Overview
This lecture covers the concept of center of mass, how to calculate it for systems of discrete masses, its connection to center of gravity, and its relevance to torque and rotational equilibrium.
Center of Mass Concept
- The center of mass is the average position of mass in an object, weighted by each element's mass.
- It can be located using a coordinate system, typically in x and y dimensions for simplicity.
- For discrete masses, the x-coordinate of the center of mass is given by ( x_{cm} = \frac{1}{M_{total}} \sum_{i=1}^n m_i x_i ).
- The y-coordinate is ( y_{cm} = \frac{1}{M_{total}} \sum_{i=1}^n m_i y_i ).
- The center of mass depends on both the masses and their positions.
Center of Mass vs. Center of Gravity
- In a uniform gravitational field, the center of mass and center of gravity are the same.
- The center of gravity is the point where gravity can be considered to act on an object as a whole.
- Objects tend to rotate around their center of mass when tossed or spun.
Calculation Example
- For two point masses (3 kg at 0 m, 5 kg at 3 m on the x-axis):
- ( x_{cm} = \frac{3 \times 0 + 5 \times 3}{3 + 5} = \frac{15}{8} = 1.875 ) m.
- The center of mass is not always at the midpoint; it depends on distribution._
Rotational Balance and Torque
- An object balances at its center of mass; if supported elsewhere, it will rotate due to torque.
- Torque (( \tau )) is produced by the force of gravity acting at the center of mass, calculated as ( \tau = mgr ).
- When torques on both sides of the pivot are equal and opposite, the object is in rotational equilibrium.
Key Terms & Definitions
- Center of Mass — The weighted average position of mass in a system.
- Center of Gravity — The point where gravity appears to act on an object.
- Torque (( \tau )) — A measure of how a force acting at a distance causes rotation, given by ( \tau = mgr ).
- Equilibrium — State where the sum of torques (rotational forces) is zero.
Action Items / Next Steps
- Practice calculating the center of mass for different mass distributions.
- Understand torque by balancing objects on pivots and identifying the role of the center of mass.