Circular Motion (radial acceleration)?

Question

A yo-yo is being spun and is moving in a circle 6.5 cm in radius at a speed of 24 cm/s
while making an unknown angle with the vertical.

a) Determine the angle .

The \rope" holding up the yo-yo is actually a rubber band whose
spring constant is 3.1 N/m. If the mass of the yo-yo is 0.10 kg,
determine the unstretched length of the rubber band.

Answer 1

The centrifugal force on the yo-yo is m*v^2/r; it is directed at the angle ø with the vertical. The vertical component of the force is m*v^2/r * cos(ø). This must overcome the gravitational force on the yo-yo which is m*g. Therefore

m*v^2/r * cos(ø) = m*g

cos(ø) = v^2/(r*g)

The stretching force on the string is the centrifugal force m*v^2/r. The string will match that force by k*∆s, where k is the spring constant and ∆s is the amount it stretches.

k*∆s = m*v^2/r

∆s = m*v^2/(r*k)

The radius was given as 6.5cm while it is spinning (string is stretched). The string length unstretched is then 6.5cm - ∆s.

Keep track of the units in solving: convert everything to kg, m, sec for computation and then convert back if necessary when done.

Answer 2

It's not clear from your question, but I am supposing that the yo-yo is moving in a circle in a horizontal plane, and the string is being held at some point above that plane. We want the angle the string makes with the vertical.

The acceleration in the horizontal plane is given by a = v^2/r
The acceration vertically upwards must be equal to g, the acceleration due to gravity, so that the yo-yo remains in the horizontal plane.

Compounding these two forces we have
tan theta = horizontal force/vertical force = v^2/gr where theta is the angle we're looking for.

For the second part we want to know the tension in the band. We get this by finding the resultant of the two forces. This is given by F = root( horizontal force squared + vertical force squared) or
F = m x root(v^4/r^2 + g^2)
Alternatively we can say F cos theta= mg, or F = mg/cos theta, as we already know theta. The length, L, of the band similarly is given by Lsin theta = r or L = r/sin theta. This length is L0 + (F/S) where L0 is the unstretched length and S is the spring constant. So,
r/sin theta = L0 + mg/Scos theta, or
L0 = r/sin theta - mg/Scos theta

I'll leave it to you to plug in the numbers.