Why the only top most position of vertical circle has been taken for speed of cane

When analyzing the motion of an object in a vertical circle, particularly in the context of a pendulum or a roller coaster, the topmost position is often of particular interest when discussing speed. This focus on the top position stems from the interplay of gravitational forces and centripetal acceleration. Let’s break this down step by step.

The Forces at Play

At the top of a vertical circle, two main forces act on the object: gravitational force and the tension (or normal force) from the string or track. The gravitational force pulls the object downward, while the tension acts in the opposite direction. At this point, the object must have enough speed to maintain its circular path, which is determined by the balance of these forces.

Centripetal Force Requirement

For an object to move in a circle, it requires a net inward force directed towards the center of the circle, known as centripetal force. At the top of the circle, the gravitational force contributes to this centripetal force. The equation for centripetal force is:

• F_c = m * v^2 / r

Where F_c is the centripetal force, m is the mass of the object, v is its speed, and r is the radius of the circle. At the top of the circle, the gravitational force (mg) must be sufficient to provide the required centripetal force:

• mg + T = m * v^2 / r

Here, T is the tension in the string or the normal force. If the object is just barely maintaining its circular motion at the top, the tension can be zero, leading to:

• mg = m * v^2 / r

Critical Speed at the Top

This simplifies to:

• v^2 = g * r

From this equation, we can derive the critical speed required at the top of the circle:

• v = √(g * r)

This indicates that the speed at the top of the vertical circle is directly related to the gravitational acceleration and the radius of the circle. If the speed is less than this critical value, the object will not be able to complete the circular motion and will fall away from the path.

Why Focus on the Top Position?

Focusing on the topmost position is crucial because it represents the minimum speed needed to maintain circular motion. If we were to analyze other points in the circle, the dynamics would change due to varying contributions of gravitational force and the required centripetal force. For instance, at the bottom of the circle, the gravitational force acts against the tension, requiring a higher speed to maintain the same circular path.

Practical Implications

This principle has real-world applications, such as in amusement park rides and roller coasters, where designers must ensure that the speed at the top of loops is sufficient to keep riders safely in their seats. Understanding this concept helps in designing safe and thrilling rides.

In summary, the topmost position of a vertical circle is critical for determining the minimum speed necessary for an object to maintain its circular path due to the balance of gravitational and centripetal forces. This principle is foundational in physics and has practical implications in various engineering fields.