Circular motion and centripetal acceleration

Here are some notes with an explanation of why a point moving in a circle is accelerating towards the centre of the circle.  Notice that the centripetal acceleration is just a geometric property of the circular motion: this derivation works out the direction and size of the centripetal acceleration by just considering the geometry of the situation, and it doesn’t need to discuss concepts like mass or force.

notes_page4

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4 thoughts on “Circular motion and centripetal acceleration

    1. That’s a good question: it’s not actually explained particularly clearly above. The diagram is showing a view of the object in polar coordinates with radius = r. Because the angular speed is ω, the angle at a time t is ωt, so this drawing ia just showing the position of the object at time t.

    1. The point is travelling with speed v so in time δt, the point travels a distance vδt round the arc of the circle.

      The point has angular velocity ω, so in the same time δt, the point sweeps out an angle ωδt.

      But the angle ωδt corresponds to a distance Rωδt around arc of the circle.

      This distance Rωδt must be the same as the distance vδt (because they are both expressions for the distance that the point has travelled round the circle), therefore Rωδt = vδt. This means Rω = v, and therefore ω = v/R.

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