# Angular acceleration of a cylinder rotating around another

• Happiness
In summary, the relation between the angular acceleration of the bottom right cylinder and its horizontal acceleration is given by ##\alpha=\frac{a_x}{\sqrt{3}R}##. The given answer of 7.84 is incorrect and there may be an approximation involved when using an infinitesimal distance in the derivation. Also, it is unlikely that ##\ddot{\theta}## is equal to the angular acceleration ##\alpha## of the bottom right cylinder. This conversation is part of a discussion on the coin rotation paradox.
Happiness

For Q11(b), what is the relation between the angular acceleration ##\alpha## of the bottom right cylinder and its horizontal acceleration ##a_x##?

I get ##\alpha=\frac{a_x}{\sqrt{3}R}##, which is half the given answer (7.84) below.

After the bottom right cylinder rolls around the top cylinder by an angle of rotation ##\theta## about its (the bottom right cylinder's) own center of mass, the cylinders will look as follows:

(The bottom left cylinder is omitted in the drawing.)

Let the axis of symmetry of the figure of 3 cylinders at ##t=0## be ##x=0##. (In other words, the point of contact of the two bottom cylinders has an ##x## coordinate of 0).

Then the ##x## coordinate of the center of the bottom right cylinder, ##x = 2R\sin\phi##, where ##\phi=30^\circ+\theta##.

##\dot{x}=2R\cos\phi\dot{\phi}##

##\ddot{x}=2R(\cos\phi\ddot{\phi}-\sin\phi\dot{\phi}^2)=2R(\cos\phi\alpha-\sin\phi\omega^2)##, where ##\alpha=\ddot{\theta}## is the angular acceleration of the bottom right cylinder and ##\omega=\dot{\theta}## is its angular velocity.

At ##t=0##, ##\phi=30^\circ## and ##\omega=0##. Thus ##\ddot{x}=2R(\cos 30^\circ\alpha)##. And we get ##\alpha=\frac{a_x}{\sqrt{3}R}##.

The given answer:

I believe the given answer is wrong because I suspect there is an approximation involved when it uses an infinitesimal distance ##d## in deriving (7.84).

Last edited:
Happiness said:
where ##\alpha=\ddot{\theta}## is the angular acceleration of the bottom right cylinder
I don't believe that ##\ddot{\theta}## equals the angular acceleration ##\alpha## of the bottom right cylinder.
https://en.wikipedia.org/wiki/Coin_rotation_paradox

Happiness

## 1. What is angular acceleration?

Angular acceleration is the rate of change of an object's angular velocity with respect to time. It is a measure of how quickly the rotational speed of an object is changing.

## 2. How is angular acceleration calculated?

Angular acceleration is calculated by dividing the change in angular velocity by the change in time. The unit of measurement for angular acceleration is radians per second squared (rad/s^2).

## 3. How does a cylinder rotating around another object experience angular acceleration?

When a cylinder rotates around another object, it experiences a change in its angular velocity due to the change in its rotational position. This change in angular velocity results in angular acceleration.

## 4. What factors can affect the angular acceleration of a cylinder rotating around another object?

The angular acceleration of a cylinder rotating around another object can be affected by the mass and size of the cylinder, the distance between the two objects, and any external forces acting on the cylinder.

## 5. How is angular acceleration related to torque?

Angular acceleration and torque are directly proportional to each other. This means that if the torque acting on a cylinder rotating around another object increases, the angular acceleration of the cylinder will also increase.

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