Question

How much energy can be stored in a spring with $k = 450N{m^{ - 1}}$ if the maximum stretch is 18cm?

Answer 1

Hint: The simple harmonic motion is defined as a special type of periodic motion where the restoring force on the moving object is directly proportional, at every instant, to the displacement of the body.
This restoring force, F is given by –
$F \propto - x$
$ \Rightarrow F = - kx$
where k = constant.

Complete answer:
When there is a displacement in a spring-mass system, it follows a simple harmonic motion. Here is a brief understanding of the simple harmonic motion.
In simple harmonic motion, the restoring force on the moving object is directly proportional, at every instant, to the displacement of the body.
This restoring force, F –
$F = - kx$
Work-energy theorem gives us the relationship between the quantities work and energy. It states that – net work done by forces acting on an object is equal to the increase in the kinetic energy of the body.
Applying the Work-Energy theorem for springs, the work done on a spring is equal to the potential energy stored in the spring since increase in the kinetic energy would only, mean the decrease in the potential energy of the spring.
Work done is equal to the product of force and displacement. Since, force is applied over every instance of the displacement, the work done is the integral of force over the displacement.
$\Rightarrow W = \int {F.dx} $
Substituting the force,
$\Rightarrow W = \int { - kx.dx} $
$ \Rightarrow W = - k\int {x.dx} $
$ \Rightarrow W = - k\dfrac{{{x^2}}}{2}$
Thus, the energy stored in the spring is equal to –
$\Rightarrow E = \dfrac{1}{2}k{x^2}$
Given –
Spring constant, $k = 450N{m^{ - 1}}$
Displacement, $x = 18cm = 0.18m$
Substituting the values,
$\Rightarrow E = \dfrac{1}{2} \times 450 \times 0.18 = \dfrac{{81}}{2} = 40.5J$

Therefore, the energy stored in the spring is equal to 40.5 J.

Note: In this question, the students can see here that the work has negative signs but the energy has positive signs. Actually, the energy should be negative because when the spring is stretched, there is a decrease in the energy stored in the object. Since, we are only considering the magnitude of energy, it is taken as positive here.