To determine the normal reaction force acting on a uniform cylinder rod just after it is released from rest, we need to analyze the forces and torques acting on the rod at that moment. Let’s break this down step by step.
Understanding the Setup
We have a uniform cylinder rod of mass m and length l. Initially, it is inclined at an angle of 30 degrees with the vertical. The lower end of the rod is in contact with a frictionless horizontal floor. When the rod is released, it will start to rotate about the point of contact with the floor.
Forces Acting on the Rod
At the moment of release, two main forces act on the rod:
- Weight (W) This acts downward through the center of mass of the rod, which is located at a distance of l/2 from the base.
- Normal Reaction (N): This acts perpendicular to the surface of the floor at the point of contact.
Components of Forces
Since the rod is inclined at an angle of 30 degrees with the vertical, we can resolve the weight into two components:
- Vertical Component (Wv): This is given by Wv = mg, where g is the acceleration due to gravity.
- Horizontal Component (Wh): This is given by Wh = mg \sin(30°) = mg \cdot 0.5 = 0.5mg.
Calculating the Normal Reaction Force
Since the floor is frictionless, the only vertical force acting on the rod is the weight and the normal reaction force. The normal force must balance the vertical component of the weight. Therefore, we can set up the following equation:
N = Wv
Substituting the expression for the vertical component of the weight:
N = mg
Conclusion
Thus, just after the rod is released, the normal reaction force from the floor is equal to the weight of the rod. Therefore, the value of the normal reaction force is:
N = mg
This means that regardless of the angle at which the rod is inclined, the normal force will always equal the weight of the rod at the moment of release, as long as there are no other vertical forces acting on it. This analysis helps us understand the dynamics of rigid bodies in motion and the role of forces in determining their behavior.
