### Video Transcript

Which of the lines on the graph
correctly shows how the angular velocity of an object varies with the radius of the
circular path followed by the object? Assume that the linear speed of the
object is constant. (A) The purple line, (B) the blue
line, (C) the gray line, (D) the orange line, (E) none of the answers are
correct.

Here, we’re looking at an object
that is moving along a circular path. We need to figure out which line on
the graph shows how the object’s angular velocity changes as the radius of the
circle changes. First, let’s recall the
relationship between the radius of the circle along which an object moves, the speed
with which it moves, and the angular velocity. 𝜔 equals 𝑣 over 𝑟, where 𝜔 is
the angular velocity, 𝑣 is the speed, and 𝑟 is the radius of the circle.

Since we are told the speed 𝑣 is
constant, we can simplify this relationship for our object. The angular velocity is inversely
proportional to the radius of the circle. In other words, if we were to
double the radius of the circle, the angular velocity would halve.

Returning to our graph then, we’re
looking for a line that shows a decrease in the angular velocity as the radius
increases. On our graph, this would be a line
with a negative slope at every point. The blue, gray, and orange lines
have a positive slope at every point, while the purple line has a zero slope at
every point. None of them even have a negative
slope, which means that none of these lines can represent the correct relationship
between an object’s angular velocity and the radius of the circle along which it
moves if the linear velocity is kept constant.

Therefore, our final answer is
option (E). None of the answers are
correct.