A long,thin rod lies along the x-axis.One end of the rod is located at x = 2.00 m and the other end of the rod is located at x = 6.00 m.The linear mass density of the rod is given by ,where a = 0.100 kg/ and b = 0.200 kg/m.What is the location of the center of mass of the rod?

Question

A long,thin rod lies along the x-axis.One end of the rod is located at x = 2.00 m and the other end of the rod is located at x = 6.00 m.The linear mass density of the rod is given by  ,where a = 0.100 kg/  and b = 0.200 kg/m.What is the location of the center of mass of the rod?

Quizplus · Accepted Answer

The position of the center of mass of an object with variable linear mass density is given by:
x_cm = (1/M) * ∫(x * dm)
where M is the total mass of the object and dm is the linear mass density element.

In this case, M is given by:
M = ∫dm = ∫(ax + b) dx = (1/2)ax^2 + bx | 2 to 6 
M = (1/2)a(6^2 - 2^2) + b(6 - 2) 
M = 1.2 kg

To find x_cm, we need to evaluate:
x_cm = (1/M) * ∫(x * dm) 
x_cm = (1/1.2) * ∫[2 + (x - 2)(0.1)] dx | 2 to 6 
x_cm = (1/1.2) * [2(x - 2) + 0.05(x - 2)^2] | 2 to 6 
x_cm = 4.82 m

Therefore, the location of the center of mass of the rod is at x = 4.82 m, which is answer C.

A Long,thin Rod Lies Along the X-Axis