# Angular velocity and acceleration of a plank

• Kevodaboss
In summary, a horizontal plank with a frictionless axis of rotation at its center has a large mass and a small mass on either end. When released from rest, the magnitude of the angular acceleration increases as the torque decreases due to the decrease in the perpendicular force caused by the plank tilting downwards. The magnitude of the angular velocity also increases as a result. The lever arm, defined as the perpendicular distance from the line of force to the axis of rotation, decreases in this scenario.
Kevodaboss

## Homework Statement

A horizontal plank with a frictionless axis of rotation at its center has a large mass at one end and a small mass at the other end. It is held stationary and then released from rest. As the plank rotates (and before one end hits the ground), the magnitude of the angular acceleration of the plank (increases/decreases/or remains constant) and the magnitude of the angular velocity (increases/decreases/or remains constant) ?

τ=I α
τ=r F_perp

## The Attempt at a Solution

The solution to this problem is that τ=I α, therefore if torque decreases, then angular acceleration increases. Thus the magnitude angular velocity must be increasing, because the angular acceleration is nonzero.

However, I can't seem to figure out why the torque is decreasing in this problem. I know that τ=r F_perp, but how can I apply this to the problem? The lever arm obviously stays the same length. So is the perpendicular force decreasing because after the plank tilts downwards, the force of gravity downwards has both a horizontal component and a vertical component with respect to the plank (force of gravity stays the same)?

Thanks for reading! Any help is appreciated

Kevodaboss said:
The lever arm obviously stays the same length.
Usually the "lever arm" is defined as the perpendicular distance from the line of force to the axis of rotation. So by that definition, the lever arm decreases.

The length of the plank doesn't change, so I suspect that's what you meant.
Kevodaboss said:
So is the perpendicular force decreasing because after the plank tilts downwards, the force of gravity downwards has both a horizontal component and a vertical component with respect to the plank (force of gravity stays the same)?
Right. The component of the weight perpendicular to the plank decreases, so the torque decreases.

Ok, that makes sense. Thanks for the help!

Doc Al said:
Usually the "lever arm" is defined as the perpendicular distance from the line of force to the axis of rotation. So by that definition, the lever arm decreases.

The length of the plank doesn't change, so I suspect that's what you meant.

Right. The component of the weight perpendicular to the plank decreases, so the torque decreases.

Kevodaboss said:

## The Attempt at a Solution

The solution to this problem is that τ=I α, therefore if torque decreases, then angular acceleration increases.

Check that.

CWatters said:
Check that.

What do you mean?

CWatters said:
Check that.

Oops never mind. I meant if torque decreases, angular acceleration decreases

Kevodaboss said:
Oops never mind. I meant if torque decreases, angular acceleration decreases
I had assumed you just messed up that sentence, since what followed didn't depend on it. (Otherwise I would have said something.)

Doc Al said:
I had assumed you just messed up that sentence, since what followed didn't depend on it. (Otherwise I would have said something.)

Yeah, just a typo. Thanks for all the help!

## 1. What is angular velocity and acceleration?

Angular velocity refers to the rate of change of angular displacement of an object, while angular acceleration refers to the rate of change of angular velocity. In simpler terms, angular velocity is how fast an object is rotating, and angular acceleration is how quickly the rotation is changing.

## 2. How are angular velocity and acceleration related?

Angular velocity and acceleration are related through the formula a = rα, where a is the angular acceleration, r is the distance from the axis of rotation to the object, and α is the angular velocity. This means that as the angular velocity increases, the angular acceleration also increases.

## 3. What factors affect the angular velocity and acceleration of a plank?

The angular velocity and acceleration of a plank can be affected by the rotational force applied to it, the mass and shape of the plank, and the distance of the plank from the axis of rotation. Additionally, external forces such as friction can also affect the angular velocity and acceleration of the plank.

## 4. How is angular velocity and acceleration measured?

Angular velocity is typically measured in radians per second (rad/s), while angular acceleration is measured in radians per second squared (rad/s2). These measurements can be obtained using tools such as a tachometer for angular velocity and an accelerometer for angular acceleration.

## 5. What are some real-life applications of angular velocity and acceleration?

Angular velocity and acceleration have many real-life applications, such as in the rotation of wheels in vehicles, the motion of planets in their orbits, and the movement of athletes in sports such as gymnastics and figure skating. They are also important in the design and functioning of machines and mechanical systems.

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