A 2kg stone is swung in a vertical circle by attaching it at the end of a string of length 2m. If the string can withstand a tension of 140.6N,the maximum speed with which the stone can be rotated is

To determine the maximum speed at which the stone can be rotated in a vertical circle, we need to consider the forces acting on the stone at the lowest point of its circular path. At this point, the tension in the string and the gravitational force both contribute to the centripetal force required to keep the stone moving in a circle.

Understanding the Forces Involved

When the stone is at the bottom of the swing, two main forces act on it:

• Tension (T): The force exerted by the string on the stone, which acts upward.
• Weight (W): The gravitational force acting on the stone, calculated as W = mg, where m is the mass and g is the acceleration due to gravity (approximately 9.81 m/s²).

Calculating the Weight of the Stone

First, we need to calculate the weight of the stone:

Given:

• Mass (m) = 2 kg
• Acceleration due to gravity (g) = 9.81 m/s²

Weight (W) = m × g = 2 kg × 9.81 m/s² = 19.62 N

Setting Up the Equation for Centripetal Force

At the lowest point of the swing, the net force providing the centripetal acceleration (Fc) is the difference between the tension in the string and the weight of the stone:

Fc = T - W

We know that the centripetal force can also be expressed as:

Fc = (m × v²) / r

Where:

• v = speed of the stone
• r = radius of the circle (length of the string) = 2 m

Combining the Equations

Now we can set the two expressions for centripetal force equal to each other:

T - W = (m × v²) / r

Substituting the known values:

140.6 N - 19.62 N = (2 kg × v²) / 2 m

Now simplify the equation:

120.98 N = (2 kg × v²) / 2 m

120.98 N = kg × v²

Now, solve for v²:

v² = 120.98 N / 2 kg = 60.49 m²/s²

Taking the square root to find v:

v = √60.49 m²/s² ≈ 7.78 m/s

Final Result

The maximum speed with which the stone can be rotated in the vertical circle is approximately 7.78 m/s. This speed ensures that the tension in the string does not exceed its maximum limit while providing the necessary centripetal force to keep the stone moving in a circular path.