# Solving yo-yo problem using only energy equations - is this ok?

• mm2424
In summary, the conversation discusses solving for the linear acceleration of a yo-yo rolling down a string using energy equations and compares it to using force equations. The approach of using energy equations is deemed sound and produces a similar answer to the force equation method.

## Homework Statement

A yo-yo has a rotational inertia of 960 gcm^2 and a mass of 120 g. Its axle radius is 3.2 mm and its string is 120 cm long. The yo-yo rolls from rest down to the end of the string. What is the magnitude of its linear acceleration?

T - mg = ma

## The Attempt at a Solution

I know there is an equation that specifically allows you to calculate the acceleration of the center of mass of a yo-yo, but I'm trying to figure it out without using that equation. I also see in the solution manual that the correct strategy is to analyze the forces acting on the yo-yo at the bottom, and then use torque expressions to find the Tension in terms of the acceleration and the radius of the axle.

My question is just whether it's ok to solve this problem using energy equations instead of force equations. I took the following stab and seem to have arrived at a similar answer to the answer key. I just want to make sure this reasoning is sound, as I have found two answers to the problem so far and neither uses these energy equations.

I set the problem up as follows, where v equals the linear velocity of the yo-yo, m equals the mass of the yo-yo, h is the length of the string, and r is the radius of its axle:

Initial Energy of System = Final Energy of System

(m)(gravity)(h) = 1/2(m)(v^2) + 1/2(I)(ω^2).

(120 g)(980 cm/s^2)(120 cm) = 1/2(120 g)(v^2) + 1/2(950gcm^2)(v^2 / r^2)

Solving this, I found that v=54.8 cm/s.

I then plugged this into (final linear velocity)^2 = 2ah and found that a was 12.51 cm/s^2.

Is this an ok approach?

Thanks!

I didn't check your arithmetic, but your method is perfectly sound. Why not solve it using force equations to double check?

Thanks! It seems to work!

## 1. What is the yo-yo problem and why is it important to solve using only energy equations?

The yo-yo problem is a physics problem that involves analyzing the motion of a yo-yo as it is released from rest and allowed to fall under the influence of gravity. It is important to solve using only energy equations because they provide a more concise and accurate way of understanding and predicting the motion of the yo-yo, without the need for complex mathematical models.

## 2. Can energy equations be used to solve any type of yo-yo problem?

Yes, energy equations can be used to solve any type of yo-yo problem as long as the motion is only influenced by the force of gravity. This is because energy equations are based on the principle of conservation of energy, which holds true for all types of motion.

## 3. How do energy equations help in solving the yo-yo problem?

Energy equations help in solving the yo-yo problem by providing a way to express the initial and final energy states of the yo-yo. This allows us to calculate the change in energy, which is equal to the work done by the force of gravity on the yo-yo. By equating this work to the change in kinetic energy, we can solve for the velocity and position of the yo-yo at any given point in time.

## 4. Is it necessary to consider other forces besides gravity when using energy equations to solve the yo-yo problem?

No, it is not necessary to consider other forces besides gravity when using energy equations to solve the yo-yo problem, as long as they are the only external forces acting on the yo-yo. However, if other forces are present (such as air resistance), they can be taken into account by adding additional terms to the energy equations.

## 5. Are there any limitations to using energy equations to solve the yo-yo problem?

Yes, there are some limitations to using energy equations to solve the yo-yo problem. Energy equations assume that the motion of the yo-yo is frictionless, which may not always be the case in real-world scenarios. Additionally, they do not take into account the rotational motion of the yo-yo, which may be significant for certain types of yo-yo problems.