In FIGURE EX10.27, what is the maximum speed of a 2.0 g particle that oscillates between x = 2.0 mm and x = 8.0 mm

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Determine the amplitude of the oscillation. The amplitude is the maximum displacement from the equilibrium position. In this case, the particle oscillates between x = 2.0 mm and x = 8.0 mm, so the equilibrium position is the midpoint: \( x_{eq} = \frac{2.0 \text{ mm} + 8.0 \text{ mm}}{2} \). The amplitude \( A \) is then \( A = 8.0 \text{ mm} - x_{eq} \).

Recall the formula for the maximum speed of a particle in simple harmonic motion: \( v_{max} = \omega A \), where \( \omega \) is the angular frequency and \( A \) is the amplitude. To calculate \( v_{max} \), we need to find \( \omega \).

The angular frequency \( \omega \) is related to the mass \( m \) and the spring constant \( k \) by the formula \( \omega = \sqrt{\frac{k}{m}} \). If the spring constant \( k \) is not provided, it must be determined or assumed based on additional context.

Substitute the values of \( \omega \) and \( A \) into the formula \( v_{max} = \omega A \). Ensure that the amplitude \( A \) is converted to meters (SI units) before performing the calculation.

Simplify the expression to find the maximum speed \( v_{max} \). Remember to keep the units consistent throughout the calculation.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. The motion is characterized by a restoring force proportional to the displacement from the equilibrium, leading to sinusoidal motion. In this context, the particle's oscillation between two positions indicates it is undergoing SHM.

Maximum Speed in SHM

In Simple Harmonic Motion, the maximum speed of an oscillating particle occurs as it passes through the equilibrium position. This speed can be calculated using the formula v_max = ωA, where ω is the angular frequency and A is the amplitude of the motion. Understanding this relationship is crucial for determining the particle's maximum speed.

Amplitude

Amplitude in the context of oscillation refers to the maximum displacement of the particle from its equilibrium position. It is half the distance between the maximum and minimum positions of the oscillation. For the given problem, the amplitude can be calculated as half the difference between the two positions (8.0 mm and 2.0 mm), which is essential for finding the maximum speed.