Energy in simple harmonic motion

Where does the energy go?

At the end of its swing the glider stops for an instant. Its kinetic energy is zero. Yet a moment later it is moving fast through the middle. Where did that energy come from? From the spring. A stretched spring stores energy, and gives it back as the glider speeds up.

The energy stored in a spring stretched or squeezed by $x$ is its potential energy:

The kinetic energy of the body is $K = \tfrac12 mv^2$. There is no friction, so their total stays the same. At the end, $x = A$ and $v = 0$. So the total energy is

The energy of an oscillator grows as the square of the amplitude. Twice the amplitude stores four times the energy.

The speed at any point

Energy gives the speed at any point without any time in the working:

We want $v$, so

At the middle, $x = 0$, the speed is largest: $v_{max} = \omega A$.

Try it. A block of $0.80\ \text{kg}$ on a spring of $80\ \text{N/m}$ swings with an amplitude of $8.0\ \text{cm}$. Then $\omega = 10\ \text{rad/s}$, $E = \tfrac12 \times 80\ \text{N/m} \times (0.080\ \text{m})^2 = 0.256\ \text{J}$, and $v_{max} = 10\ \text{rad/s} \times 0.080\ \text{m} = 0.80\ \text{m/s}$.

Watch the two dots. When one rises, the other falls by the same amount. Where do they meet?

Where are the two energies equal?

Set the potential energy to half the total:

So the kinetic energy and the potential energy are equal at about $71\%$ of the amplitude, not at half way. At $x = A/2$ the potential energy is only a quarter of the total, and the kinetic energy is three quarters.

Energy against time

Look at the right graph of the figure. In one period the body passes the middle twice and stops at an end twice. So the kinetic energy reaches its largest value twice in each period. Each energy repeats every $T/2$: it changes at twice the frequency of the motion.

Over a whole period, the kinetic energy spends as much time high as low. On average each energy is half of the total:

A spring that hangs

For a hanging spring, gravity also stores energy. Measure $y$ from the rest point, not from the natural length. Then the sum $\tfrac12 mv^2 + \tfrac12 ky^2$ stays the same, just as on a floor. The work of gravity is already taken care of by the shift of the rest point.

Practise with new numbers

Try it in Python

The program finds the kinetic and the potential energy at ten moments of one period. It adds them up, and it finds the average of each.

Common mistakes

• Forgetting the square. The energy is $\tfrac12 kA^2$. Twice the amplitude gives four times the energy.
• Putting the equal energies at $A/2$. They are equal at $A/\sqrt2$.
• Saying the energy changes at the frequency of the motion. Each energy repeats every half period, so at twice the frequency.
• Using $v = \omega(A - x)$. The right form is $v = \omega\sqrt{A^2 - x^2}$.

Remember these

• $U = \tfrac12 kx^2$, $K = \tfrac12 mv^2$ and $E = \tfrac12 kA^2 = \tfrac12 m\omega^2A^2$.
• $v = \omega\sqrt{A^2 - x^2}$ and $v_{max} = \omega A$.
• $K = U$ at $x = A/\sqrt2$.
• Each energy repeats every $T/2$ and averages to $E/2$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. The amplitude of an oscillator is tripled. What happens to its total energy?

2. An oscillator has an amplitude of $10\ \text{cm}$. Where is its potential energy a quarter of the total?

3. A body in simple harmonic motion has a period of $0.6\ \text{s}$. How often does its kinetic energy reach its largest value?

4. A block on a spring has a total energy of $0.40\ \text{J}$. What is its kinetic energy on average over a period?