A mass m is attached to a weightless string of length L, cross section S, and tensile strength ( the maximum tension it can withstand ) T . The mass is suddenly released from a point near the fixed end of the string. What condition should be on the value of Young's modulus Y so that the string does not break?

To determine the condition on the value of Young's modulus, we need to consider the maximum tension that can be exerted on the string without it breaking.

The force acting on the string is the weight of the mass attached to it, given by the formula F = mg, where m is the mass and g is the acceleration due to gravity.

When the mass is released, it will start to move in a circular path due to the tension in the string, creating a centripetal force. The tension in the string is equal to the centripetal force required to keep the mass moving in a circle.

The centripetal force is given by the formula Fc = (mv^2) / r, where m is the mass, v is the velocity of the mass, and r is the radius of the circular path (equal to the length of the string, L).

Since the mass is released from rest, its initial velocity is zero. Therefore, the centripetal force required to keep the mass moving in a circle can be written as Fc = (m(0^2)) / L = 0.

Now, for the string not to break, the tension T in the string must be greater than or equal to the maximum force it can withstand. This can be expressed as T >= Fc.

Therefore, T >= 0.

Since we know that Young's modulus, Y, is defined as the ratio of stress to strain, we can use this information to determine the condition on Y. Stress is given by the formula stress = force / area, and strain is given by the formula strain = change in length / original length.

Since the string is weightless and has no cross-sectional area, we can conclude that the stress is zero, as there is no force acting on the string. Thus, to prevent the string from breaking, Young's modulus Y should be such that it allows for zero stress.

In other words, the value of Young's modulus Y can be any value, as long as it allows for zero stress in the string.