Question

The stress induced in a body, when suddenly loaded, is when the same load is applied gradually.
A. Twice
B. Equals To
C. One-half
D. Four Times

Answer 1

Hint: We can start by taking the two as two cases. Initially, in the first case, we take the equation of force applied on a spring. We obtain the value of extension. In the second case we take the work – energy equation, energy stored in the bar = work done by force. This helps us establish a relation between the extension in the first and second case.

Complete step by step answer:
The stress induced in a body depends on the amount of time in which the force is being applied to the body. If the force is applied to the body very fast then the stress will also increase as the time taken is inversely proportional to the stress applied on the body. It is calculated that if the force is applied very slowly then it takes half the stress then it will take when the force is applied on the body very fast.

When the load is loaded gradually: When the maximum load P is attained, the internal force will be in equilibrium with the external force
\[F = kx = P \\
\Rightarrow x = \dfrac{P}{k}\]
When the force is applied suddenly: As the load is applied suddenly internal force in the bar will not be in equilibrium with applied force. Thus, because of the net force, there will be some acceleration so the particles and hence the entire bar will undergo accelerated extension. The internal force in the bar will increase up to the applied load P with extension.

For extending beyond the point $x$, the internal force will be more than P and hence the particles will undergo retarded motion and ultimately will come to a stop. The maximum extension as mentioned can be obtained using the work-energy equation, energy stored in the bar is equal to the work done by force. As soon as the bar stops moving (extending) all the stored energy will be in the form of strain energy.
\[\dfrac{1}{2}k{\left( {x'} \right)^2} = Px'\]
We can cancel the values and maintain \[x' = \dfrac{{2P}}{k}\]
From the first case, we can see that \[x = \dfrac{P}{k}\]
That means, \[x' = 2x\]
We know that \[\text{stress = Energy} \times \text{strain}\]
And the formula of strain is \[\text{strain} = \dfrac{\text{change in length}}{\text{original length}}\]
This says that stress and length are dependent linearly. This proves that the maximum stress induced in a body due to suddenly applied load is twice the stress induced when the same load is applied gradually.

Therefore, the correct answer is A.

Note: When load is applied gradually it is assumed that equilibrium between internal force and applied force is maintained all the time till we reach the final value of applied load and corresponding deformation. Whereas when a sudden load is applied the equilibrium is not maintained except at one point during deformation.