Mass on a Spring

Spring Energy

stored stretch becomes speed

A stretched spring stores energy that, when released, trades into kinetic energy, fastest as the block passes through the middle.

Example

A stretched spring stores energy that, when released, trades into kinetic energy, fastest as the block passes through the middle. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A stretched spring stores energy

A stretched or squashed spring stores energy, one half the stiffness times the displacement squared. At full stretch, 3 metres, that is one half times 2 times 9, or 9 joules.

PE=12 k x2=12 ⋅ 2 N/m ⋅ 9=9 JPE = \tfrac{1}{2}\,k\,x^{2} = \tfrac{1}{2} \,\cdot\, 2\ \text{N/m} \,\cdot\, 9 = \hl{9}\ \text{J}
At full stretch: all energy is in the springA block held at the far right with the spring fully stretched.m

Same spring, stretch changes energy by a square

Hold stiffness fixed. Stretch twice as far and the stored spring energy grows by four times, because the displacement is squared.

kxPE2 N/m1 m1 J2 N/m2 m4 J2 N/m3 m9 J\begin{array}{c|c|c}k & x & PE \\ \hline 2\ \text{N/m} & 1\ \text{m} & 1\ \text{J} \\ 2\ \text{N/m} & 2\ \text{m} & 4\ \text{J} \\ 2\ \text{N/m} & 3\ \text{m} & 9\ \text{J}\end{array}

Same stretch, stiffer spring stores more energy

Hold stretch fixed. A stiffer spring stores more energy at the same displacement.

kxPE2 N/m3 m9 J4 N/m3 m18 J6 N/m3 m27 J\begin{array}{c|c|c}k & x & PE \\ \hline 2\ \text{N/m} & 3\ \text{m} & 9\ \text{J} \\ 4\ \text{N/m} & 3\ \text{m} & 18\ \text{J} \\ 6\ \text{N/m} & 3\ \text{m} & 27\ \text{J}\end{array}

Released, the spring energy becomes motion

Let go and the spring energy turns into kinetic energy. At the middle the spring is relaxed, so all 9 joules are kinetic and the block is fastest. Setting one half times the mass times the speed squared equal to 9 gives a top speed of 3 metres per second. Partway out, at 2 metres, the energy is split: 4 joules still stored and 5 in motion, still adding to 9.

12 m v2=9  ⇒  v=3 m/s\tfrac{1}{2}\,m\,v^{2} = 9 \;\Rightarrow\; v = \hl{3}\ \text{m}/\text{s}
At the middle: all energy is motionThe block passing through the middle where the spring is relaxed, with a velocity arrow showing its top speed.mv

Spring energy plus kinetic energy stays fixed

Read three positions as an energy ledger. Spring energy falls, kinetic energy rises, and the total stays fixed.

PEKEE9 J0 J9 J4 J5 J9 J0 J9 J9 J\begin{array}{c|c|c}PE & KE & E \\ \hline 9\ \text{J} & 0\ \text{J} & 9\ \text{J} \\ 4\ \text{J} & 5\ \text{J} & 9\ \text{J} \\ 0\ \text{J} & 9\ \text{J} & 9\ \text{J}\end{array}
mechanics Storing 9 J at full stretch turns entirely into motion at the middle, a clean 3 m/s top speed.