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MHT-CET Physics · Formula sheet

Gravitation formulas

8 formulas and 13 common traps for MHT-CET Physics Gravitation, grouped by subtopic.

Full notes with worked examples

Newton's Law and the Field of a Sphere

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Inverse-Square Force, Shells and Spheres

Newton's law and the sphere

F=Gm1m2r2,gout=GMr2,gin=GMrR3F = \frac{Gm_1m_2}{r^2}, \qquad g_{\text{out}} = \frac{GM}{r^2}, \qquad g_{\text{in}} = \frac{GMr}{R^3}

Common traps

Using GM/r² inside a solid sphere

Inside, only the mass closer to the centre pulls, and the field falls to zero at the centre: g = GMr/R³. The ratio of an outside and an inside field is R³/(r₁²r₂).

Forgetting the shell's mass outside it

Inside a shell its own pull cancels, but outside it the shell's mass adds to everything within. Between a sphere and a surrounding shell only the sphere counts; beyond the shell both do.

How g Changes: Height, Depth, Density and Spin

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Height and depth

gh=gR2(R+h)2,gd=g(1−dR)g_h = g\frac{R^2}{(R + h)^2}, \qquad g_d = g\left(1 - \frac{d}{R}\right)

g on Other Planets

Surface gravity

g=GMR2=43πGρRg = \frac{GM}{R^2} = \frac{4}{3}\pi G\rho R

The Earth's Spin and Latitude

Effective g with spin

gλ=g−Rω2cos⁡2λg_\lambda = g - R\omega^2\cos^2\lambda

Common traps

Using the small-height formula at large heights

g(1 − 2h/R) only works for h ≪ R; at h = R it would give −g. Use g R²/(R + h)² whenever h is comparable to R.

Reading 'reduced by' as 'reduced to'

Reduced BY 64% leaves 36%; reduced TO 64% leaves 64%. The two give 2R/3 and R/4.

Using g ∝ 1/R² at fixed density

1/R² holds for a fixed MASS. At a fixed density the mass grows as R³, so g grows as R.

Using cos λ instead of cos²λ

The reduction is Rω²cos²λ: one cos for the smaller circle's radius, one for the component along gravity. At 30° that is 3/4 of Rω², not √3/2.

Gravitational Potential Energy and Escape Velocity

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Energy Conservation Between Two Distances

Gravitational potential energy

U=−GMmr,12mv2=GMm(1r2−1r1)U = -\frac{GMm}{r}, \qquad \tfrac{1}{2}mv^2 = GMm\left(\frac{1}{r_2} - \frac{1}{r_1}\right)

Escape Velocity and the Height Below It

Escape velocity

ve=2GMR=2gR,h=n2R1−n2v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}, \qquad h = \frac{n^2R}{1 - n^2}

Common traps

Measuring from the surface instead of the centre

Potential energy uses the distance from the centre. '3R above the surface' is 4R from the centre, and the answers differ by a factor of two.

Using mgh far from the earth

mgh assumes g does not change. For heights comparable to R, use −GMm/r at both ends.

Using h = v²/2g for a large launch speed

At a third of the escape velocity, v²/2g gives R/9; the true height, with g falling off, is R/8.

Scaling escape velocity with √R at fixed density

v_e = R√(8πGρ/3) is proportional to R when the density is fixed, so twice the radius doubles it.

Satellites, Orbits and Kepler's Laws

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Orbital Speed, Period and Kepler's Third Law

Circular orbit

v=GMr,T=2πr3GMv = \sqrt{\frac{GM}{r}}, \qquad T = 2\pi\sqrt{\frac{r^3}{GM}}

A Satellite's Energy

Orbital energy

E=−GMm2r=U2=−KE = -\frac{GMm}{2r} = \frac{U}{2} = -K

Common traps

Using the height instead of the orbit radius

A satellite at height R above the surface orbits at radius 2R. The period is 2π√((2R)³/gR²), not 2π√(R/g).

Taking the separation as the orbit radius

Two equal masses circling their midpoint are 2r apart. The force uses (2r)², the circular motion uses r.

Launching with only the orbit's kinetic energy

From the surface the satellite must also be lifted. The launch energy is the total orbital energy minus the surface energy −GMm/R, not GMm/2r alone.

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