The string in a yo-yo is wound around an axle of radius 0.547 cm. The yo-yo has

Question

The string in a yo-yo is wound around an axle of radius 0.547 cm. The yo-yo has both rotational and translational motion, like a rolling object, and has mass 0.344 kg and outer radius 1.99 cm. Starting from rest, it rotates and falls a distance of 1.16 m (the length of the string). Assume for simplicity that the yo-yo is a uniform circular disk and that the string is thin compared to the radius of the axle. (a) What is the speed of the yo-yo when it reaches the distance of 1.16 m? m/s (b) How long does it take to fall? [Hint: The translational and rotational kinetic energies are related, but the yo-yo is not rolling on its outer radius.] s

Explanation / Answer

a)

m g h = 0.5 m v^2 + 0.5 I w^2

m g h = 0.5 m v^2 + 0.5 I (v^2/R^2)

m g h = 0.5 m v^2 + 0.5 (0.5 m R^2) (v^2/R^2)

g h = 0.75 v^2

9.8 * 1.16 = 0.75 v^2

v = 3.8932 3.89 m/s

b)

v^2 = 2 a h

a = v^2/2h = 3.8932*3.8932/2/1.16 = 6.5332

t = (2h/a) = (2*1.16/6.5332) = 0.596 s

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