A 200 g block hangs from a spring with spring constant 10 N/m. At t = 0 s the block is 20 cm below the equilibrium point and moving upward with a speed of 100 cm/s. What are the block's a. Oscillation frequency?

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Convert the given quantities into SI units: The mass of the block is 200 g, which is equivalent to 0.2 kg. The spring constant is already in SI units (10 N/m). The displacement from equilibrium is 20 cm, which is 0.2 m, and the speed is 100 cm/s, which is 1 m/s.

Recall the formula for the angular frequency of a mass-spring system: \( \omega = \sqrt{\frac{k}{m}} \), where \( k \) is the spring constant and \( m \) is the mass of the block. Substitute the values of \( k = 10 \ \text{N/m} \) and \( m = 0.2 \ \text{kg} \) into the formula.

Calculate the oscillation frequency \( f \) using the relationship between angular frequency and frequency: \( f = \frac{\omega}{2\pi} \). Use the value of \( \omega \) obtained in the previous step.

Interpret the result: The oscillation frequency \( f \) represents the number of complete oscillations the block makes per second. Ensure the units of the final answer are in hertz (Hz).

Verify the assumptions: Confirm that the system is ideal (no damping or external forces) and that the spring follows Hooke's law, as these are necessary conditions for the formulas used to be valid.

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주요 개념

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Spring Constant

The spring constant, denoted as 'k', is a measure of a spring's stiffness. It quantifies the force required to stretch or compress the spring by a unit distance. In this case, a spring constant of 10 N/m means that a force of 10 Newtons is needed to stretch the spring by 1 meter. This property is crucial for understanding the dynamics of oscillatory motion.

Oscillation Frequency

Oscillation frequency refers to the number of complete cycles of motion that occur in a unit of time, typically measured in Hertz (Hz). For a mass-spring system, the frequency can be calculated using the formula f = (1/2π)√(k/m), where 'k' is the spring constant and 'm' is the mass. This concept is essential for determining how quickly the block will oscillate around its equilibrium position.

Equilibrium Position

The equilibrium position is the point at which the net force acting on the block is zero, meaning the spring is neither compressed nor stretched. In this scenario, the block's equilibrium position is where the gravitational force is balanced by the spring force. Understanding this position is vital for analyzing the motion of the block as it oscillates above and below this point.