Gravitation — JEE Mains Physics

Formulas (3)

• Newton's law and adding the pulls of several masses · Newton's law of gravitation
  $$F = \frac{Gm_1m_2}{r^{2}} \qquad \vec F_{\text{net}} = \sum_i \vec F_i$$
• Field and potential of a sphere, a shell and a cavity · Uniform sphere of mass M and radius R
  $$r \ge R:\ E = \frac{GM}{r^{2}},\ V = -\frac{GM}{r} \qquad r < R:\ E = \frac{GMr}{R^{3}},\ V = -\frac{GM(3R^{2} - r^{2})}{2R^{3}}$$
• Potential energy of a system and work to rearrange it · Potential energy
  $$U = -\sum_{\text{pairs}} \frac{Gm_im_j}{r_{ij}} \qquad \Delta U_{R \to R+h} = \frac{mgh}{1 + h/R}$$

Watch out for (9)

• Opposite masses leave only their difference
  → Newton's law and adding the pulls of several masses
• A semicircle does not cancel
  → Newton's law and adding the pulls of several masses
• Moving mass between two bodies weakens the pull
  → Newton's law and adding the pulls of several masses
• Zero field inside a shell, not zero potential
  → Field and potential of a sphere, a shell and a cavity
• A cavity ratio is always more than 1
  → Field and potential of a sphere, a shell and a cavity
• Measure the cavity's distance to its own centre
  → Field and potential of a sphere, a shell and a cavity
• Count the diagonals
  → Potential energy of a system and work to rearrange it
• mgh fails when h is comparable to R
  → Potential energy of a system and work to rearrange it
• Maximum potential energy usually means maximum size
  → Potential energy of a system and work to rearrange it

Formulas (2)

• g on the surface: mass, radius and density · Surface gravity
  $$g = \frac{GM}{R^{2}} = \frac{4}{3}\pi G\rho R$$
• g at a height above the surface · g at height h
  $$g_h = \frac{g}{(1 + h/R)^{2}} \approx g\left(1 - \frac{2h}{R}\right)\ \ (h \ll R)$$

Watch out for (5)

• Same density is not same mass
  → g on the surface: mass, radius and density
• A diameter changes in the same ratio as the radius
  → g on the surface: mass, radius and density
• From the centre or from the surface?
  → g at a height above the surface
• A height of one diameter is 2R
  → g at a height above the surface
• The 2h/R rule fails for large heights
  → g at a height above the surface

Formulas (3)

• g at a depth below the surface · g at depth d
  $$g_d = g\left(1 - \frac{d}{R}\right)$$
• Comparing g at a depth with g at a height · Equal g below and above
  $$1 - \frac{d}{R} = \frac{1}{(1 + h/R)^{2}} \quad\Rightarrow\quad d \approx 2h\ \ (h \ll R)$$
• Effect of the earth's spin on g · Effective g at latitude λ
  $$g' = g - \omega^{2}R\cos^{2}\lambda$$

Watch out for (6)

• Depth is linear, height is inverse square
  → g at a depth below the surface
• g is largest at the surface
  → g at a depth below the surface
• d = 2h is only for small distances
  → Comparing g at a depth with g at a height
• Both directions reduce g
  → Comparing g at a depth with g at a height
• No effect at the poles, most at the equator
  → Effect of the earth's spin on g
• The floating day equals a grazing orbit's period
  → Effect of the earth's spin on g

Formulas (2)

• Escape velocity and how it scales between planets · Escape velocity
  $$v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR} = R\sqrt{\frac{8\pi G\rho}{3}}$$
• Energy conservation for launches and falls · Mechanical energy is conserved
  $$\tfrac{1}{2}mv_1^{2} - \frac{GMm}{r_1} = \tfrac{1}{2}mv_2^{2} - \frac{GMm}{r_2} \qquad r_{\max} = \frac{R}{1 - \lambda^{2}}$$

Watch out for (6)

• Equal escape velocity needs equal M/R
  → Escape velocity and how it scales between planets
• With density given, v_e grows with R, not √R
  → Escape velocity and how it scales between planets
• Direction does not matter
  → Escape velocity and how it scales between planets
• A fall from height R gives √(gR), not √(2gR)
  → Energy conservation for launches and falls
• Escape energy is mgR, not ½mgR
  → Energy conservation for launches and falls
• r is from the centre
  → Energy conservation for launches and falls

Formulas (3)

• Orbital speed, period and angular momentum · Circular orbit of radius r
  $$v_o = \sqrt{\frac{GM}{r}} \qquad T = 2\pi\sqrt{\frac{r^{3}}{GM}} \qquad L = m\sqrt{GMr}$$
• Energy of a satellite and the energy to change orbit · Energies in a circular orbit
  $$KE = \frac{GMm}{2r} \quad U = -\frac{GMm}{r} \quad E = -\frac{GMm}{2r} \qquad \Delta E = \frac{GMm}{2}\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$$
• Bodies orbiting each other · Binary system
  $$\frac{Gm_1m_2}{d^{2}} = m_1\omega^{2}r_1 = m_2\omega^{2}r_2 \qquad T = 2\pi\sqrt{\frac{d^{3}}{G(m_1 + m_2)}}$$

Watch out for (7)

• r is R + h
  → Orbital speed, period and angular momentum
• Speed does not depend on the satellite's mass; L does
  → Orbital speed, period and angular momentum
• PE is 2E, not E/2
  → Energy of a satellite and the energy to change orbit
• Higher orbit: more energy, less speed
  → Energy of a satellite and the energy to change orbit
• Launching is not the same as changing orbits
  → Energy of a satellite and the energy to change orbit
• The separation is 2r, not r
  → Bodies orbiting each other
• Two different distances in one equation
  → Bodies orbiting each other

Formulas (2)

• Third law: period and orbit radius · Kepler's third law
  $$T^{2} = \frac{4\pi^{2}}{GM}r^{3} \qquad \frac{T_2}{T_1} = \left(\frac{r_2}{r_1}\right)^{3/2}\sqrt{\frac{M_1}{M_2}}$$
• First and second laws: ellipses and equal areas · Kepler's second law
  $$\frac{dA}{dt} = \frac{L}{2m} = \text{constant} \qquad v_{\max}r_{\min} = v_{\min}r_{\max}$$

Watch out for (5)

• The central mass matters; the planet's mass does not
  → Third law: period and orbit radius
• Height or radius?
  → Third law: period and orbit radius
• Raise the ratio to the power 3/2, not 3
  → Third law: period and orbit radius
• Fastest when nearest
  → First and second laws: ellipses and equal areas
• Constant areal velocity, not constant speed
  → First and second laws: ellipses and equal areas