pls help. A massless pulley is suspended by a massless spring and string. a block of mass m is connected to the pulley as shown. find the frequency of the block

To determine the frequency of a block connected to a massless pulley suspended by a massless spring and string, we need to analyze the system's dynamics. This involves understanding how the spring's properties and the mass of the block interact to create oscillatory motion. Let's break this down step by step.

Understanding the System

In this scenario, we have a block of mass \( m \) attached to a pulley, which is in turn suspended by a spring. When the block is displaced from its equilibrium position and released, it will oscillate due to the restoring force provided by the spring. The key to finding the frequency of this oscillation lies in the spring's properties and the mass of the block.

Spring Force and Hooke's Law

The force exerted by a spring can be described by Hooke's Law, which states:

• F = -kx

Here, \( F \) is the restoring force, \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position. The negative sign indicates that the force exerted by the spring opposes the displacement.

Equation of Motion

When the block is displaced by a distance \( x \), the net force acting on the block can be expressed as:

• F_{net} = -kx = m \frac{d^2x}{dt^2}

This equation represents a simple harmonic motion (SHM) where the acceleration of the block is proportional to its displacement and directed towards the equilibrium position.

Finding the Frequency

The general solution for the frequency of a mass-spring system undergoing SHM is given by:

• f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}

In this formula, \( f \) represents the frequency, \( k \) is the spring constant, and \( m \) is the mass of the block. The frequency indicates how many oscillations occur per second.

Example Calculation

Let's say the spring constant \( k \) is 200 N/m and the mass \( m \) is 2 kg. Plugging these values into the frequency formula gives:

• f = \frac{1}{2\pi} \sqrt{\frac{200}{2}} = \frac{1}{2\pi} \sqrt{100} = \frac{10}{2\pi} \approx 1.59 \text{ Hz}

This means the block will oscillate with a frequency of approximately 1.59 Hz.

Conclusion

In summary, to find the frequency of a block connected to a massless pulley and spring system, you apply the principles of simple harmonic motion, using Hooke's Law and the relationship between mass and spring constant. By substituting the appropriate values into the frequency formula, you can determine how quickly the block oscillates. This understanding is crucial in various applications, from engineering to physics, where oscillatory systems are common.