System of Particles and Rotational Motion formulas
Class 11 physics formula sheet for NEET and JEE: the key equations of NCERT chapter 6, the special cases questions are built on, and diagrams where they help.
By Sreeraj P, M.Sc Physics · 10+ years teaching NEET and JEE
Most used formulasOther formulas and cases
1Centre of mass
Two particles: $d_1=\dfrac{m_2d}{m_1+m_2}$ from $m_1$ (closer to the heavier). $\sum m_i\vec r_i'=0$ about the CM. Mass need not be present at the CM.
Internal forces (explosion, mutual attraction) do not move the CM: exploding shell's CM follows the same parabola; two bodies attracting each other meet at the CM. Atwood: $a_{cm}=\left(\dfrac{m_1-m_2}{m_1+m_2}\right)^2g$.
| Case | Result |
|---|---|
| Masses $m,2m,\dots,nm$ at $x=1,2,\dots,n$ | $\dfrac{2n+1}3$; at $x=1,4,9,\dots$: $\dfrac{n(n+1)}2$ |
| Rod with $\lambda=A+Bx$ / $\lambda\propto x^n$ | $\dfrac{L(3A+2BL)}{3(2A+BL)}$ / $\dfrac{(n+1)L}{n+2}$ |
| Mass $m_1$ moved $d$, keep CM fixed | move $m_2$ by $\dfrac{m_1d}{m_2}$ the other way |
| Man walks $l$ on boat (mass $M$) | boat moves $\dfrac{ml}{M+m}$ back; man moves $\dfrac{Ml}{M+m}$ |
| Disc radius $r$ removed from disc $R$ (centres $d$ apart) | CM shifts $\dfrac{r^2d}{R^2-r^2}$ away; hole touching the edge: $\dfrac{r^2}{R+r}$ (for $R=2r$: $R/6$) |
| Sphere $r$ removed from sphere $R$ | shift $\dfrac{r^3d}{R^3-r^3}$ |
| Mass added / removed at distance $d$ | shift $\dfrac{m_{\text{added}}d}{M+m_{\text{added}}}$ / $\dfrac{m_{\text{removed}}d}{M-m_{\text{removed}}}$ |
| Stacked identical blocks (length $l$), max overhang | $n$-th from top overhangs $\dfrac l{2n}$ |
| Body | CM from centre / base | Body | CM |
|---|---|---|---|
| Semicircular ring | $\dfrac{2R}\pi$ | Semicircular disc | $\dfrac{4R}{3\pi}$ |
| Arc of angle $\alpha$ | $\dfrac{2R\sin(\alpha/2)}\alpha$ | Quarter ring | $\dfrac{2\sqrt2R}\pi$ from centre |
| Hemispherical shell | $\dfrac R2$ | Solid hemisphere | $\dfrac{3R}8$ |
| Solid cone | $\dfrac h4$ from base | Hollow cone (surface) | $\dfrac h3$ from base |
| Triangle | centroid | Semicircular annulus | $\dfrac{4(R_1^2+R_1R_2+R_2^2)}{3\pi(R_1+R_2)}$ |
2Torque, equilibrium, toppling
Couple: $\tau=Fd$ (rotation without translation). Equilibrium: $\sum\vec F=0$ and $\sum\vec\tau=0$ (about any point). Lever: load × load arm = effort × effort arm; MA $=\dfrac{d_{\text{effort}}}{d_{\text{load}}}$. Torque about $\vec r_2$ of force at $\vec r_1$: $(\vec r_1-\vec r_2)\times\vec F$.
| Case | Result |
|---|---|
| Ladder on smooth wall, rough floor (angle $\theta$ with floor) | slips unless $\mu\ge\dfrac{\cot\theta}2$; wall force $=$ friction $=\dfrac{mg}2\cot\theta$ |
| Force $F$ at height $b$ on block (base $2a$) | topples about front edge if $Fb>mga$ |
| Cube pushed at top edge | topples before sliding if $\mu>\frac12$; pushed at $\frac34$ height: $F_{\min}=\frac23mg$ |
| Cylinder (radius $r$, height $h$) on tilting incline | slides first if $\mu<\dfrac{2r}h$; topples first if $\mu>\dfrac{2r}h$ |
| Rod held by strings at ends, one cut | other tension becomes $\dfrac{mg}4$, $\alpha=\dfrac{3g}{2l}$ |
4 more sections and 19 formulas in the full chapter
- 3Moment of inertia3 formulas · 1 case table
- 4Rotational dynamics and energy8 formulas · 1 case table
- 5Angular momentum3 formulas · 1 case table
- 6Rolling5 formulas · 1 case table · 1 diagram
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