To determine how the fundamental frequency of a plate changes when the load is doubled, we need to consider the relationship between the load applied and the frequency of vibration. The fundamental frequency of a vibrating plate is influenced by several factors, including the material properties, geometry, and the mass of the load applied to it.

Understanding the Relationship Between Load and Frequency

The fundamental frequency (f) of a plate can be expressed in a simplified form as:

• f ∝ √(k/m)

In this equation:

• f is the fundamental frequency.
• k is the stiffness of the plate, which remains constant for a given plate geometry and material.
• m is the mass of the load applied to the plate.

When a load is applied to the plate, it effectively increases the mass (m) that the plate has to support. As the mass increases, the fundamental frequency decreases because the plate becomes "heavier" and vibrates more slowly.

Calculating the New Frequency

Initially, we have a load of 12 kg resulting in a fundamental frequency of 500 Hz. When the load is doubled to 24 kg, we can analyze the effect on frequency:

• Let the initial mass (m1) = 12 kg and the initial frequency (f1) = 500 Hz.
• Let the new mass (m2) = 24 kg.

Using the relationship between frequency and mass, we can set up a ratio:

• f1/f2 = √(m2/m1)

Substituting the known values:

• 500 Hz / f2 = √(24 kg / 12 kg)
• 500 Hz / f2 = √(2)
• 500 Hz / f2 = 1.414

Now, solving for f2:

• f2 = 500 Hz / 1.414
• f2 ≈ 353.55 Hz

Final Thoughts

When the load on the plate is doubled from 12 kg to 24 kg, the fundamental frequency decreases to approximately 353.55 Hz. This illustrates how increasing the mass of the load affects the vibrational characteristics of the plate, leading to a lower frequency of vibration. Understanding this relationship is crucial in fields such as structural engineering and materials science, where the dynamic behavior of components under load is a key consideration.