Askiitians Tutor Team
Last Activity: 14 Days ago
It seems like you're grappling with the concepts of torque, angular acceleration, and how they relate to equilibrium conditions. Let's break this down step by step to clarify the confusion.
Understanding Torque and Equilibrium
Torque () is defined as the product of the force applied and the distance from the pivot point (or axis of rotation) at which the force is applied. Mathematically, it can be expressed as:
Torque (τ) = r × F
Where is the distance from the pivot point and is the force applied. In your case, it seems you're considering the torque due to a weight (mg) acting at a distance (R).
Analyzing the Situation
When you mention the equation or , you're essentially describing a linear relationship where the torque increases with the distance from the pivot. This is correct when the system is in motion or when a force is applied. However, when the system reaches a state of rest, the dynamics change.
- At Rest: When the worm is at rest, the net torque acting on it is zero. This means that all the torques acting on the system must balance out.
- Torque Equation: If you set up the torque equation at equilibrium, you would have . This implies that any applied torque must be countered by an equal and opposite torque.
Angular Acceleration and Torque
Now, regarding the moment of inertia (M.O.I) and angular acceleration, the relationship is given by:
Torque (τ) = I × α
Where is the moment of inertia and is the angular acceleration. When the system is at rest, as you correctly pointed out, the angular acceleration is zero. Thus, the torque must also be zero:
τ = I × 0 = 0
Reconciling the Two Perspectives
Your initial thought process regarding the torque being a function of is valid, but it applies to the dynamic situation where forces are acting. Once the worm is at rest, the torque indeed becomes zero because the system is in equilibrium. Therefore, the confusion arises from mixing the conditions of motion and rest.
Final Thoughts
In summary, when analyzing the torque in a dynamic situation, you can express it as a function of . However, when the system is at rest, the torque must equal zero, leading to the conclusion that the answer should reflect the equilibrium condition. This is where your reasoning shifted from A to B. Understanding the context of motion versus rest is crucial in these types of problems.
Feel free to ask if you have more questions or need further clarification on any specific point!