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

priya , 11 Years ago
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Askiitians Tutor Team

Last Activity: 21 Days ago

To determine the minimum speed of a particle at the horizontal position of a vertical circle that allows it to complete the full circular motion, we need to consider the forces acting on the particle at the top of the circle. This scenario is a classic problem in physics, particularly in the study of circular motion and dynamics.

Understanding Forces in Circular Motion

When the particle is at the top of the vertical circle, two main forces act on it: the gravitational force (weight) and the tension in the string. For the particle to maintain its circular path, it must have sufficient centripetal force directed towards the center of the circle.

Key Concepts

  • Centripetal Force: This is the net force required to keep an object moving in a circular path, directed towards the center of the circle.
  • Gravitational Force: The weight of the particle, which acts downwards and is equal to the mass of the particle multiplied by the acceleration due to gravity (mg).
  • Tension in the String: The force exerted by the string on the particle, which can vary depending on the position of the particle in the circle.

Calculating Minimum Speed

At the top of the circle, the centripetal force required to keep the particle moving in a circle is provided by the combination of the gravitational force and the tension in the string. The equation for centripetal force can be expressed as:

F_c = m * v^2 / r

Where:

  • F_c: Centripetal force
  • m: Mass of the particle
  • v: Speed of the particle
  • r: Radius of the circle (length of the string)

At the top of the circle, the centripetal force is provided by the weight of the particle and the tension in the string:

F_c = mg + T

For the minimum speed at the top of the circle, we can assume that the tension in the string (T) is zero (this is the critical condition where the particle is just about to lose contact with the string). Thus, we can set up the equation:

m * v^2 / r = mg

Solving for Minimum Speed

Now, we can simplify this equation:

Canceling the mass (m) from both sides (assuming m is not zero), we get:

v^2 / r = g

Rearranging gives us:

v^2 = g * r

Taking the square root of both sides, we find:

v = √(g * r)

Final Thoughts

In this equation, g represents the acceleration due to gravity (approximately 9.81 m/s²), and r is the length of the string. Therefore, the minimum speed of the particle when the string is horizontal, which allows it to complete the circle, is given by the formula:

v = √(g * L)

where L is the length of the string. This result highlights the relationship between gravitational force, circular motion, and the critical conditions necessary for maintaining that motion. Understanding these principles is essential in physics, especially in mechanics.

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