Center OF Mass formulas

Every formula JEE Advanced expects you to know for Center OF Mass, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 30
Subject: Physics
Exam: JEE Advanced
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Center OF Mass, every formula

30 formulas, typeset and free. Print it, or keep it open beside your practice.

CM of two particles
$r_{c m} = \frac{m _{1} r _{1} + m _{2} r _{2}}{m _{1} + m _{2}}$
CM of N particles
$r_{c m} = \frac{1}{M} i \sum m_{i} r_{i}$
CM of a continuous body
$r_{c m} = \frac{\int r d m}{\int d m}$
Two-particle balance
$m_{1} r_{1} = m_{2} r_{2}$
distances from CM
CM of a half ring
$y_{c m} = \frac{2 R}{π}$
CM of a half disc
$y_{c m} = \frac{4 R}{3 π}$
CM of a hollow hemisphere
$y_{c m} = \frac{R}{2}$
CM of a solid hemisphere
$y_{c m} = \frac{3 R}{8}$
CM of a solid cone
$y_{c m} = \frac{h}{4}$
CM of a hollow cone
$y_{c m} = \frac{h}{3}$
CM of a triangular plate
$y_{c m} = \frac{h}{3}$
at the centroid
Velocity of CM
$v_{c m} = \frac{\sum m _{i} v _{i}}{M}$
Acceleration of CM
$a_{c m} = \frac{F _{e x t}}{M}$
Momentum of a system
$P = M v_{c m}$
Conservation of momentum
$F_{e x t} = 0 \Rightarrow P = constant$
Impulse–momentum theorem
$J = \int F d t = Δ p$
Coefficient of restitution
$e = \frac{velocity of separation}{velocity of approach}$
Elastic collision: v1
$v_{1} = \frac{m _{1} - m _{2}}{m _{1} + m _{2}} u_{1} + \frac{2 m _{2}}{m _{1} + m _{2}} u_{2}$
Elastic collision: v2
$v_{2} = \frac{m _{2} - m _{1}}{m _{1} + m _{2}} u_{2} + \frac{2 m _{1}}{m _{1} + m _{2}} u_{1}$
Equal masses (elastic)
$velocities are interchanged$
Perfectly inelastic velocity
$v_{co m} = \frac{m _{1} u _{1} + m _{2} u _{2}}{m _{1} + m _{2}}$
Loss in KE (perfectly inelastic)
$Δ K = \frac{1}{2} \frac{m _{1} m _{2}}{m _{1} + m _{2}} (u_{1} - u_{2})^{2}$
Reduced mass
$μ = \frac{m _{1} m _{2}}{m _{1} + m _{2}}$
Elastic vs inelastic
$e = 1 elastic, e = 0 perfectly inelastic$
Rebound height after n bounces
$h_{n} = e^{2 n} h_{0}$
Speed after nth rebound
$v_{n} = e^{n} v_{0}$
Total distance before stopping
$H = h_{0} \frac{1 + e ^{2}}{1 - e ^{2}}$
Oblique elastic (equal masses)
$θ + ϕ = 9 0^{\circ}$
scattering angle
Thrust force (variable mass)
$F_{t} = v_{r e l} \frac{d m}{d t}$
Rocket equation (thrust)
$F = μ ∣ v_{r e l} ∣$
μ = burn rate

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Center OF Mass formulas

Center OF Mass, every formula

CM of two particles

CM of N particles

CM of a continuous body

Two-particle balance

CM of a half ring

CM of a half disc

CM of a hollow hemisphere

CM of a solid hemisphere

CM of a solid cone

CM of a hollow cone

CM of a triangular plate

Velocity of CM

Acceleration of CM

Momentum of a system

Conservation of momentum

Impulse–momentum theorem

Coefficient of restitution

Elastic collision: v1

Elastic collision: v2

Equal masses (elastic)

Perfectly inelastic velocity

Loss in KE (perfectly inelastic)

Reduced mass

Elastic vs inelastic

Rebound height after n bounces

Speed after nth rebound

Total distance before stopping

Oblique elastic (equal masses)

Thrust force (variable mass)

Rocket equation (thrust)

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