Internal forces for massless rods in rotation

[euprax]

Lets say we have a mass attached to a rigid, massless rod that is fixed at the other end, so we have a system that can only undergo rotation. My question is, if we apply a force to the rigid rod (and not to the actual mass), how can we derive (using only F=Ma) the shear force distribution in the rod so that the forces applied to the ends will produce the expected angular acceleration.

It seems like I can get this answer using conservation of angular momentum or conservation of energy, but I cannot see how I could use only free body diagrams and F=Ma to detemine that, for example, applying a force midway through the rod will result in only half the force being transmitted to the mass at the end of the rod.

Again, I know how to get the answer, but I cannot visualize what is happening with the internal forces to make such a result come about.

Thanks!

 

[LightHero]

The answer to this question can be found using the concept of moment of inertia. Moment of inertia is a measure of how difficult it is to change the rotational motion of an object. In this case, when you apply a force to the rod, it will cause a torque (or rotational force) to be applied to the mass. This torque is equal to the product of the force applied and the distance from the point of application to the point of rotation (in this case, the end of the rod). The amount of torque applied to the mass depends on the mass of the object and its moment of inertia. The moment of inertia of an object (in this case, the mass at the end of the rod) is proportional to its mass and to the square of its distance from the point of rotation (in this case, the end of the rod). Therefore, if you apply a force to the rod midway between the end and the mass, the torque applied to the mass will be half of what it would have been if you had applied the same force at the end of the rod. This is because the distance from the point of application to the point of rotation is doubled. In summary, the shear force distribution in the rod can be determined using F=Ma and the concept of moment of inertia. The force applied at any point along the rod will result in a proportionally smaller torque being applied to the mass at the end of the rod as the distance from the point of application to the point of rotation increases.