Question

For a body in simple harmonic motion. Potential energy (PE), Kinetic energy (KE), Total energy (TE) are measured as a function of displacement x. Which of the following statements is true?

• A. KE is maximum when x = 0
• B. TE is zero when x = 0
• C. KE is maximum when x is maximum
• D. PE is maximum when x = 0

Hint:

At the center, speed is peak (KE max); at the ends, the spring is stretched (PE max).

Answer 1

The Correct Option is A

Step 1: How energy moves in SHM.
In simple harmonic motion the total energy stays fixed, but it keeps swapping between kinetic energy (KE) and potential energy (PE) as the body moves.

Step 2: Write the two energies.
If $A$ is the amplitude and $x$ the displacement, \[ KE = \tfrac{1}{2}k\left(A^2 - x^2\right), \qquad PE = \tfrac{1}{2}k x^2 \] Their sum is $\tfrac{1}{2}kA^2$, a constant.

Step 3: Look at the centre, $x = 0$.
Put $x = 0$ into the KE formula: $KE = \tfrac{1}{2}kA^2$, which is the largest it can ever be. At this same point $PE = 0$. So the body is fastest right at the middle.

Step 4: Look at the ends, $x = A$.
At the extreme positions the speed is zero, so $KE = 0$ there and $PE$ is at its maximum. This is the opposite of the centre.

Step 5: Check the statements.
"KE is maximum when $x = 0$" is exactly what we found, so it is true. The total energy is never zero, KE is not maximum at the ends, and PE is smallest (not largest) at the centre.

Step 6: Conclusion.
The correct statement is that KE is maximum when $x = 0$. \[ \boxed{KE \text{ is maximum when } x = 0} \]