A spring is standing upright on a table with its bottom end fastened to the table. A block is dropped from a height 3.0 cm above the top of the spring. The block sticks to the top end of the spring and then oscillates with an amplitude of 10 cm. What is the oscillation frequency?

Verified step by step guidance

1. Identify the key parameters of the problem: The amplitude of oscillation is 10 cm (0.1 m), and the block is dropped from a height of 3.0 cm (0.03 m) above the spring. The goal is to find the oscillation frequency, which depends on the spring constant and the mass of the block.
2. Use the relationship between the angular frequency \( \omega \) and the oscillation frequency \( f \): \( \omega = 2 \pi f \). The angular frequency is also related to the spring constant \( k \) and the mass \( m \) by \( \omega = \sqrt{\frac{k}{m}} \). Rearrange this to express \( f \): \( f = \frac{1}{2 \pi} \sqrt{\frac{k}{m}} \).
3. Determine the spring constant \( k \) using energy conservation. The block's potential energy when dropped (\( m g h \)) is converted into the spring's elastic potential energy (\( \frac{1}{2} k A^2 \)) at maximum compression. Set \( m g h = \frac{1}{2} k A^2 \) and solve for \( k \): \( k = \frac{2 m g h}{A^2} \).
4. Substitute \( k \) into the frequency formula. Replace \( k \) with \( \frac{2 m g h}{A^2} \) in \( f = \frac{1}{2 \pi} \sqrt{\frac{k}{m}} \). This gives \( f = \frac{1}{2 \pi} \sqrt{\frac{2 g h}{A^2}} \).
5. Substitute the known values: \( g = 9.8 \ \text{m/s}^2 \), \( h = 0.03 \ \text{m} \), and \( A = 0.1 \ \text{m} \). Simplify the expression to find the oscillation frequency \( f \).

Key Concepts

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 block 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. The motion is characterized by a constant frequency and amplitude.

Spring Constant (k)

The spring constant, denoted as 'k', is a measure of a spring's stiffness. It is defined by Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement from the equilibrium position (F = -kx). The value of 'k' is crucial for determining the frequency of oscillation in a mass-spring system.

Frequency of Oscillation

The frequency of oscillation refers to how many complete cycles of motion occur in a unit of time, typically measured in Hertz (Hz). For a mass-spring system undergoing SHM, the frequency can be calculated using the formula f = (1/2π)√(k/m), where 'm' is the mass of the block and 'k' is the spring constant. This relationship shows how the mass and spring constant influence the oscillation rate.