the pendulum approaches infinity because there is no effective force to restore the pendulum to its original position.

Key concepts

Period of Oscillation

The period of oscillation refers to the time it takes for a pendulum to complete one full swing, returning to its starting position. It is an essential characteristic of oscillatory motion. For a simple pendulum, which is an idealized pendulum with a weightless, stiff rod and a mass that acts as a point particle, the period is given by the formula:

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Effective Gravitational Acceleration

Effective gravitational acceleration (dynamic typing: dynamic typing: dynamic typing: ) is crucial for determining the period of a pendulum. It represents the total acceleration acting on the pendulum's mass due to gravity, taking into account any additional accelerative forces present, such as those in an accelerating system like an elevator. In a static environment, dynamic typing: dynamic typing: dynamic typing: would simply be the standard acceleration due to gravity, commonly approximated as dynamic typing: dynamic typing: dynamic typing: on Earth. However, in non-static systems, dynamic typing: dynamic typing: dynamic typing: must be adjusted to include these additional forces, resulting in a modified period for the pendulum.

For example, when a pendulum is in an elevator accelerating upwards, the effective gravitational acceleration experienced by the pendulum increases because the elevator’s acceleration adds to Earth's gravity. Conversely, if the elevator accelerates downward, the effective gravitational acceleration decreases as the elevator’s acceleration works against gravity. Understanding this concept is key when calculating the pendulum's period in different scenarios.

Pendulum in Accelerating Systems

A pendulum in an accelerating system behaves differently than one in a stationary or uniform motion system due to changes in effective gravitational acceleration. When a pendulum is in an elevator that accelerates, the force experienced by the pendulum changes because of the net acceleration of the system.

Imagine being in an elevator with a simple pendulum. If the elevator moves upwards or downwards with a constant acceleration, the pendulum experiences a modified gravity-like effect. This change causes the pendulum's period to adjust accordingly. In an upward accelerating elevator, gravity effectively becomes stronger, leading to a shorter period of oscillation. On the other hand, if the elevator accelerates downward, the effective gravity weakens, which results in a longer period of oscillation. These effects are critical when analyzing pendulum motion in elevators or other accelerating systems and have practical implications in engineering and science.

Pendulum in Free Fall

When discussing a pendulum in free fall, we are examining a scenario where the entire system is falling under the influence of gravity alone, with no other forces acting upon it. In this unique case, both the pendulum and the elevator experience the same acceleration due to gravity. Because of this, the pendulum doesn't 'feel' gravity in the usual sense—it does not experience a restorative force that would propel it back towards equilibrium. As a result, the effective gravitational acceleration is zero, and the concept of a period of oscillation loses its meaning because the pendulum does not oscillate.

Technically, the period of a pendulum in free fall approaches infinity, a concept that might seem counterintuitive. Practically, if you were to release a pendulum in a freely falling elevator, you would observe the pendulum remaining motionless relative to you, as both you and the pendulum are accelerating downward at the same rate due to gravity. This phenomenon has fascinating implications in understanding gravitational effects in different frames of reference and is a beautiful demonstration of the principles of relativity and equivalence.