Oscillations of a spring

A spring hung from a hook

Hang a toy on a spring. The spring stretches by some length $d$ and the toy comes to rest. Now pull the toy down a little and let go. It bobs up and down. Is this still simple harmonic motion? Does gravity change the period?

At rest, the forces balance. The spring pulls up with $kd$ and gravity pulls down with $mg$:

Now pull the toy down a further distance $y$ below the rest point. The spring is stretched by $d + y$, so it pulls up with $k(d + y)$. Gravity still pulls down with $mg$. Take up as positive and $y$ as the distance below the rest point. The net upward force is

This upward force is $ky$, toward the rest point. So the net force is $-ky$, measured from the rest point, just like a spring on a smooth floor. Gravity only moves the middle of the motion down. It does not change the period:

From $kd = mg$ we also get $\dfrac{m}{k} = \dfrac{d}{g}$. So

You only need the stretch to know the period. A toy that stretches its spring by $4.0\ \text{cm}$ bobs with $T = 2\pi\sqrt{\dfrac{0.040\ \text{m}}{9.8\ \text{m/s}^2}} = 0.40\ \text{s}$.

Make the block heavier. The rest point moves down, and the period grows. Do the two readouts of the period always agree?

Two springs

A car rests on four springs. A bed rests on many. A spring balance may use two springs. How do springs work together?

Side by side, in parallel. Both springs stretch by the same $x$. Their forces add: $k_1x + k_2x = (k_1 + k_2)x$. So the pair acts like one spring with

End to end, in series. The same force $F$ passes through both springs. Their stretches add: $\dfrac{F}{k_1} + \dfrac{F}{k_2}$. So the pair acts like one spring with

A block between two walls, with a spring on each side. Move the block by $x$. One spring stretches by $x$ and the other is squeezed by $x$. Both push the block back the same way. So this is parallel too: $k = k_1 + k_2$.

Two equal springs of $40\ \text{N/m}$ give $80\ \text{N/m}$ in parallel and $20\ \text{N/m}$ in series. Parallel springs are stiffer than either one. Series springs are softer than either one.

Cutting a spring

Cut a spring in half. Pull one half with a force $F$. Only half as many coils stretch, so it stretches half as much. Its spring constant is twice as large.

In general, the spring constant times the length stays the same. A piece that is a fraction of the length has the spring constant divided by that fraction. A third of a spring of $k$ has $3k$.

Two bodies on one spring

Two carts on a smooth track are joined by a spring. Pull them apart and let go. Both carts move, toward each other and away again. The middle point between them, weighted by mass, stays still. The period is

Here $\mu$ is the reduced mass. Two carts of $2.0\ \text{kg}$ each have $\mu = 1.0\ \text{kg}$. If one cart is fixed to a wall, $\mu$ becomes the mass of the other cart, as before.

Practise with new numbers

Try it in Python

The program finds the spring constant and the period of one block on two springs, taken alone, in series and in parallel.

Common mistakes

• Thinking gravity changes the period of a hanging spring. It only moves the rest point. The period is still $2\pi\sqrt{m/k}$.
• Mixing up the two rules. Side by side: add the spring constants. End to end: add the inverses.
• Calling a block between two walls a series pair. Both springs push it back the same way, so they are in parallel.
• Thinking a cut spring is softer. A shorter piece is stiffer.

Remember these

• Hanging spring: $kd = mg$ at rest, and $T = 2\pi\sqrt{m/k} = 2\pi\sqrt{d/g}$.
• Parallel: $k = k_1 + k_2$. Series: $\dfrac{1}{k} = \dfrac{1}{k_1} + \dfrac{1}{k_2}$.
• A piece of a spring has a larger spring constant. $k \times$ length stays the same.
• Two bodies on one spring: use the reduced mass $\mu = \dfrac{m_1m_2}{m_1 + m_2}$.

Check yourself

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

1. A load stretches a spring by $9.8\ \text{cm}$ at rest. What is the period of its small oscillations?

2. Two springs, each $50\ \text{N/m}$, hold a block. What is the spring constant of the pair in parallel, and in series?

3. A spring of $k$ is cut into four equal pieces. What is the spring constant of one piece?

4. A block on a spring is taken to the Moon and hung there. Its spring stretches less. Does the period change?