Extended Bodies at Rest

Center of Mass

The center of mass of an object is the point at which the mass of the system can be considered to be concentrated for the purposes of analyzing translational motion. For a system of particles, the center of mass is determined by the weighted average of their positions.

• Definition: The center of mass is the point where the object balances in all directions.

• Formula (Discrete Particles):

• Key Point: Particles with higher mass contribute more to the location of the center of mass than those with lower mass.

• Application: The center of mass is crucial in analyzing the motion of extended bodies and systems of particles.

Example 1: 1D Center of Mass Problem

The sun has a mass of kg, while Jupiter has a mass of about kg. If Jupiter is about km from the sun, the approximate radius of the sun's motion about the solar system's center of mass due to its interaction with Jupiter can be calculated using the center of mass formula. Compare this to the radius of the sun, which is about 700,000 km.

Example 2: Center of Mass of Three Masses

Three masses (300 g, 200 g, and 100 g) are connected by massless, rigid rods forming a triangle. The coordinates of the center of mass relative to mass A can be found using the above formulas.

Additional info: The center of mass coordinates can be calculated as follows:

Torque

Torque measures the effectiveness of a force in causing an object to rotate about a pivot point. It is a vector quantity and depends on the magnitude of the force, the distance from the pivot, and the angle at which the force is applied.

• Definition: Torque () is the rotational equivalent of force.

• Formula:

• Key Point: The direction of torque is determined by the right-hand rule; counterclockwise (CCW) torque is considered positive.

• Example: When pedaling a bicycle, your foot exerts a torque on the crank, causing it to rotate.

Rigid-Body Model

The rigid-body model is an idealization in which an extended object maintains its size and shape as it moves. This model simplifies the analysis of rotational and translational motion.

• Definition: A rigid body does not deform under applied forces.

• Limitation: The model fails if the object changes shape or is deformed.

Static Equilibrium

Static equilibrium occurs when an object is at rest and both the net force and net torque acting on it are zero. This ensures that the object does not undergo translational or rotational motion.

• Translational Equilibrium: The sum of all external forces is zero.

• Rotational Equilibrium: The sum of all external torques about any axis is zero.

• Application: Used to analyze structures, beams, and objects at rest, such as a meter stick balanced on a fulcrum or a ladder leaning against a wall.

Example: Meter Stick with Attached Masses

Masses are attached to a uniform meter stick (mass = 150.0 g) at specified positions. The force at the fulcrum when the system is balanced can be found by applying the conditions for static equilibrium.

Example: Ladder Against a Wall

A ladder of length 2 m and mass 2 kg leans against a frictional wall at an angle of 60°. The maximum distance a person (mass 100 kg) can climb without the ladder slipping can be determined using static equilibrium and frictional force analysis.

Additional info: For such problems, consider both the forces and torques acting on the system, including gravitational force, normal force, friction, and tension.