Oscillations formulas
Class 11 physics formula sheet for NEET and JEE: the key equations of NCERT chapter 13, 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
1Simple harmonic motion
Restoring force ∝ displacement. Any small oscillation: $\omega^2=\dfrac{\text{restoring force per unit displacement}}{\text{inertia}}$; from energy $U=\tfrac12k_{\text{eff}}x^2$.
$v_{\max}=A\omega$ (mean), $a_{\max}=A\omega^2$ (extremes). $v$ leads $x$ by $\pi/2$, $a$ leads $x$ by $\pi$. $\dfrac{x^2}{A^2}+\dfrac{v^2}{A^2\omega^2}=1$ (ellipse); $v$–$x$: ellipse, $a$–$x$: straight line, slope $-\omega^2$. Two positions: $A^2=\dfrac{v_1^2x_2^2-v_2^2x_1^2}{v_1^2-v_2^2}$, $\omega^2=\dfrac{v_1^2-v_2^2}{x_2^2-x_1^2}$.
| Time from mean to | $A/2$ | $A/\sqrt2$ | $\sqrt3A/2$ | $A$ |
|---|---|---|---|---|
| (fraction of $T$) | $T/12$ | $T/8$ | $T/6$ | $T/4$ |
- From extreme to $A/2$: $T/6$. Average speed over a cycle $\dfrac{4A}T=\dfrac{2A\omega}\pi$. $x=A\sin^2\omega t$: SHM of amplitude $A/2$, angular frequency $2\omega$. $x=a\sin\omega t+b\cos\omega t$: amplitude $\sqrt{a^2+b^2}$.
- SHM is the projection of uniform circular motion on a diameter.
$K=U$ at $x=A/\sqrt2$. KE and PE oscillate with frequency $2f$; their averages over a cycle are $E/2$ each. $U(x)=U_0+\tfrac12kx^2$ near a minimum: $\omega=\sqrt{U''(x_0)/m}$.
Two SHMs of same frequency along one line. Perpendicular with same $\omega$: $\delta=0$ line, $\pi/2$ with equal $A$: circle, otherwise ellipse (Lissajous; ratio 1:2 gives a figure-of-eight).
2Spring systems
Series: $T^2=T_1^2+T_2^2$; parallel: $\dfrac1{T^2}=\dfrac1{T_1^2}+\dfrac1{T_2^2}$. Block between two springs (either side): $k_1+k_2$. Vertical spring: same $T$; stretch at equilibrium $mg/k$, $T=2\pi\sqrt{\delta/g}$.
| Case | Result |
|---|---|
| Spring cut into $n$ equal parts | each $nk$; into lengths $l_1:l_2$: $k_i\propto1/l_i$ |
| Two masses joined by spring (free) | $T=2\pi\sqrt{\mu/k}$, $\mu=\dfrac{m_1m_2}{m_1+m_2}$ |
| Spring with mass $m_s$ | $T=2\pi\sqrt{\dfrac{m+m_s/3}{k}}$ |
| Mass $m$ added: period $T_1\to T_2$ | $k=\dfrac{4\pi^2 m}{T_2^2-T_1^2}$ |
| Spring over a pulley (rolling disc, $I$) | $T=2\pi\sqrt{\dfrac{m+I/R^2}k}$ |
| Rolling cylinder/sphere tied to spring | disc $2\pi\sqrt{3m/2k}$, sphere $2\pi\sqrt{7m/5k}$ |
| Block on smooth incline with spring | same $T$; only equilibrium shifts |
| Upper block on oscillating lower block (friction $\mu$) | no slip if $A\omega^2\le\mu g$ |
| Object on a platform in vertical SHM | leaves at top if $A\omega^2>g$ |
| Mass $m$ placed on block $M$ at mean position | new amplitude $A'=A\sqrt{\dfrac{M}{M+m}}$ (mass dropped on at extreme: $A$ unchanged) |
3 more sections and 8 formulas in the full chapter
- 3Pendulums4 formulas · 1 case table
- 4Other SHM systems1 case table
- 5Damped and forced oscillations4 formulas
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