Answer:
[tex](1/2)\, (k)\, (a^{2})[/tex], where [tex]k[/tex] denotes the spring constant.
Explanation:
Assume that the motion of this object is horizontal.
The total energy of this system is the sum of:
The energy of this system is converted between the two forms.
When the mass is at the center of motion, the spring is neither stretched or compressed. EPE of the spring would is zero, but KE of the mass is maximized and the mass is moving at maximum speed.
As the object moves away from the center of motion, the mass would slow down as it compresses or stretches the spring. When the displacement of the spring is maximized, EPE is maximized and KE of the mass becomes zero. The total energy of the system at that point is equal to the EPE of the spring.
By definition, the amplitude [tex]a[/tex] of a simple harmonic motion is the maximum possible displacement. In this example, when displacement is maximized, EPE is also maximized and is equal to total energy of this system. In other words, the total energy of this system would be equal to the EPE in the spring when the spring is stretched to a displacement of "[tex]a[/tex]".
When a spring of constant [tex]k[/tex] is displaced by [tex]x[/tex], the EPE in that spring is [tex](1/2)\, (k)\, (x^{2})[/tex]. When the spring in this question is stretched to a displacement of "[tex]a[/tex]", the EPE of this spring would be equal to [tex](1/2)\, (k)\, (a^{2})[/tex]. This value would be equal to the total energy in this system.