A particle is rotated in a vertical circle by connecting it to a light rod of length l and keeping the other end of the rod fixed. The minimum speed of the particle when the light rod is horizontal for which the particle will complete the circle is?

Question

Answer

Minimum Speed for Completing Vertical Circle

Understanding the Problem: A particle is attached to a light rod of length l and rotated in a vertical circle. We need to find the minimum speed of the particle when the rod is horizontal for it to complete the circle.

Applying Physics Concepts: To solve this problem, we need to apply the concept of centripetal force and gravitational force acting on the particle at different points of the circle.

Calculating Centripetal Force: When the particle is at the topmost point of the circle, the centripetal force acting on it is equal to the gravitational force acting downwards. Hence, we can equate the two forces:

mv^2/l = mg

Calculating Minimum Speed: Solving the above equation for v, we get:

v = sqrt(gl)

Final Answer: Therefore, the minimum speed of the particle for it to complete the circle when the rod is horizontal is sqrt(gl).

Explanation:

When a particle is rotated in a vertical circle, it experiences a change in direction and hence, a change in velocity. The centripetal force acting on the particle is provided by the tension in the rod. At the topmost point of the circle, the particle experiences maximum gravitational force acting downwards. Therefore, the minimum speed required for the particle to complete the circle is when the centripetal force is equal to the gravitational force acting downwards.

Using the formula for centripetal force, we can equate it to the gravitational force acting downwards. Solving the equation for v, we get the minimum speed required for the particle to complete the circle when the rod is horizontal.

It is important to note that any speed higher than the minimum speed will also allow the particle to complete the circle. However, the minimum speed is the threshold below which the particle will not be able to complete the circle.

Answer

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