A body of mass m at the end of string is whirled around in a vertical circle of radius R.Find the critical speed below which the string would become slack at the highest point.

To determine the critical speed below which the string would become slack at the highest point of a vertical circle, we need to analyze the forces acting on the mass as it moves through the circular path. This involves understanding the concepts of centripetal force and gravitational force.

Understanding Forces at the Highest Point

At the highest point of the vertical circle, the mass experiences two main forces: the gravitational force acting downward and the tension in the string acting upward. For the string to remain taut, the tension must be sufficient to provide the necessary centripetal force to keep the mass moving in a circular path.

Key Equations

Let's break down the forces involved:

• The gravitational force acting on the mass is given by F_gravity = mg, where g is the acceleration due to gravity.
• The centripetal force required to keep the mass moving in a circle is given by F_centripetal = \frac{mv^2}{R}, where v is the speed of the mass and R is the radius of the circle.

At the highest point, the tension in the string (T) and the gravitational force must provide the centripetal force required to keep the mass in circular motion. Therefore, we can express this relationship as:

Force Balance Equation

At the highest point, the equation can be set up as follows:

T + mg = \frac{mv^2}{R}

For the string to become slack, the tension (T) must be zero. Thus, we can simplify the equation:

0 + mg = \frac{mv^2}{R}

Deriving the Critical Speed

Now, we can solve for the critical speed (v) at which the string becomes slack:

Rearranging the equation gives:

mg = \frac{mv^2}{R}

We can cancel the mass (m) from both sides (assuming m is not zero), leading to:

g = \frac{v^2}{R}

Now, multiplying both sides by R results in:

v^2 = gR

Taking the square root of both sides, we find:

v = \sqrt{gR}

Conclusion

The critical speed below which the string would become slack at the highest point of the vertical circle is given by the formula:

v = \sqrt{gR}

In practical terms, this means that if the mass is moving slower than this critical speed, the tension in the string will not be enough to keep it taut, and the string will go slack. This concept is crucial in understanding circular motion and the forces involved in such systems.