# Falling Weight Attached to a Wheel and a Sphere

• Como Bluff
In summary, the problem involves calculating the acceleration of a falling weight connected to a wheel and a sphere with thin walls. The net moment of inertia for the system needs to be determined, taking into account the moments of inertia for each object involved. The tensions in the string and their relation to torques need to be considered. By creating unknowns for the tensions and using free body equations, the accelerations and angular accelerations can be related through the radii. The solution involves three equations with three unknowns.
Como Bluff
How do I calculate the acceleration of the falling weight? It is hanging from a string which goes through a wheel, and is attached to a sphere with thin walls. The string doesn't stretch and the wheel and the sphere spin without friction.

The fact that the weight is connected to multiple masses is the problem. How does one calculate the net moment of inertia that is affecting the acceleration of the falling weight?

For the sake of simplicity, I left out the values given in the original statement of the problem; these are of course available upon request.

The moments of inertia for the objects in the problem:

Jsphere = $\frac{2}{3}$mr2
Jwheel = $\frac{1}{2}$mr2

Thanks!

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Deal with the tensions in the string, relating them to the torques.

Thanks for quick reply haruspex! I don't quite get it yet. Could you elaborate your answer a little more?

Como Bluff said:
Thanks for quick reply haruspex! I don't quite get it yet. Could you elaborate your answer a little more?
Create unknowns for the tensions in the two sections of string. (They will not be the same.) Do the usual free body equations for each of the three massive components. You can relate all the accelerations and angular accelerations through the radii. You should have three equations with three unknowns (the two tensions and the acceleration).

I got it now, thanks!

(I had also made the classic mistake of using the given diameter as the radius, which was needed to calculate the mass of the wheel from the moment of inertia..)

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## 1. What is the purpose of attaching a weight to a wheel and a sphere?

The purpose of attaching a weight to a wheel and a sphere is to demonstrate the effects of gravitational force and rotational inertia on different shapes and objects. This experiment can also help in understanding the principles of physics and motion.

## 2. How does the weight affect the motion of the wheel and sphere?

The weight attached to the wheel and sphere increases the overall mass of the objects, leading to an increase in inertia. This makes it harder for the objects to accelerate and change their direction of motion.

## 3. What factors can influence the rate at which the wheel and sphere fall?

The rate at which the wheel and sphere fall can be influenced by the weight of the objects, the height from which they are dropped, and the shape and size of the objects. The presence of air resistance and surface friction can also affect the rate of falling.

## 4. How does the shape of the objects affect their motion?

The shape of the objects can greatly affect their motion. Objects with a larger surface area, such as a sphere, experience more air resistance and therefore have a slower rate of falling compared to objects with a smaller surface area, such as a wheel.

## 5. What can be learned from this experiment?

This experiment can teach us about the relationship between gravitational force, mass, and inertia. It also helps in understanding the effects of air resistance and surface friction on the motion of objects. Additionally, this experiment can be used to compare and contrast the motion of different shapes and objects, and how they interact with their surroundings.

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