Question

Learning a half-circle of wire of length l and mass m is placed on a coordinate system such that, if the circle were complete, its center would be at the origin. The wire semicircle begins on the negative x-axis, crosses the negative y-axis at the wire's midpoint, and ends at the symmetrically located position on the positive x-axis, as shown. Find the position vector cm for the center of mass of the wire in the given coordinate system. Express your answer in terms of the given information, using ij unit-vector notation.

Answer 1

The position vector for the center of mass of a semicircular wire, which is placed on a coordinate system with its center at the origin, is cm = 0(i) + (Ø2l/(\pi^2))(j).

Step-by-step explanation:

To find the position vector for the center of mass of a semicircular wire, we need to utilize the symmetry of the problem. The semicircular wire has a constant linear density, meaning the mass per unit length is the same throughout the wire. Due to the symmetry of the semicircle about the y-axis, the x-coordinate of the center of mass is 0.

The y-coordinate of the center of mass can be calculated using the formula for a semicircle which ends up as Ø = (2R/\pi)(j), where R is the radius of the semicircle. Since we are given the length (l) of the semicircle, we can find the radius by using the relation R = l/\pi.

So, the position vector in ij unit-vector notation is cm = 0(i) + (Ø2l/(\pi^2))(j). This represents the center of mass of the wire in the coordinate system given.