The keys on a computer keyboard contain small springs that compress by 4.8 millimeters when a key is pressed. On a particular day, 12500 keystrokes are made on the keyboard. The springs in the keys each have a constant of 2.5 newtons per meter. How much energy was supplied to make all the keystrokes performed that day?
Okay, so we know that we’ve got a computer keyboard. And in that computer keyboard, the keys contain small springs that compress by 4.8 millimeters when a key is pressed. Now on a particular day, we’re told that 12500 keystrokes are made on the keyboard. In other words, whoever was typing on that keyboard has pressed keys 12500 times during that day.
Now we’re told that the springs in the keys each have a constant of 2.5 newtons per meter. So each key has a spring with a spring constant of 2.5 newtons per meter. And we’re asked to work out how much energy was supplied to make all the keystrokes perform that day.
Let’s start by drawing a diagram. Here is a side-on view of the keyboard. And here’s somebody’s hand coming to press one of the keys on the keyboard. So now that they have pressed the key, the key’s gone down by 4.8 millimeters. In other words, that’s how much the spring is compressed by. And this happens every time you press a key on the keyboard.
Now if we look inside the keyboard when we’ve pressed the key, we can see that there’s the compressed spring in there. And we’ve been told that the spring has a constant of 2.5 newtons per meter. And this is true for all of the springs in the keyboard. They’re all basically identical springs.
Now we can recall that the amount of energy stored in a spring, in other words, the amount of elastic potential energy in a spring, is given by half 𝑘𝑥 squared. 𝐸 represents the elastic potential energy stored in a compressed or extended spring. 𝑘 is the spring constant. And 𝑥 is the extension or compression of the spring, in other words, how much is the spring being stretched or compressed by relative to its natural length.
Now in this case, we’ve been given the compression of the spring every time we press a key. That value is 4.8 millimeters. So this is the value of 𝑥. Now every time we press a key, we store an elastic potential energy 𝐸 in that spring. So where does that energy come from?
Well, whoever is typing is supplying that energy by pushing the key. So the typist supplies the energy. And the spring stores this energy. In other words, the amount of energy supplied by the typist to press a key is the same as the elastic potential energy stored in the spring when the key is pressed down.
So to work out the amount of energy supplied per key press, all we need to do is say that 𝐸 sub press, the amount of energy supplied per key press, is equal to half 𝑘𝑥 squared.
However, we’re not just pressing one key once. We’re pressing keys that day a total of 12500 times. So the total energy supplied to press keys, which we’ll call 𝐸 sub tot, is equal to 12500 times 𝐸 sub press. And so this happens to be 12500 times half 𝑘𝑥 squared.
At this point, we can plug in the values of 𝑘 and 𝑥. But just before we do that, we need to realize that the value of 𝑥 that we’ve been given is in millimeters. However, if we want to work in standard units, we need to convert this to meters. So we can recall that the conversion factor is that one millimeter is equal to 10 to the power of minus three meters. And so 4.8 millimeters is the same as 4.8 times 10 to the negative three meters. And this is the value we need to be using when we’re plugging it into our calculations.
So let’s replace 𝑥 with 4.8 times 10 to the power of negative three meters. And let’s start subbing in some values. We find that the total energy supplied is equal to 12500 times half times 𝑘 times 𝑥 squared. And we can plug this into our calculator to give us a value of 0.36 joules. And we know that this value is in joules because we use standard units for all of the quantities in the equation. We use newtons per meter for 𝑘. And we use meters for 𝑥.
So the energy will also come out in its standard units, which is joules. And at this point, we have our final answer. The total energy supplied to make all the keystrokes perform that day was 0.36 joules.