A SPHERE HAS A CHAIN TIED AT THE TOP OF IT .ITS LENGTH L IS L

GIVE FULL ANSWER WITH A FIGURE EXPLAINING ALLL THE BASICS HAPPENING

To analyze the motion of a sphere with a chain tied at its top, we need to consider the forces acting on the sphere once the chain is released. The scenario involves gravitational force, tension in the chain, and the motion of the sphere itself. Let’s break this down step by step, including a visual representation to clarify the concepts involved.

Understanding the Setup

Imagine a sphere suspended by a chain from its top point. The length of the chain is given as \( L < \frac{\pi r}{2} \), where \( r \) is the radius of the sphere. When the chain is released, the sphere will begin to fall due to gravity. The key forces at play are:

• Gravitational Force (Weight): This acts downward and is equal to \( mg \), where \( m \) is the mass of the sphere and \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).
• Tension in the Chain: This force acts upward and opposes the weight of the sphere.

Free Body Diagram

To visualize the forces, consider the following diagram:

Analyzing the Motion

When the chain is released, the sphere will start to accelerate downwards. The net force acting on the sphere can be expressed as:

Net Force (F_net) = Weight - Tension (T)

According to Newton's second law, this net force is also equal to the mass of the sphere multiplied by its acceleration (a):

F_net = ma

Setting Up the Equation

Combining these two equations gives us:

mg - T = ma

Rearranging this, we find:

T = mg - ma

Now, if we assume that the chain is taut and the sphere is in free fall, the tension will be zero at the moment of release. Thus, we can simplify our equation to:

0 = mg - ma

Finding the Acceleration

From the equation above, we can derive the acceleration of the sphere:

ma = mg

Dividing both sides by \( m \) (assuming \( m \neq 0 \)), we get:

a = g

This means that the sphere accelerates downward at the same rate as free fall, which is \( 9.81 \, \text{m/s}^2 \).

Conclusion

In summary, when the chain is released, the sphere accelerates downward with an acceleration equal to \( g \). The tension in the chain does not affect the initial acceleration since it is zero at the moment of release. This scenario illustrates the principles of forces and motion effectively, demonstrating how gravity influences the motion of objects.