Drive the formula for the potential energy of compressed /stretched spring
Answers
Answer:
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Explanation:
Elastic Potential Energy
Elastic potential energy is Potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring. It is equal to the work done to stretch the spring, which depends upon the spring constant k as well as the distance stretched.Spring potential energy
When you compress or stretch a spring, as soon as the stress is relieved, the spring attains its normal shape instantly. Its Elastic potential energy helps it do so. Generally, these elastic substances follow the Hooke’s law.Hooke’s Law
Before unveiling the mechanism of spring potential energy, we need to understand the Hook’s Law. According to this Law, the force needed to change the shape of spring is proportional to the displacement of the spring. The displacement referred here is how far the spring is compressed or stretched from its normal shape. Mathematically, Hook’s Law can be summarised as F= – k x.
Spring potential energy
To find the Spring potential energy, we need to use the Hooke’s law. Since the potential energy is equal to the work done by a spring and work, in turn, is the product of force and distance, we get our force from Hooke’s law. Distance here is the displacement in the position of the spring.
In the figure, x is the displacement from the equilibrium position. When we pull the spring to a displacement of x as shown in the figure, the work done by the spring is :
W = 0∫xm Fdx = -∫kx dx = -k(xm)2/2
The work done by pulling force Fp is :
Fp = k (xm)2 / 2
The work done by the pulling force Fp is in positive as it has overcome the force of spring. Therefore,
W= k xm2 / 2
When displacement is less than 0, the work done by the springs force is
Ws = – kxc 2 / 2
and the work done by the external force F is = + kxc2/2. In the process of displacement of the object from initial displacement xi to final displacement, xf the work done is,
Ws = – xf∫xi kx dx = k xi2/2 – k xf 2/2
From the equation, it is clear that the work done by the force of spring depends only on the endpoints of displacement. Also, we can see that in a cyclic process, the work done by the springs force is zero. Hence, we can say that the spring force is a conservative force because it depends on the initial and final positions only. Therefore, this work done is in the form of the Spring potential energy: If a spring extends by x on loading, then the energy stored by the spring is (Given T is the spring force and K is force constant):
A) 2x/T2 B) T2/2k C) 2k/T2 D) T2/2x
Solution: B) Force or tension, T = kx [Hooke’s Law]. Hence, x = T/k
Energy stored in the spring = k x2/2 = T2/2k.