At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.12 m/s. Determine the period and frequency of the motion.

검증된 단계별 안내

Convert the mass of the object from grams to kilograms, as SI units require mass in kilograms. Use the conversion: \( m = 885 \; \text{g} = 0.885 \; \text{kg} \).

Recall the formula for the angular frequency of a spring-mass system: \( \omega = \sqrt{\frac{k}{m}} \), where \( k \) is the spring constant and \( m \) is the mass. Substitute \( k = 184 \; \text{N/m} \) and \( m = 0.885 \; \text{kg} \) into the formula.

Use the relationship between angular frequency and the period of motion: \( T = \frac{2\pi}{\omega} \). Substitute the value of \( \omega \) obtained in the previous step to calculate the period \( T \).

Determine the frequency of the motion using the relationship \( f = \frac{1}{T} \), where \( T \) is the period. Substitute the value of \( T \) to find \( f \).

Summarize the results: The period \( T \) represents the time it takes for one complete oscillation, and the frequency \( f \) represents the number of oscillations per second.

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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. In this case, the mass attached to the spring will move back and forth due to the restoring force exerted by the spring, which is proportional to the displacement from the equilibrium position. Understanding SHM is crucial for analyzing the motion of the mass-spring system.

Period and Frequency

The period of a motion is the time it takes to complete one full cycle, while frequency is the number of cycles per unit time, typically measured in Hertz (Hz). For a mass-spring system undergoing SHM, the period (T) can be calculated using the formula T = 2π√(m/k), where m is the mass and k is the spring constant. Frequency (f) is the reciprocal of the period, f = 1/T.

Spring Constant (k)

The spring constant (k) is a measure of a spring's stiffness, defined as the force required to compress or extend the spring by a unit distance. In this problem, the spring constant is given as 184 N/m, indicating how much force is needed to stretch or compress the spring. This value is essential for calculating the period of oscillation in the mass-spring system.