Torque, force and rotational acceleration

[The_Journey]

This isn't a homework question.

I'm really confused at this, say you're holding a massless ruler / stick and there is a mass (m) on it at a distance d from your hand. If you move the mass to 2d from your hand, would you have to apply twice the force as before, or quadruple to keep it stable?

I know torque = r cross F. F is constant (mg), so the force you would have to apply should be twice if you move it to 2d. But my teacher said it is quadruple because the rotational inertia of the stick would be m(2d)^2 which is 4 times the rotational inertia as before.

Can anybody explain to me if the force would have to be twice or 4 times as before?

Again NOT a homework problem, I just thought of this.

Some equations and math would be nice too.

 

[The_Journey]

Nvm I'll just put this in the homework section.

 

[Naty1]

NVM...you have two different situations, one static, one dynamic, hence two different answers...if you know the formula for rotational inertia you can work out that result using
v = wr

 

[The_Journey]

NVM...you have two different situations, one static, one dynamic, hence two different answers...if you know the formula for rotational inertia you can work out that result using
v = wr

Can you explain what you mean by static and dynamic? The stick is always stable.

 

[PJC01]

I can provide a response to this question. The concept of torque, force, and rotational acceleration is closely related to the study of rotational motion and mechanics.

To understand this scenario, we need to first define some key terms. Torque is the measure of the force that can cause an object to rotate around an axis. It is calculated as the product of the force applied and the distance from the axis of rotation. Force, on the other hand, is a physical quantity that can cause an object to accelerate or deform. In this case, the force being applied is the weight of the mass (mg). Rotational acceleration refers to the rate of change of angular velocity of an object in rotational motion.

Now, let's analyze the scenario mentioned. If you move the mass from d to 2d, the distance from the axis of rotation is doubled. As torque is directly proportional to the distance from the axis of rotation, the torque will also double. This means that to keep the mass stable, you would need to apply twice the force as before.

However, your teacher's explanation about the rotational inertia of the stick is also correct. Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to rotational motion. It depends on the mass of the object and the distribution of that mass around the axis of rotation. In this case, when the mass is moved to 2d, the rotational inertia of the stick would increase by a factor of 4. This is because the mass is now at a greater distance from the axis of rotation, resulting in a larger moment of inertia.

So, to summarize, the force required to keep the mass stable would increase by a factor of two due to the increased torque. However, the rotational inertia of the stick would also increase by a factor of four, resulting in a total force of four times the original force.

Mathematically, we can represent this scenario using the equation for torque, which is τ = r x F. In this equation, τ represents torque, r represents the distance from the axis of rotation, and F represents the force applied. As the distance (r) is doubled, the torque (τ) also doubles. However, the rotational inertia is represented by the equation I = mr^2, where m is the mass and r is the distance from the axis of rotation. As r is squared, the rotational inertia will increase by a factor of four when the