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.

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

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
mechanics Storing 9 J at full stretch turns entirely into motion at the middle, a clean 3 m/s top speed.