All surfaces are smooth. Initially the spring is held unstretched and then the system is released from rest. Find the angular frequency of oscillations of the small block.

To determine the angular frequency of oscillations for a small block attached to a spring system, we need to analyze the forces acting on the block and how they relate to the spring's properties. Let's break this down step by step.

Understanding the System

Imagine a block resting on a frictionless surface, attached to a spring. When the spring is unstretched, the block is at its equilibrium position. Once the system is released, the block will start to oscillate due to the restoring force exerted by the spring.

Key Concepts

• Hooke's Law: The force exerted by a spring is proportional to its displacement from the equilibrium position. Mathematically, this is expressed as F = -kx, where k is the spring constant and x is the displacement.
• Mass of the Block: Let’s denote the mass of the block as m.
• Angular Frequency: The angular frequency (ω) of a mass-spring system is given by the formula: ω = √(k/m).

Deriving the Angular Frequency

When the block is displaced from its equilibrium position and released, it experiences a restoring force that pulls it back toward the equilibrium. This force causes the block to accelerate, leading to oscillatory motion. To find the angular frequency, we can use the relationship derived from Newton's second law and Hooke's law.

Step-by-Step Calculation

1. Start with Hooke's Law: F = -kx.
2. According to Newton's second law, F = ma, where a is the acceleration of the block. Thus, we can set these equal: ma = -kx.
3. Rearranging gives us a = -(k/m)x.
4. This equation resembles the standard form of simple harmonic motion, a = -ω²x, where ω² = k/m.
5. From this, we can derive the angular frequency: ω = √(k/m).

Example Calculation

Let’s say the spring constant k is 200 N/m and the mass m of the block is 2 kg. Plugging these values into our formula:

ω = √(k/m) = √(200 N/m / 2 kg) = √(100 s⁻²) = 10 rad/s.

Conclusion

The angular frequency of oscillations for the small block in this system is 10 rad/s. This means that the block will oscillate back and forth around the equilibrium position at this frequency, demonstrating the principles of harmonic motion. Understanding these concepts is crucial for analyzing various physical systems involving springs and oscillations.