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Circular motion

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Instantaneous acceleration

Let us first calculate the average acceleration over an interval of time Δt. After that we will shrink the interval down to zero. When we do that the average acceleration approaches the instantaneous acceleration.

In fact, we define the instantaneous acceleration as the limit of the average acceleration as the time interval approaches zero.

Draw the velocity vectors at two different instants of time (during which the particle has turned through Δθ) then move the vectors so they are tail to tail so that we may draw Δv, the change in velocity.

Figure 4

The initial velocity, the final velocity and the change in velocity vectors form an isosceles triangle, since v1 and v2 have the same length. We are talking about uniform circular motion.

The average acceleration for this interval is Δv/Δt . Its direction is along Δv. The angle between Δv and v1 is (π- Δθ)/2.

Get ready! We are now going to make Δt smaller and smaller. As Δt shrinks, Δθ shrinks. In fact, Δθ = ωΔt.

What about Δv? It too shrinks in size. As it shrinks, we can approximate its length by the length of the arc of a circle of radius v subtending an angle Δθ at the center.

Figure 5

The length of the arc is vΔθ.

The angle is measured in radians. In fact, the angle in radians is defined as the arc length divided by the radius. Here we are making an arc by rotating the velocity vector. So the radius of the arc is v.

The thing that I want you to take on faith is that as Δθ shrinks, using vΔθ for the length of Δv becomes more and more accurate.

We can now see that the magnitude of the average acceleration approaches vω.

(1)

We have included other forms of vω in brackets.

The direction of the acceleration is the direction of Δv. The angle between Δv and v1 is

As Δt → 0, Δθ → 0, the angle between Δv and v1 approaches π/2. That is, the acceleration becomes perpendicular to the velocity. Since the velocity is tangential, the acceleration points along the radius, towards the center (look at the drawings above).

To summarize: We start with the average acceleration over a time interval Δt. We let Δt → 0. In this limit, the acceleration approaches vω in magnitude and becomes perpendicular to the velocity.

Figure 6

The instantaneous acceleration of a particle moving at constant speed around a circle of radius R points towards the center of the circle and has the magnitude vω. Note that vω = v2/R.


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