Question

A uniform rod of length \(30.0 \mathrm{cm}\) is bent into the shape of an inverted U. Each of the three sides is of length \(10.0 \mathrm{cm} .\) Find the location, in \(x\) - and \(y\) -coordinates, of the CM as measured from the origin.

Short answer

Answer: The center of mass (CM) for the bent rod can be found at coordinates (6.67 cm, 5 cm).

Step-by-step solution

Label the rods and find their individual centers of mass

First, let's label the three rods: Rod A is the left vertical side, Rod B is the horizontal top part, and Rod C is the right vertical side. As each rod is uniform, the center of mass of each rod is at its midpoint. Here are the CM coordinates for each rod: - For Rod A: CM_A = (0, 5 cm) - For Rod B: CM_B = (5 cm, 10 cm) - For Rod C: CM_C = (10 cm, 5 cm)

Calculate the moment of each individual rod's CM about the x- and y-axes

To find the overall center of mass of the system, we need to compute the moment of each rod's CM about the x and y-axes. The moment of a rod's center of mass about an axis is given by the product of the distance of its CM from the given axis and the mass of the rod. Let m be the mass per unit length of the uniform rod. Thus, the mass of each 10 cm rod is M = 10m. We can now compute the moments of each rod's CM as follows: - For Rod A: - Moment about x-axis: M * CM_A_y = 10m * 5 cm = 50m cm - Moment about y-axis: M * CM_A_x = 10m * 0 cm = 0m cm - For Rod B: - Moment about x-axis: M * CM_B_y = 10m * 10 cm = 100m cm - Moment about y-axis: M * CM_B_x = 10m * 5 cm = 50m cm - For Rod C: - Moment about x-axis: M * CM_C_y = 10m * 5 cm = 50m cm - Moment about y-axis: M * CM_C_x = 10m * 10 cm = 100m cm

Compute the total moment about the x- and y-axes

We can now add up the moments of each rod's CM for the total moment about x and y axes: - Total moment about x-axis: Moment_A_x + Moment_B_x + Moment_C_x = 50m + 100m + 50m = 200m cm - Total moment about y-axis: Moment_A_y + Moment_B_y + Moment_C_y = 0 + 50m + 100m = 150m cm

Calculate the overall center of mass of the system

The overall center of mass of the system is determined by using the moment balance equation: CM_x = (Total moment about x-axis) / (Total mass) CM_y = (Total moment about y-axis) / (Total mass) The total mass of the uniform rod is 30m (10m for each segment). Therefore, we can find the overall CM as follows: CM_x = (200m cm) / (30m) = 20/3 cm ≈ 6.67 cm CM_y = (150m cm) / (30m) = 5 cm Finally, the location of the center of mass (CM) of the bent rod, expressed in x- and y-coordinates, is given by: CM = (6.67 cm, 5 cm)