In the situation as shown in figure time period of small vertical oscillation of block will be.

To determine the time period of small vertical oscillations of a block, we need to consider the principles of simple harmonic motion (SHM). When a block is attached to a spring and is displaced vertically, it can oscillate up and down. The time period of these oscillations depends on the mass of the block and the spring constant. Let's break this down step by step.

Understanding the System

Imagine a block of mass \( m \) attached to a spring with a spring constant \( k \). When the block is at rest, the spring is neither stretched nor compressed. If you pull the block down slightly and then release it, the block will start to oscillate vertically. This motion can be modeled as simple harmonic motion.

Key Variables

• Mass (m): The weight of the block affects how quickly it can accelerate.
• Spring Constant (k): This measures the stiffness of the spring. A stiffer spring (higher \( k \)) will result in faster oscillations.

Formula for Time Period

The time period \( T \) of a mass-spring system undergoing simple harmonic motion is given by the formula:

T = 2π√(m/k)

Breaking Down the Formula

In this formula:

• \( T \) is the time period of the oscillation.
• \( m \) is the mass of the block.
• \( k \) is the spring constant.

This equation shows that the time period increases with the mass of the block and decreases with a stiffer spring. Essentially, a heavier block takes longer to oscillate, while a stiffer spring allows for quicker oscillations.

Example Calculation

Let’s say we have a block with a mass of 2 kg attached to a spring with a spring constant of 50 N/m. We can plug these values into the formula:

T = 2π√(2 kg / 50 N/m)

Calculating this gives:

T = 2π√(0.04) = 2π(0.2) ≈ 1.26 seconds

Visualizing the Motion

Think of the oscillation like a swing. If you have a heavier person on the swing, it will take longer to complete a full swing back and forth compared to a lighter person. Similarly, a stiffer spring will allow the block to oscillate faster, just like a swing with shorter chains moves quicker.

Final Thoughts

In summary, the time period of small vertical oscillations of a block attached to a spring can be calculated using the mass of the block and the spring constant. This relationship is fundamental in understanding oscillatory motion in physics, and it can be applied to various real-world systems, from pendulums to mechanical watches. If you have any specific figures or additional parameters, feel free to share, and we can delve deeper into the calculations or concepts!