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### Course: Physics archive>Unit 4

Lesson 1: Circular motion and centripetal acceleration

# Visual understanding of centripetal acceleration formula

Visual understanding of how centripetal acceleration relates to velocity and radius. Created by Sal Khan.

## Want to join the conversation?

• If the velocity is constant, then wouldn't the acceleration be 0? So how can Ac = v^2/r when the velocity is constant?
• The MAGNITUDE of the velocity is constant although velocity has speed AND direction so the direction of the velocity is changing thus the velocity is changing, thus the object is accelerating, even when the magnitude of the velocity / the speed is the same.
• Can somebody explain in details as how v can be the radius of the second circle?
• If you take an arbitrary vector that always has the same magnitude, but varying directions, you can create a circle showing all possible variations of the vector. The radius of the circle will be equal to the magnitude of the vector.

Sal took the velocity vector, which has a fixed magnitude, but changes direction constantly, and made a circle showing all "versions" of it. Therefore the radius of the circle must be equal to v.
• At Sal translates those acceleration vectors but they seem to be in the incorrect position. I do not see how they become center seeking.
• The acceleration vectors he translates are the ones that correspond to each colored vector. For example, the acceleration vector that corresponds to the green vector in the circle on the right becomes translated to the green vector in the circle on the left. Both of the acceleration vectors are the same for that green vector, Sal just moved it to a different place.
• at why does he put the velocity vectors in the middle? and how is the radius of the same circle equal to v wouldn't that imply that v=r?
• I think the second diagram shows the forces acting on the object in the object's viewpoint or reference frame (I think), with the center representing the object itself. If you were somehow on the object, you would observe a velocity vector that points upwards initially but changes direction in a clockwise manner because of an acceleration acting perpendicularly to the velocity vector.

In contrast, the first diagram shows everything in a different reference point (i.e. we are observing the object from a fixed position away from the object).
• I have noticed that all tangents are 90 degrees to the radius... Does that apply for all tangents? I understood everything else.
• Good Question Shak Kataev :)
Since, it is hard to prove that without visual representation, I would recommend you to take a look at this link :-https://proofwiki.org/wiki/Radius_at_Right_Angle_to_Tangent
Hope it helps..
• Hi Sal, can you explain me why does the velocity vector and radius vector takes equal time period to travel 1/4circle ??
• I'm not 100% certain about this answer but if you imagine a race car that took 60 seconds to go around a track, you can think of the length of the track in terms of time. You can plug in values of time and receive the position of the car at that time. You can also plug in values for time and receive the velocity of the car at that time. Both the velocity and position paths are 60 seconds long which means that the length of each of their paths should be the same.
• what is a position vector?
• why is the position vector measured from centre of the circle? It could also be measured from an external point.
• Yes of course, it is your choice of coordinate systems. And the results should be the same.
But the calculations simplify a lot if you observe the symmetry of the problem (something going around in a circle) and chose a coordinate systems that reflects this symmetry.
Here you can specify the position of the object by just one number (the angle variable) because the radius stays the same. If you take another coordinate system (eg x-y axes) you need two numbers to specify any position: the x-coordinate and the y-coordinate.
• at , how did he write T=(1/4*pi*v)a
• V is the new radius of the smaller circle representing with the magnitudes of the acceleration. For this circle time is the change in velocity over acceleration. Watch it again when he is making the smaller circle. Hope that helps.
• why is it assumed that angle between r1 and r3 is 90 degrees? shouldn't the formula be T = angle/360 X 2pir / V
• He is just using this as an example, and since he drew the diagram he can assume what he wants. However, I agree that he should have started with the most general form of the this formula, as you wrote.