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A particle has position vector as a function of time given by R = 4 sin(2πt) î + 4 cos(2πt) ĵ, where R is measured in metres, t in seconds, and î and ĵ are unit vectors along the x- and y-axes. Which of the following statements is incorrect for this motion?

1. The magnitude of the acceleration vector is v²/R, where v is the particle’s speed
2. The magnitude of the particle’s velocity is 8 m/s
3. The particle moves along a circle of radius 4 m
4. The acceleration vector is directed along −R

Correct answer: The magnitude of the particle’s velocity is 8 m/s

Solution

R = 4 sin(2*pi*t) i + 4 cos(2*pi*t) j describes uniform circular motion of radius 4 m with omega = 2*pi rad/s, so speed v = omega*R = 8*pi ~ 25.1 m/s. The statement 'velocity is 8 m/s' is therefore the incorrect one; the centripetal results (a = v^2/R, directed along -R) are all correct.

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