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?
The effective spring constant of the string is
k = Y*S/L Newtons/meter
When the mass m is dropped from the fixed end, the string will stretch a maximum amount dL, so that
m g L = (1/2) k *dL^2 = (1/2)Y*S/L*dL^2
dL^2 = 2 m g L^2/(Y*S)
dL/L = sqrt[2 m g/(Y*S)]
If the string is about to break at maximum stretch, the tensile stress will then be
sigma = T = Y*(dL/L)
T = sqrt[2 m g Y/S]
Solve for Y. A larger value of Y will cause the string to break.
To determine the condition on the value of Young's modulus (Y) so that the string does not break, we need to consider the maximum tension (T) that the string can withstand.
Young's modulus (Y) is a measure of the stiffness of a material. It relates stress (force per unit area) to the corresponding strain (the change in dimensions due to the applied force).
The tension force in the string is given by T = m*g, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2). This is the maximum tension force the string can experience before breaking.
The stress experienced by the string can be calculated using the formula: Stress = Force / Area. In this case, the force is T and the area is the cross-sectional area of the string (S).
Therefore, the stress on the string is Stress = T / S.
To ensure that the string does not break, the stress on the string should be less than or equal to the tensile strength (T) of the string: Stress <= T.
Substituting the values, we have T/S <= T.
Simplifying the inequality, we get 1/S <= 1.
Since 1 is less than or equal to 1, we can conclude that the condition for the value of Young's modulus (Y) is that it should be greater than or equal to infinity (Y >= ∞) to prevent the string from breaking.
Essentially, any value of Young's modulus greater than or equal to infinity will satisfy the condition, as the string will be able to withstand any amount of tension without breaking.