Centers of mass and centroids practice problems

Find the center of mass of the system: \( \displaystyle m_1 = 1 \) at \( \displaystyle x_1 = 1 \), \( \displaystyle m_2 = 3 \) at \( \displaystyle x_2 = 4 \), \( \displaystyle m_3 = 6 \) at \( \displaystyle x_3 = 0 \), \( \displaystyle m_4 = 5 \) at \( \displaystyle x_4 = -2 \).

Find the center of mass of the system: \( \displaystyle m_1 = 5 \) at \( \displaystyle x_1 = 3 \), \( \displaystyle m_2 = 5 \) at \( \displaystyle x_2 = -3 \).

Find the centroid of the region bounded by \( \displaystyle y = 5 - x \) and \( \displaystyle y = 0 \) for \( \displaystyle 0 \le x \le 2 \).

Find the centroid of the region bounded by \( \displaystyle y = \sqrt{x} \) and \( \displaystyle y = 0 \) for \( \displaystyle 0 \le x \le 1 \).

Find the centroid of the region bounded by \( \displaystyle y = 2 x \) and \( \displaystyle y = x^{2} \).

Find the centroid of the region bounded by \( \displaystyle y = x \) and \( \displaystyle y = x^{2} \).

A rod on \( \displaystyle 0 \le x \le 1 \) has density \( \displaystyle \rho(x) = x + 1 \). Find its center of mass.

A rod on \( \displaystyle 0 \le x \le 2 \) has density \( \displaystyle \rho(x) = x^{2} + 1 \). Find its center of mass.

Find the centroid of the region bounded by \( \displaystyle y = 4 - x^{2} \) and \( \displaystyle y = 0 \) for \( \displaystyle -2 \le x \le 2 \).

Find the centroid of the region bounded by \( \displaystyle y = \sqrt{x} \) and \( \displaystyle y = 0 \) for \( \displaystyle 0 \le x \le 4 \).