Question

What is a planar lamina? Describe what is meant by the center of mass \((\bar{x}, \bar{y})\) of a planar lamina.

Short answer

A planar lamina is a 2D object with mass and negligible thickness. The center of mass of a planar lamina \((\bar{x}, \bar{y})\) is the point where the entire mass of the object is considered to be concentrated and can be calculated using the relevant formulas for \(\bar{x}\) and \(\bar{y}\).

Step-by-step solution

Definition of Planar Lamina

A planar lamina is a type of object in physics that is two-dimensional, meaning it has length and width but negligible thickness. These objects also have mass which may be uniformly distributed or vary across its area.

Explanation of the Center of Mass

The center of mass of a planar lamina, or of any object, is the point at which the entire mass of the object is considered to be concentrated. This means that if a force is applied at this point, it will move the entire object without causing it to rotate.

Mathematical Representation of The Center of Mass

The center of mass \((\bar{x}, \bar{y})\) of a planar lamina can be calculated using the formulas \(\bar{x} = \frac{1}{M} \int x dm\) and \(\bar{y} = \frac{1}{M} \int y dm\) where M is the total mass of the lamina and x and y are the coordinates of a differential mass element dm.