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MHT-CET Physics · Formula sheet

Oscillations formulas

11 formulas and 12 common traps for MHT-CET Physics Oscillations, grouped by subtopic.

Full notes with worked examples

SHM: Displacement, Velocity, Acceleration and Phase

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Velocity, Acceleration and Displacement

SHM velocity

v=ωA2−x2,a=−ω2xv = \omega\sqrt{A^2 - x^2}, \qquad a = -\omega^2 x

Phase, and Time to Reach a Point

Phase

x=Asin⁡(ωt+α),ω=2πTx = A\sin(\omega t + \alpha), \qquad \omega = \frac{2\pi}{T}

Restoring Forces, Detachment and Damping

Damped amplitude

A=A0e−λt,ω′=km−(b2m)2A = A_0e^{-\lambda t}, \qquad \omega' = \sqrt{\frac{k}{m} - \left(\frac{b}{2m}\right)^2}

Common traps

Subtracting the accelerations instead of adding

The two positions are on the same side, so their distance is x₂ − x₁ = (u² − V²)/(a₁ + a₂). Dividing by a₁ − a₂ gives x₁ + x₂ instead.

Using cos when the motion starts at the mean

cos ωt starts at the extreme. A particle released from the mean position follows sin ωt — mixing them swaps T/12 with T/6.

Thinking damping only shrinks the amplitude

Damping also lowers the angular frequency: ω′ = √(k/m − b²/4m²). Adding the damping term, or leaving out the square root, gives the wrong options.

Spring-Mass Systems, Spring Combinations and Other Oscillators

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The Spring-Mass Period

Spring-mass

T=2πmkT = 2\pi\sqrt{\frac{m}{k}}

Springs in Series, in Parallel, and Cut

Combining springs

k∥=k1+k2,1kseries=1k1+1k2k_{\parallel} = k_1 + k_2, \qquad \frac{1}{k_{\text{series}}} = \frac{1}{k_1} + \frac{1}{k_2}

Other Oscillators: Bowls, Floating Blocks, Liquid Columns, Magnets

Liquid column and floating block

Tcolumn=2πM2Adg,Tblock=2πadgT_{\text{column}} = 2\pi\sqrt{\frac{M}{2Adg}}, \qquad T_{\text{block}} = 2\pi\sqrt{\frac{ad}{g}}

Common traps

Scaling the period with the mass itself

T grows as √m. Doubling the mass makes the period √2 times, not twice — the ratios in these questions are always squared.

Putting g into a spring's period

Hanging the spring vertically only moves the equilibrium point down by mg/k; the period is still 2π√(m/k). g enters only when the question gives you the static stretch instead of m and k.

Treating springs on opposite sides as series

A block between two walls stretches one spring and compresses the other by the SAME x, so both forces act on it: the constants add, as in parallel.

Using Adg for a liquid column

Displacing the liquid by y lowers one side by y and raises the other by y, a level difference of 2y. The restoring force is 2Aydg, which puts the 2 inside the square root.

Energy in Simple Harmonic Motion

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Kinetic and Potential Energy at a Point

Energy split

UE=x2A2,KE=1−x2A2\frac{U}{E} = \frac{x^2}{A^2}, \qquad \frac{K}{E} = 1 - \frac{x^2}{A^2}

Total Energy and What It Depends On

Total energy

E=12mω2A2E = \tfrac{1}{2}m\omega^2A^2

Common traps

Putting the energies equal at half the amplitude

At A/2 the potential energy is only a quarter of the total. The half-and-half point is A/√2 ≈ 0.71A.

Forgetting the frequency in the energy

Same amplitude does not mean same energy. Energy goes as ω²A², so a stiffer or shorter oscillator at the same amplitude carries more.

The Simple Pendulum: Period, Effective g, Speed and Tension

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Period and Length

Simple pendulum

T=2πLgT = 2\pi\sqrt{\frac{L}{g}}

Effective g: Lifts, Moving Supports and Orbit

Accelerating support

T′=2πLg±aT' = 2\pi\sqrt{\frac{L}{g \pm a}}

Speed and Tension Along the Swing

Tension along the swing

Tmax⁡=mg(3−2cos⁡θ),Tmin⁡=mgcos⁡θT_{\max} = mg(3 - 2\cos\theta), \qquad T_{\min} = mg\cos\theta

Common traps

Changing the bob to change the period

A heavier or denser bob of the same size leaves T unchanged. Only the length (to the centre of mass) and g matter.

Adding the acceleration in the wrong direction

Accelerating UP presses the bob down harder — larger g_eff, shorter period. Accelerating down (or falling) lightens it — longer period, infinite in free fall.

Taking the minimum tension as mg

At the extreme the bob is still but the string is slanted: only mg cos θ is balanced by the tension. mg is the tension of a pendulum at rest.

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