Question

Which one of the following statements correctly expresses the requirement(s) for a system to be in static equilibrium? a) The center of mass of the system must be at rest, but the system can be rotating. b) The center of mass of the system must be at rest or moving with a constant speed. c) The center of mass of the system must be at rest, and the system must not be rotating. d) The system must be rotating about its center of mass. e) The system must be rotating at a constant angular speed around its center of mass, which is at rest.

Short answer

Answer: The correct statement is "The center of mass of the system must be at rest, and the system must not be rotating."

Step-by-step solution

Statement a

In this statement, the center of mass is at rest, but the system is still rotating. Since there is rotational motion involved, the system is not in static equilibrium.

Statement b

According to this statement, the center of mass is either at rest or moving with a constant speed. However, it does not mention any conditions for the rotational motion of the system. Since we need conditions for both linear and angular motion to determine if static equilibrium exists, we cannot choose statement b.

Statement c

Here, it is stated that the center of mass must be at rest and the system must not be rotating. Both linear and angular motion requirements are met - center of mass at rest, and no rotation. Thus, the system is in static equilibrium.

Statement d

In this statement, the system is indeed rotating about its center of mass. As there's rotational motion involved, the system cannot be in static equilibrium.

Statement e

According to statement e, the system is both rotating at a constant angular speed and has its center of mass at rest. Since the system is rotating, it is not in static equilibrium.

Conclusion

Based on our analysis, statement c) "The center of mass of the system must be at rest, and the system must not be rotating." correctly expresses the requirements for a system to be in static equilibrium.

Key concepts

Center of Mass

The center of mass of an object is crucial in understanding its equilibrium and motion. It is essentially the point at which the mass of an object or system is thought to be concentrated. When defining static equilibrium, we consider whether the center of mass is at rest or in motion.

It's important to remember that the center of mass does not actually have to be located within the physical boundary of the object. For instance, the center of mass of a hollow spherical shell, like a basketball, is at its center - a point that contains no actual mass. To achieve static equilibrium, the center of mass must be stable; this means it should either be at rest or moving uniformly in a straight line, with no acceleration in any direction. In other words, no external net force should act upon it, which could result in a change in its state of motion.

In the context of the exercise, statement c correctly identifies that for static equilibrium, the center of mass must not be affected by unbalanced forces or torques, meaning it should be at rest to satisfy equilibrium conditions.

Rotational Motion

Rotational motion refers to the movement of an object around a fixed axis. This is different from translational motion, where an object moves from one place to another without necessarily rotating. An easy way to visualize rotational motion is to think of a spinning top or the rotation of a planet around its axis.

Understanding rotational motion involves a range of physical parameters like angular velocity, angular acceleration, and moment of inertia. It also involves force, but in terms of torque, which is a measure of how much a force acting on an object causes that object to rotate. In terms of static equilibrium, it's essential that there is no net torque acting on the system; this would otherwise cause it to start rotating or to change its rotational speed.

As indicated in step 4 of the solution, a system in static equilibrium cannot be rotating, since rotation implies the presence of angular motion, which would mean the object is not in a state of rest or constant linear motion.

Angular Speed

Angular speed is a measure of the rate at which an object rotates about an axis. It is analogous to linear speed but instead of covering a straight distance, it describes how fast an object travels through an angle. It is most often measured in radians per second.

In scenarios involving rotating objects, even if the center of mass remains stationary, if there is a constant angular speed, there is still kinetic energy involved due to rotational motion. This indicates that the object is not in static equilibrium, as there is an ongoing change in the angular position of points within the system.

Statement e mistakenly suggests that an object can be in static equilibrium if it is rotating at a constant angular speed. However, for static equilibrium, an absence of both linear motion (apart from steady straight-line movement) and angular motion is necessary, underlining why the correct answer relates to the system being entirely at rest, inclusive of zero angular speed.