Centre of Mass & Collisions
Centre of mass of standard shapes, composite bodies, motion of CM, CM frame, and rocket propulsion.
Centre of Mass — Discrete System
→ DerivationPosition of centre of mass of a system of point masses.
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Centre of Mass — Continuous Body
→ DerivationPosition of centre of mass of a continuous mass distribution.
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Centre of Mass of a Uniform Rod
→ DerivationCentre of mass of a uniform rod of length L lies at its midpoint.
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Centre of Mass of a Triangle
→ DerivationCentre of mass of a uniform triangular lamina lies at one-third the height from the base.
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Centre of Mass of a Circular Arc
→ DerivationCM of a circular arc of radius R subtending half-angle α at centre. For semicircular arc: y_cm = 2R/π.
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Centre of Mass of a Semicircular Ring
→ DerivationCentre of mass of a thin semicircular wire of radius R.
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Centre of Mass of a Semicircular Disc
→ DerivationCentre of mass of a uniform solid semicircular disc of radius R.
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Centre of Mass of a Hemispherical Shell
→ DerivationCentre of mass of a thin hollow hemispherical shell of radius R.
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Centre of Mass of a Solid Hemisphere
→ DerivationCentre of mass of a uniform solid hemisphere of radius R.
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Centre of Mass of a Hollow Cone
→ DerivationCentre of mass of a thin hollow conical shell of height h, measured from the base.
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Centre of Mass of a Solid Cone
→ DerivationCentre of mass of a uniform solid cone of height h, measured from the base.
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Centre of Mass of a Composite Body
→ DerivationBreak composite body into standard shapes, find CM of each part, combine. Covers L-shape, T-shape, and similar.
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Negative Mass Method — Excluded Portion
→ DerivationCM when a portion is removed. Treat removed mass as negative. Covers disc with hole, sphere with cavity.
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Velocity of Centre of Mass
→ DerivationVelocity of CM is the mass-weighted average of individual velocities.
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Acceleration of Centre of Mass
→ DerivationThe CM of a system accelerates as if all external forces act on total mass M at the CM.
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Velocity in Centre of Mass Frame
→ DerivationVelocity of a particle in the CM frame. Total momentum in CM frame is always zero.
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Kinetic Energy in CM Frame
→ DerivationTotal KE = KE of CM motion + KE of motion relative to CM. The internal KE is minimum in CM frame.
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Rocket Equation (Tsiolkovsky)
→ DerivationSpeed gained by rocket. u = exhaust speed relative to rocket, m₀ = initial mass, m = current mass.
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Thrust Force on Rocket
→ DerivationForce on rocket due to expulsion of exhaust gas. u = exhaust speed, dm/dt = rate of mass loss (negative).
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