Figure 1 shows a metal block that is suspended from a surface by a spring. Initially, the metal block is held in place so that the spring is not extended. The metal block is then allowed to fall, and the spring extends.
In this initial picture of the metal block then, we see that it’s suspended from a
spring which is at its natural length; it’s not stretched or compressed at all. The block in this snapshot sits motionless because it’s being held in place. But it has the potential to stretch out the spring.
That’s what happens in the second snapshot. Here, the block has been released. And under its weight, the spring gets stretched out and extended we’re told by 24
centimeters. This is the longest extension that the spring reaches. So here again, just like in the first picture, the metal block is at rest.
Knowing all this, let’s turn to our first question.
Complete the sentences to describe the energy transfers that occur. Use expressions in the box. Thermal, sound, gravitational potential, elastic potential, kinetic. Before the block is allowed to fall, it has a store of blank energy. As the block is falling, the block’s store of blank energy increases. When the spring is extended, the spring stores energy as blank energy.
Looking back to Figure 1, we see it shows us that this block suspended from a spring
descends over some distance over time. During this motion, there are energy transfers that occur between one type of energy
In this box here, we have five different energy types shown to us. These five types of energy are all candidates for filling in the blanks in these
sentences about the energy transfers in this process. Our task is to read through these sentences. And based on what they say as well as what we see in Figure 1, we want to figure out
what words go in these blanks. And the words we’ll choose will come from the different types of energy shown in this
Knowing all that, let’s start with our first sentence. Before the block is allowed to fall, it has a store of blank energy.
If we locate this moment in Figure 1, we see that it is shown on the left. This is where the block is being held in place and before it’s allowed to fall. At this point, the block isn’t in motion in the spring isn’t stretched at all.
The fact that the block isn’t moving tells us that it can’t have any kinetic
energy. And the fact that the spring is unstretched tells us that the spring has no elastic
potential energy to it. But there is a type of potential energy that this block has. And it has to do with this elevation.
Notice that when we release the block, it’s free to descend; it’s free to fall
towards the Earth. Now that descend is slowed down and ultimately stopped by the spring. But the fact remains that initially this block does have some potential energy due to
The energy that an object has because it’s some distance off the ground has to do
with energy due to gravity. Even before it’s in motion, the block has gravitational potential energy. That’s what goes in the first blank. So our first sentence reads “Before the block is allowed to fall, it has a store of
gravitational potential energy.”
Now, let’s move on to the second sentence. As the block is falling, the block’s store of blank energy increases.
Let’s think about what happens to this block as it falls. We know it starts from rest and it ends at rest. But in between, it’s in motion. Now, here’s the question: if you take a ball in your hand and then drop it, what
happens to the ball? It speeds up, right? Well, the same thing happens to our block as it starts to fall; its speed peaks
What type of energy from the different energy types listed here has to do with
speed? We know it’s not necessarily thermal or sound; it’s not gravitational potential or
elastic potential. It’s kinetic energy that has to do with the speed on an object’s motion. It’s that kind of energy then that goes in this blank. And the sentence now reads “As the block is falling, the block’s store of kinetic
Just as a side note, we know that as the block continues to fall though, its kinetic
energy decreases because the block ends up at rest as we’ve seen. It’s this position of the block that’s the subject of our third sentence “When the
spring is extended, the spring stores energy as blank energy.”
Let’s compare for a second the length of our spring initially and then finally. We see that clearly the spring has stretched out. We’re told it’s extended by 24 centimeters. That extension of the spring is itself a way for energy to be stored in the
Because if you stretch out a spring, what does it want to do? It wants to return to its natural length, right? And in order to do that, in order to return due to its natural length, the spring is
capable of exerting a force.
We can feel that force as a pulling on our hand when the spring wants to get back to
its normal length. That pull — that force — is enabled by the energy stored in the spring which is in
the form of elastic potential energy.
It doesn’t have to do with gravity or the spring’s motion and it has nothing to do
with thermal energy or sound energy either. Rather it’s the stretched material of the spring itself that creates this elastic
potential energy. So our third and final sentence reads “When the spring is extended, the spring stores
energy as elastic potential energy.”
These are the various types of energy involved in these energy transfer
Next, let’s look at just how much elastic potential energy the spring has when it is
The middle block reaches its lowest point when the spring has extended by 24
centimeters. The spring has a spring constant of 200 newtons per meter. Calculate the energy stored in the stretched spring. Use the correct equation from the Physics Equation Sheet.
So knowing that when the spring is extended, the metal block is 24 centimeters below
its original point, in other words that the spring has extended 24 centimeters
beyond its original length. We want to calculate how much energy is stored in the stretched spring.
We saw earlier that this type of energy is elastic potential energy. That’s the energy that is stored in a spring that’s either stretched, beyond its
normal length, or compressed, shorter than its normal length. We’ll call this stored energy capital 𝐸.
And on our Physics Equation Sheet, we’ll find the equation that gives us the elastic
potential energy of a stretched spring. And here is that equation. It tells us that the energy stored in a stretched spring is equal to one-half times
what’s called the spring constant, symbolized by 𝑘, multiplied by the extension of
the spring; we’ve called it 𝑒 squared.
In our problem statement, we’re told what the spring constant, which is 𝑘 in this
equation, is. It’s 200 newtons per meter. If we insert that value for 𝑘 into our equation for the spring energy, notice the
units of that value: newtons per meter. What’s especially important here is that we’ve been given length units in units of
meters, while our spring extension is given to us in units of centimeters.
When we plug in the value for our spring extension in 𝑒, we want the units of that
value to match the units of length in our spring constant 𝑘. In other words, we’re going to convert 24 centimeters into whatever that value is in
Here is what we can recall to help us do that. We know that 100 centimeters is equal to one meter. And another way of saying the exact same thing is to say that one centimeter is equal
to 0.01 meters. And we can use this fact to help us convert 24 centimeters into some distance in
Here is how we’ll do that. We’ll take our length, 24 centimeters, and we’ll replace the centimeters here with
0.01 meters. Our length then becomes 24 times 0.01 meters, which gives us a result of 0.24
meters. That’s the spring extension in units of meters.
It’s this value in these units that we’ll plug in for 𝑒 in our equation for the
elastic potential energy of the stretched spring. And with that value inserted, we’re now ready to multiply through and solve for
capital 𝐸. When we do, we find a result of 5.76 newton meters.
And a newton meter can be rewritten as another unit, the unit of joules. One newton meter is equal to a joule. So our energy is 5.76 joules. That’s how much elastic potential energy is stored in the spring when the block is at
its lowest point.
For our next question, let’s look at another type of energy involved in this
The equation that relates the change in the gravitational potential energy of an
object to its mass, the acceleration due to gravity, and the vertical distance it
moves, is change in gravitational potential energy equals mass times acceleration
due to gravity times vertical distance. Use this equation to work out the mass of the metal block. The acceleration due to gravity is 9.8 meters per second squared. Give your answer to two significant figures.
So in this question, we’re going to use the fact that the elevation of our metal
block changes. And therefore, it experiences a change in gravitational potential energy. That change can be used to work backwards via this equation that we’re given to solve
for the mass of the metal block and it’s that we want to solve for.
To do this, let’s clear a little bit of space and write some of the information we’ve
been given in this problem statement in some shorthand. First things first, what we’ve done is we’ve rewritten our equation in symbolic form,
where here our change in gravitational potential energy is Δ𝐸 sub 𝑔, the mass of
the object we’re measuring is 𝑚, the acceleration due to gravity is lowercase 𝑔,
and the change in the vertical distance of this object is Δℎ.
We’re told in the problem statement that the acceleration due to gravity is to be
treated as 9.8 meters per second squared. What we want to do is use this value along with this equation and some other
information we’ll pick up in a minute to solve for the mass 𝑚 of our metal block in
We see that this equation as is lets us solve for Δ𝐸 sub 𝑔, the change in
gravitational potential energy. But we want to solve for 𝑚. And so that’s gonna take a bit of rearranging. To get there, to have 𝑚 by itself on one side of this equation, here’s what we’ll
We’ll divide both sides of the equation by the acceleration due to gravity multiplied
by the change in vertical distance Δℎ. When we do that, notice on the right-hand side of this equation that the factor of
𝑔, the acceleration due to gravity, as well as Δℎ cancels out.
And then, we have this wonderful equation that does exactly what we want. In terms of some scenario parameters, we’re able to solve for the mass 𝑚 of our
metal block. The only question now is what is Δ𝐸 sub 𝑔, what’s 𝑔, and what’s Δℎ.
Let’s start with 𝑔 the acceleration due to gravity. We’re given that value as 9.8 meters per second squared. So we can plug in for that value right away. Now, what about Δℎ? Remember that this is the vertical change in height of our metal block. So the initial height of the block down to the final height of the block, that change
in height, is Δℎ.
We’re told in Figure 1 that that height change is equal to 24 centimeters. And we might be tempted to simply plug that value in for Δℎ in our equation. But here’s the thing: just like in our last question, we want the units in our
overall equation that we used to be consistent with one another.
To do that, it means that our lengths and distances will all be expressed in units of
meters. So just like before, we’ll take this distance in centimeters and convert it to
meters. We saw that 24 centimeters is equal to 0.24 meters and that’s the value that we’ll
plug in for Δℎ in our equation.
We’re making great progress and just one more term to go, Δ𝐸 sub 𝑔, the change in
gravitational potential energy of our metal block as it descends. We’re not told directly what this value is in our problem statement. But from a solution we found earlier, we can infer what this change in gravitational
potential energy is.
Remember that earlier we solved for the elastic potential energy of our stretched
spring and found it to be 5.76 joules. Going back to Figure 1 because our block started and ended at rest, that means that
this change in the elastic potential energy of the spring is also equal to the
magnitude of the change in the gravitational potential energy of our metal
Through the conservation of energy, these two values are equal. So we’ll use 5.76 joules for Δ𝐸 sub 𝑔 in our equation. With that in place, we’re, now, ready to calculate 𝑚, the mass of our metal
When we calculate this expression, we’re careful to rounded to two significant
figures, like we’re directed in the problem statement. The mass of the block ends up being 2.4 kilograms to two significant figures. This then is the mass of the metal block that’s on the end of the stretched