### Video Transcript

Which are the lines on the graph correctly shows how the linear speed of an object varies with the radius of the circular path followed by the object? Assume that the angular velocity of the object is constant. (A) The yellow line, (B) the gray line, (C) the blue line, (D) the orange line.

Our graph shows us the linear speed of an object, in units of centimeters per second, plotted against the radius of the circular path in which that object travels. So imagine we have an object, thatโs our blue dot, moving around a circular path. As it does so, this object keeps a constant linear speed, weโll call it ๐ฃ, and this speed depends on the radius of the circle, weโll call it ๐, that the object moves around.

What we want to do in this example is find the mathematical relationship in graphical form of linear speed ๐ฃ against radius ๐. As our object moves around circles of different radii, weโre to assume that the angular velocity of the object remains constant. Thatโs an important bit of information because the angular velocity ๐ of an object moving in a circular path is equal to the linear speed of that object ๐ฃ divided by the radius of the circle around which it moves. Weโre told in this example that regardless of the value of ๐, ๐ is the same always. Notice that if we multiply both sides of this equation by the radius ๐, then that factor cancels on the right. And we find that ๐ times ๐ equals ๐ฃ, or reversing the sides of the equation ๐ฃ equals ๐ times ๐.

On our graph, the radius ๐ is the independent variable. Thatโs represented by ๐ in this equation. The dependent variable is our objectโs linear speed. Thatโs ๐ฃ. We see that these two variables are related mathematically by a constant value ๐. Typically, ๐ is not constant, but in this case weโre told that it is. So hereโs what we can write. We can say that the linear speed of our object thatโs moving in a circular path of some radius ๐ is equal to a constant, weโll call it ๐ถ, times ๐. Whichever of the four lines on our graph is correct will follow this general functional shape.

The first thing we can notice about this equation is that itโs linear. That is, all the variables on the right-hand side, in this case thatโs just ๐, appear to the first power. This means that any curve on our graph that is not linear canโt be our answer. We see that both the blue curve here and the yellow curve here do not follow this linear form. Therefore, we wonโt choose answer option (A) or option (C), corresponding to those two curves.

To see whether itโs the orange curve here or the gray curve here that is our correct answer, letโs look again at the equation that we want to plot: ๐ฃ is equal to a constant times ๐. Letโs imagine that at a given value of ๐, that value increases, say, by a factor of two. If we multiply ๐ by two, then since ๐ถ is a constant, the entire right-hand side of our expression will increase by a factor of two. For this equation to hold true then, ๐ฃ on the left must also increase by a factor of two. This shows us that as ๐ changes, say, by doubling ๐, the linear speed ๐ฃ must change in response for our equation to hold true. We see that the orange curve is a flat curve, meaning that ๐ฃ is always the same. Based on our equation though, ๐ฃ canโt stay the same while ๐ changes. We wonโt choose answer option (D), the orange curve.

That leaves the gray curve. Notice that this line passes through the point 0.5 on the radius axis and 0.50 on the linear speed axis. We would expect then that if we double ๐ to 1.0, based on our equation, the value of linear speed ๐ฃ would also double. And indeed, we see that that happens along the gray curve. At a value of 1.0 centimeters for the radius ๐, we have a value of 1.00 centimeters per second for linear speed. This confirms to us that itโs the gray curve that correctly shows how linear speed varies with the radius of the circular path followed by our object. We note that this graph is known to be correct because of the condition that angular velocity is constant. We choose then answer option (B).