Centre of Mass — Prelims Questions

Three particles of masses 1 kg, 2 kg, and 3 kg are placed at the vertices of an equilateral triangle of side 1 m. If the 1 kg mass is at the origin (0,0) and the 2 kg mass is on the x-axis, what are the coordinates of the Centre of Mass?

Four particles of masses 1 kg, 2 kg, 3 kg, and 4 kg are placed at the vertices of a square of side 1 m. The 1 kg mass is at (0,0), 2 kg at (1,0), 3 kg at (1,1), and 4 kg at (0,1). What are the coordinates of the Centre of Mass?

A uniform circular disc of radius $R$ has a circular hole of radius $R/2$ cut out from it. The center of the hole is at a distance $R/2$ from the center of the original disc. Where is the Centre of Mass of the remaining portion?

A system consists of two particles of masses $m_1$ and $m_2$. If the first particle is pushed towards the Centre of Mass by a distance $d$, by what distance should the second particle be moved to keep the Centre of Mass at the same position?

A uniform rod of length $L$ and mass $M$ is bent into a semicircle. What is the distance of its Centre of Mass from the center of the semicircle?

A bomb at rest explodes into two fragments of masses $m_1$ and $m_2$. If the fragments move with velocities $\vec{v}_1$ and $\vec{v}_2$ respectively, what is the velocity of the Centre of Mass of the system immediately after the explosion?

A man of mass $m$ stands at one end of a boat of mass $M$ and length $L$. The boat is initially at rest on a frictionless pond. If the man walks to the other end of the boat, by what distance does the boat move relative to the ground?