Gravitation — MHT-CET Physics

Formulas (1)

• Inverse-Square Force, Shells and Spheres · Newton's law and the sphere
  $$F = \frac{Gm_1m_2}{r^2}, \qquad g_{\text{out}} = \frac{GM}{r^2}, \qquad g_{\text{in}} = \frac{GMr}{R^3}$$

Watch out for (2)

• Using GM/r² inside a solid sphere
  → Inverse-Square Force, Shells and Spheres
• Forgetting the shell's mass outside it
  → Inverse-Square Force, Shells and Spheres

Formulas (3)

• Height and Depth · Height and depth
  $$g_h = g\frac{R^2}{(R + h)^2}, \qquad g_d = g\left(1 - \frac{d}{R}\right)$$
• g on Other Planets · Surface gravity
  $$g = \frac{GM}{R^2} = \frac{4}{3}\pi G\rho R$$
• The Earth's Spin and Latitude · Effective g with spin
  $$g_\lambda = g - R\omega^2\cos^2\lambda$$

Watch out for (4)

• Using the small-height formula at large heights
  → Height and Depth
• Reading 'reduced by' as 'reduced to'
  → Height and Depth
• Using g ∝ 1/R² at fixed density
  → g on Other Planets
• Using cos λ instead of cos²λ
  → The Earth's Spin and Latitude

Formulas (2)

• Energy Conservation Between Two Distances · Gravitational potential energy
  $$U = -\frac{GMm}{r}, \qquad \tfrac{1}{2}mv^2 = GMm\left(\frac{1}{r_2} - \frac{1}{r_1}\right)$$
• Escape Velocity and the Height Below It · Escape velocity
  $$v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}, \qquad h = \frac{n^2R}{1 - n^2}$$

Watch out for (4)

• Measuring from the surface instead of the centre
  → Energy Conservation Between Two Distances
• Using mgh far from the earth
  → Energy Conservation Between Two Distances
• Using h = v²/2g for a large launch speed
  → Escape Velocity and the Height Below It
• Scaling escape velocity with √R at fixed density
  → Escape Velocity and the Height Below It

Formulas (2)

• Orbital Speed, Period and Kepler's Third Law · Circular orbit
  $$v = \sqrt{\frac{GM}{r}}, \qquad T = 2\pi\sqrt{\frac{r^3}{GM}}$$
• A Satellite's Energy · Orbital energy
  $$E = -\frac{GMm}{2r} = \frac{U}{2} = -K$$

Watch out for (3)

• Using the height instead of the orbit radius
  → Orbital Speed, Period and Kepler's Third Law
• Taking the separation as the orbit radius
  → Orbital Speed, Period and Kepler's Third Law
• Launching with only the orbit's kinetic energy
  → A Satellite's Energy