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Center OF Mass formulas

Master Center OF Mass through 30 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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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
$r_{c m} = \frac{m _{1} r _{1} + m _{2} r _{2}}{m _{1} + m _{2}}$
Q1MCQCM two particles
The CM of masses $1$ kg at $x=0$ and $3$ kg at $x=4$ m is at:
- A
  $x=3$
- B
  $x=2$
- C
  $x=1$
- D
  $x=4$
CM of N particles
$r_{c m} = \frac{1}{M} i \sum m_{i} r_{i}$
Q1MCQCM N particles
The CM position of a system is:
- A
  $\dfrac{1}{M}\sum m_i\vec r_i$
- B
  $\sum\vec r_i$
- C
  $\dfrac{\sum\vec r_i}{n}$
- D
  $M\sum\vec r_i$
CM of a continuous body
$r_{c m} = \frac{\int r d m}{\int d m}$
Q1MCQCM continuous
For a continuous body, the CM is:
- A
  $\dfrac{\int\vec r\,dm}{\int dm}$
- B
  $\int\vec r\,dm$
- C
  $\dfrac{\int dm}{\int\vec r\,dm}$
- D
  $\int dm$
Two-particle balance
$m_{1} r_{1} = m_{2} r_{2}$
distances from CM
Q1MCQTwo-particle balance
The CM of two particles lies:
- A
  closer to the heavier mass
- B
  closer to the lighter mass
- C
  at the midpoint always
- D
  outside the system
CM of a half ring
$y_{c m} = \frac{2 R}{π}$
Q1MCQHalf ring
The CM of a half ring of radius $R$ is at height:
- A
  $\dfrac{2R}{\pi}$
- B
  $\dfrac{4R}{3\pi}$
- C
  $\dfrac{R}{2}$
- D
  $\dfrac{3R}{8}$
CM of a half disc
$y_{c m} = \frac{4 R}{3 π}$
Q1MCQHalf disc
The CM of a half disc of radius $R$ is at height:
- A
  $\dfrac{4R}{3\pi}$
- B
  $\dfrac{2R}{\pi}$
- C
  $\dfrac{R}{2}$
- D
  $\dfrac{3R}{8}$
CM of a hollow hemisphere
$y_{c m} = \frac{R}{2}$
Q1MCQHollow hemisphere
The CM of a hemispherical shell of radius $R$ is at height:
- A
  $\dfrac{R}{2}$
- B
  $\dfrac{3R}{8}$
- C
  $\dfrac{4R}{3\pi}$
- D
  $\dfrac{2R}{\pi}$
CM of a solid hemisphere
$y_{c m} = \frac{3 R}{8}$
Q1MCQSolid hemisphere
The CM of a solid hemisphere of radius $R$ is at height:
- A
  $\dfrac{3R}{8}$
- B
  $\dfrac{R}{2}$
- C
  $\dfrac{4R}{3\pi}$
- D
  $\dfrac{2R}{\pi}$
CM of a solid cone
$y_{c m} = \frac{h}{4}$
Q1MCQSolid cone
The CM of a solid cone of height $h$ is at height (from base):
- A
  $\dfrac{h}{4}$
- B
  $\dfrac{h}{3}$
- C
  $\dfrac{h}{2}$
- D
  $\dfrac{2h}{3}$
CM of a hollow cone
$y_{c m} = \frac{h}{3}$
Q1MCQHollow cone
The CM of a hollow cone of height $h$ is at height (from base):
- A
  $\dfrac{h}{3}$
- B
  $\dfrac{h}{4}$
- C
  $\dfrac{h}{2}$
- D
  $\dfrac{2h}{3}$
CM of a triangular plate
$y_{c m} = \frac{h}{3}$
at the centroid
Q1MCQTriangular plate
The CM of a triangular plate is at:
- A
  its centroid
- B
  a vertex
- C
  the midpoint of a side
- D
  the orthocentre
Velocity of CM
$v_{c m} = \frac{\sum m _{i} v _{i}}{M}$
Acceleration of CM
$a_{c m} = \frac{F _{e x t}}{M}$
Q1MCQAcceleration of CM
The acceleration of the CM equals:
- A
  $\dfrac{\vec F_{ext}}{M}$
- B
  $\vec F_{ext}$
- C
  $\dfrac{\vec F_{int}}{M}$
- D
  $0$ always
Momentum of a system
$P = M v_{c m}$
Q1MCQMomentum of system
The total momentum of a system equals:
- A
  $M\vec v_{cm}$
- B
  $M\vec a_{cm}$
- C
  $\sum m_i$
- D
  $0$
Conservation of momentum
$F_{e x t} = 0 \Rightarrow P = constant$
Q1MCQMomentum conservation
When no external force acts, the system's momentum:
- A
  stays constant
- B
  increases
- C
  decreases
- D
  is zero
Impulse–momentum theorem
$J = \int F d t = Δ p$
Coefficient of restitution
$e = \frac{velocity of separation}{velocity of approach}$
Q1MCQRestitution
The coefficient of restitution is:
- A
  $\dfrac{\text{velocity of separation}}{\text{velocity of approach}}$
- B
  $\dfrac{\text{velocity of approach}}{\text{velocity of separation}}$
- C
  $v_1 v_2$
- D
  $m_1/m_2$
Elastic collision: v1
$v_{1} = \frac{m _{1} - m _{2}}{m _{1} + m _{2}} u_{1} + \frac{2 m _{2}}{m _{1} + m _{2}} u_{2}$
Q1MCQElastic v1
For an elastic head-on collision, $v_1$ is:
- A
  $\dfrac{m_1-m_2}{m_1+m_2}u_1+\dfrac{2m_2}{m_1+m_2}u_2$
- B
  $\dfrac{m_1+m_2}{m_1-m_2}u_1$
- C
  $u_2$
- D
  $\dfrac{m_1u_1+m_2u_2}{m_1+m_2}$
Elastic collision: v2
$v_{2} = \frac{m _{2} - m _{1}}{m _{1} + m _{2}} u_{2} + \frac{2 m _{1}}{m _{1} + m _{2}} u_{1}$
Q1MCQElastic v2
For an elastic head-on collision, $v_2$ is:
- A
  $\dfrac{m_2-m_1}{m_1+m_2}u_2+\dfrac{2m_1}{m_1+m_2}u_1$
- B
  $u_1$
- C
  $\dfrac{m_1u_1+m_2u_2}{m_1+m_2}$
- D
  $\dfrac{m_1-m_2}{m_1+m_2}u_1$
Equal masses (elastic)
$velocities are interchanged$
Q1MCQEqual masses elastic
In an elastic collision of equal masses, the velocities:
- A
  are interchanged
- B
  are equal
- C
  both become zero
- D
  double
Perfectly inelastic velocity
$v_{co m} = \frac{m _{1} u _{1} + m _{2} u _{2}}{m _{1} + m _{2}}$
Loss in KE (perfectly inelastic)
$Δ K = \frac{1}{2} \frac{m _{1} m _{2}}{m _{1} + m _{2}} (u_{1} - u_{2})^{2}$
Q1MCQLoss in KE
The KE loss in a perfectly inelastic collision is:
- A
  $\tfrac12\dfrac{m_1m_2}{m_1+m_2}(u_1-u_2)^{2}$
- B
  $\tfrac12(m_1+m_2)u_1^{2}$
- C
  $0$
- D
  $m_1u_1$
Reduced mass
$μ = \frac{m _{1} m _{2}}{m _{1} + m _{2}}$
Elastic vs inelastic
$e = 1 elastic, e = 0 perfectly inelastic$
Q1MCQe values
A perfectly elastic collision has:
- A
  $e=1$
- B
  $e=0$
- C
  $e=0.5$
- D
  $e>1$
Rebound height after n bounces
$h_{n} = e^{2 n} h_{0}$
Q1MCQRebound height
After $n$ bounces (restitution $e$), a ball dropped from $h_0$ rises to:
- A
  $e^{2n}h_0$
- B
  $e^{n}h_0$
- C
  $e h_0$
- D
  $\dfrac{h_0}{n}$
Speed after nth rebound
$v_{n} = e^{n} v_{0}$
Q1MCQSpeed after rebound
The speed after the $n$th rebound is:
- A
  $e^{n}v_0$
- B
  $e^{2n}v_0$
- C
  $e v_0$
- D
  $\dfrac{v_0}{n}$
Total distance before stopping
$H = h_{0} \frac{1 + e ^{2}}{1 - e ^{2}}$
Q1MCQTotal distance
The total distance travelled by a bouncing ball before stopping is:
- A
  $h_0\dfrac{1+e^{2}}{1-e^{2}}$
- B
  $h_0$
- C
  $\dfrac{h_0}{1-e}$
- D
  $2h_0$
Oblique elastic (equal masses)
$θ + ϕ = 9 0^{\circ}$
scattering angle
Q1MCQOblique elastic
After an elastic oblique collision of equal masses (one at rest), the angle between the two final velocities is:
- A
  $90^{\circ}$
- B
  $60^{\circ}$
- C
  $45^{\circ}$
- D
  $180^{\circ}$
Thrust force (variable mass)
$F_{t} = v_{r e l} \frac{d m}{d t}$
Q1MCQThrust force
The thrust force from a variable-mass system is:
- A
  $\vec v_{rel}\dfrac{dm}{dt}$
- B
  $m\dfrac{dv}{dt}$
- C
  $\dfrac{dm}{dt}$
- D
  $mg$
Rocket equation (thrust)
$F = μ ∣ v_{r e l} ∣$
μ = burn rate
Q1MCQRocket thrust
A rocket ejects gas at rate $\mu$ with relative speed $v_{rel}$. The thrust is:
- A
  $\mu\,v_{rel}$
- B
  $\dfrac{\mu}{v_{rel}}$
- C
  $\mu$
- D
  $v_{rel}$

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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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