Aparticle travels in a circular orbit of radius 21 m. its speed is changing at a rate of 23.1 m/s2 at an instant when its speed is 37.2 m/s. what is the magnitude of the acceleration (in m/s? ) of the particle?

The particle has an acceleration vector with one component directed toward the center of its orbit, and the other directing tangentially to its orbit. Call these components ( for center) and ( for tangent). Then its acceleration vector has magnitude

We have

where is the particle's speed and is the radius of orbit, so

We're given that the particle's speed changes at a rate of 23.1 m/s^2. Its velocity vector points in the same direction as , i.e. perpendicular to , so

Then the magnitude of the particle's acceleration is

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