Yo-Yo Motion Puzzle - Find the Speed & Time of Fall

[lfabb]

Homework Statement

The string in a yo-yo is wound around an axle of radius 0.512 cm. The yo-yo has both rotational and translational motion, like a rolling object, and has mass 0.206 kg and outer radius 1.88 cm. Starting from rest, it rotates and falls a distance of 1.20 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. What is the speed of the yo-yo when it reaches the distance of 1.20 m? How long does it take to fall?

Homework Equations

mgh=1/2mv^2+1/2Iw^2
I=1/2mr^2
w=v/r

The Attempt at a Solution

r1=.512
r2=1.88
mgh=1/2mv^2+1/2(1/2mr2^2)(v/r2)^2

The answer is obviously incorrect. What exactly am I doing wrong? Thanks!

 

[Redbelly98]

You're almost there, there's just a small change to make in your equation.

w depends on the radius at which the string is located.

The string isn't wound around the outside of the yo-yo (r2 that you used in w=v/r), so think about where the string is wound on the yo-yo.

p.s. welcome to PF.

 

[thomate1]

I would like to offer some suggestions to help you find the correct solution to this problem.

Firstly, it is important to understand the different types of motion involved in this problem. The yo-yo has both rotational and translational motion. The rotational motion is caused by the torque created by the force of gravity acting on the mass of the yo-yo. The translational motion is the motion of the center of mass of the yo-yo as it falls.

Secondly, when solving problems in physics, it is important to clearly define and label all variables. In this case, we have the mass of the yo-yo (m), the radius of the axle (r1), the outer radius of the yo-yo (r2), the distance it falls (h), the speed of the yo-yo when it reaches the distance of 1.20 m (v), and the time it takes to fall (t).

Now, let's look at the equations you have used in your attempt at a solution. The first equation, mgh=1/2mv^2+1/2Iw^2, is the conservation of energy equation. This equation applies to objects that have only translational motion. In this problem, the yo-yo has both translational and rotational motion, so we cannot use this equation.

The second equation, I=1/2mr^2, is the moment of inertia for a uniform circular disk. This equation is correct, but it does not take into account the rotational motion caused by the torque of gravity.

To solve this problem, we need to use the conservation of angular momentum equation, L=Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. This equation takes into account the rotational motion of the yo-yo.

We can also use the equation for the torque of gravity, τ=mgRsinθ, where m is the mass of the yo-yo, g is the acceleration due to gravity, R is the distance from the center of mass to the point where the force is applied, and θ is the angle between the force and the radius.

By combining these equations and using the given values, we can find the speed and time of fall for the yo-yo.

So, in summary, to find the correct solution to this problem, we need to consider both the