### Video Transcript

On the graph shown, which of the lines correctly shows how the linear speed of a rotating object varies with the radius of the circular path followed by the object? Assume that the centripetal force acting on the object is the same for any radius of the circular path. Is it (A) the red line, (B) the gray line, (C) the blue line, or (D) the orange line?

We see these four colored lines on our graph, which shows us the linear speed of our moving object against the radius of that object’s circular path. Let’s clear some space now so we can sketch what this might look like. Say we have a circular path like this. If an object follows this path, then at any instant, we could sketch in its linear speed; we’ll call it 𝑣. And we could also sketch in the radial distance of that object from the center of the circular path; we’ll call it 𝑟. Our graph relates linear speed, what we’ve called 𝑣, to radius 𝑟.

Now, what if instead of this first circular path, our object moved on this second one? Because of this change, the radius of the circle around which the object moves has increased. And we want to pick which of the four curves on our graph shows us how the linear speed 𝑣 changes with 𝑟. A clue to help us figure this out is in this statement here; we assume that the centripetal force that acts on the object is the same, regardless of the radius of the circular path it travels in. For an object like the one we have here traveling in a circle of radius 𝑟 at a linear speed 𝑣, the center-seeking force that acts on that object, also called the centripetal force, is by Newton’s second law of motion equal to the mass of that object times its centripetal acceleration.

Centripetal acceleration, 𝑎 sub 𝑐, can be written in a couple of different ways. One way to write it though involves the linear speed 𝑣 of the object moving in a circle and the radius 𝑟 of that object’s circular path. Using this expression for 𝑎 sub 𝑐, we can substitute it into our expression for the centripetal force, 𝐹 sub 𝑐. And we now have an equation for centripetal force in terms of linear speed 𝑣 and the radius of the object’s circular path 𝑟.

Now we’re told that the centripetal force that acts on our object is the same, regardless of the radius of the circle around which that object moves. What we can write then is that the centripetal force that acts on our rotating object is equal to a constant we’ll call 𝑐. That is, no matter the value of 𝑟, somehow that value combines with 𝑣 squared so that this fraction is a constant value. If we multiply that fraction by the mass of the object, which is also constant, we get the constant we’ve called 𝑐.

Let’s now focus on just this part of our equation. Say that we divide both sides of this equation by the object’s mass 𝑚. We see that the mass will cancel out on the left, and on the right we have our constant 𝑐 divided by a constant value 𝑚. A constant divided by a constant is simply a constant. And let’s call this new constant 𝑘. In this equation, we have linear speed 𝑣, which does vary with 𝑟, we have 𝑟, which we know can vary, and we have our constant 𝑘. If we multiply both sides of our equation by the radius 𝑟, then 𝑟 cancels out on the left. We have that 𝑣 squared equals a constant times 𝑟. And this means we’re very close to getting an equation for the linear speed 𝑣 in terms of the radius 𝑟.

If we take the square root of both sides of this expression, then the square root and the square on the left cancel one another out. On the right-hand side, we have the square root of a constant, which is also a constant, times the square root of 𝑟. So here’s what we can finally say. The linear speed 𝑣 of our object as it moves in a circular path is directly proportional to the square root of the radius of that circular path. This means that whichever of the four graphs is the correct one to choose will have the same shape as the line 𝑦 is equal to the square root of 𝑥.

Here, the value on our vertical axis is not 𝑦 but 𝑣. And the value in our horizontal axis is not 𝑥 but 𝑟. Considering possible values of 𝑟, we see that if 𝑟 is equal to zero, then since the square root of zero is zero, 𝑣 will be zero as well. Therefore, we can cross the red curve off of our list of contenders. Our answer must pass through the origin at zero, zero. Considering the gray curve on our graph, we see that this is a linear curve. That doesn’t agree with the shape of the curve 𝑦 is equal to the square root of 𝑥. So we won’t choose this gray line.

Comparing the two remaining curves, the orange and the blue, we see that the orange line, as 𝑟 increases, is increasing in value at an increasing rate. The blue line on the other hand increases but seemingly at a decreasing rate. The shape of the blue line is consistent with the shape of the line plotting 𝑦 is equal to the square root of 𝑥. This agrees with the general form of our function 𝑣 is directly proportional to the square root of 𝑟. Of the four lines on our graph then, it’s the blue line that correctly represents the linear speed of our object being considered against its radius. We choose answer option (C).