Confusion About Net External Force Acting On Centre Of Mass

Mar 12, 2025 by ADMIN 60 views

**Understanding the Confusion about Net External Force Acting on Centre of Mass**

Introduction

In the realm of Newtonian Mechanics and Rotational Dynamics, understanding the concept of net external force acting on the centre of mass is crucial. However, this concept can sometimes lead to confusion, especially when dealing with complex systems. In this article, we will delve into the scenario of a perfect disk of uniform mass density and a point object attached to its periphery, and explore the net external force acting on the centre of mass.

The Scenario

Consider a perfect disk of uniform mass density and of mass $m$ , radius $R$ , and a point object of the same mass $m$ attached to its periphery. The point object is located at a distance $R$ from the centre of the disk. We will assume that the disk is rotating about its central axis, and the point object is also rotating with the disk.

Free Body Diagram

To analyze the net external force acting on the centre of mass, we need to create a free body diagram. A free body diagram is a graphical representation of the forces acting on an object. In this case, the free body diagram will include the following forces:

The weight of the disk, $mg$ , acting downwards
The weight of the point object, $mg$ , acting downwards
The normal force exerted by the ground on the disk, $N$ , acting upwards
The frictional force exerted by the ground on the disk, $f$ , acting tangentially

Net External Force

The net external force acting on the centre of mass is the vector sum of all the external forces acting on the system. In this case, the net external force is given by:

F_{net} = mg + mg - N - f

However, since the disk is rotating about its central axis, the normal force $N$ and the frictional force $f$ are not acting on the centre of mass. Therefore, the net external force acting on the centre of mass is:

F_{net} = 2mg

Analysis

The net external force acting on the centre of mass is $2mg$ , which is twice the weight of the disk. This result may seem counterintuitive, as one might expect the net external force to be equal to the weight of the disk. However, this is not the case, as the point object is also contributing to the net external force.

Rotational Dynamics

To understand why the net external force is $2mg$ , we need to consider the rotational dynamics of the system. The point object is attached to the periphery of the disk, and is rotating with the disk. As the disk rotates, the point object experiences a centrifugal force, which is directed away from the centre of the disk. This centrifugal force is given by:

F_c = \frac{mv^2}{R}

where $v$ is the velocity of the point object. Since the point object is rotating with the disk, its velocity is given by:

v = \omega R

where $\omega$ is the angular velocity of the disk. Substituting this expression into the equation for the centrifugal force, we get:

F_c = \frac{m\omega^2 R^2}{R} = m\omega^2 R

The centrifugal force is acting on the point object, and is contributing to the net external force acting on the centre of mass. Therefore, the net external force is given by:

F_{net} = 2mg

Conclusion

In conclusion, the net external force acting on the centre of mass of a perfect disk of uniform mass density and a point object attached to its periphery is $2mg$ . This result may seem counterintuitive, but it can be understood by considering the rotational dynamics of the system. The point object is experiencing a centrifugal force, which is contributing to the net external force acting on the centre of mass.

Additional Considerations

There are several additional considerations that need to be taken into account when analyzing the net external force acting on the centre of mass. These include:

Torque: The torque exerted by the point object on the disk is also contributing to the net external force acting on the centre of mass.
Angular momentum: The angular momentum of the system is also affected by the point object.
Non-uniform mass distribution: If the mass distribution of the disk is non-uniform, the net external force acting on the centre of mass will be different.

References

Goldstein, H. (1980). Classical Mechanics. Addison-Wesley Publishing Company.
Landau, L. D., & Lifshitz, E. M. (1976). Mechanics. Pergamon Press.
Marion, J. B., & Thornton, S. T. (1992). Classical Dynamics of Particles and Systems. Harcourt Brace Jovanovich Publishers.

Glossary

Centre of mass: The point where the entire mass of an object can be considered to be concentrated.
Net external force: The vector sum of all the external forces acting on an object.
Rotational dynamics: The study of the motion of objects that are rotating or revolving around a central axis.
Centrifugal force: A force that is directed away from the centre of rotation, and is experienced by an object that is rotating with a central axis.
Q&A: Understanding the Net External Force Acting on Centre of Mass ====================================================================

Introduction

In our previous article, we explored the concept of net external force acting on the centre of mass, and how it can be affected by the presence of a point object attached to the periphery of a perfect disk of uniform mass density. However, we know that there are many more questions that need to be answered. In this article, we will address some of the most frequently asked questions related to this topic.

Q: What is the net external force acting on the centre of mass when the point object is not attached to the disk?

A: When the point object is not attached to the disk, the net external force acting on the centre of mass is simply the weight of the disk, which is $mg$ .

Q: How does the net external force acting on the centre of mass change when the disk is rotating at a constant angular velocity?

A: When the disk is rotating at a constant angular velocity, the net external force acting on the centre of mass remains the same, which is $2mg$ . The rotation of the disk does not affect the net external force acting on the centre of mass.

Q: What is the effect of the point object's mass on the net external force acting on the centre of mass?

A: The mass of the point object does not affect the net external force acting on the centre of mass. The net external force acting on the centre of mass is determined by the mass of the disk, and not by the mass of the point object.

Q: How does the net external force acting on the centre of mass change when the disk is accelerating?

A: When the disk is accelerating, the net external force acting on the centre of mass changes. The net external force acting on the centre of mass is now given by:

F_{net} = 2mg + ma

where $a$ is the acceleration of the disk.

Q: What is the effect of friction on the net external force acting on the centre of mass?

A: Friction does not affect the net external force acting on the centre of mass. The net external force acting on the centre of mass is determined by the weight of the disk and the point object, and not by the frictional forces acting on the disk.

Q: How does the net external force acting on the centre of mass change when the disk is rotating in a non-circular path?

A: When the disk is rotating in a non-circular path, the net external force acting on the centre of mass changes. The net external force acting on the centre of mass is now given by:

F_{net} = 2mg + m\omega^2 r

where $r$ is the radius of the non-circular path, and $\omega$ is the angular velocity of the disk.

Q: What is the effect of the point object's velocity on the net external force acting on the centre of mass?

A: The velocity of the point object does not affect the net external force acting on the centre of mass. The net external force acting on the centre of mass is determined by the mass of the disk and the point object, and not by their velocities.