Variation of g — With altitude, depth, latitude
Gravitation
10
JEE Qs
8%
Hard
75
min
Master the derivation of each variation and the conditions under which approximations are valid to avoid conceptual and calculation errors.
🧮 Key Formulas
✅ Key Points for JEE
- 1Acceleration due to gravity (g) decreases with increasing altitude (h). The exact formula g_h = g(R/(R+h))^2 must be used for large h; the approximation g_h ≈ g(1-2h/R) is valid only when h << R.
- 2Acceleration due to gravity (g) decreases linearly with increasing depth (d) from the surface, becoming zero at the Earth's center. g_d = g(1-d/R).
- 3Due to Earth's rotation, g varies with latitude (λ). g is maximum at the poles (λ=90°, g_pole = g) and minimum at the equator (λ=0°, g_eq = g - Rω^2).
- 4For the same distance from the surface, g decreases more rapidly with altitude than with depth.
- 5The linear approximations for altitude and depth are derived using the binomial approximation (1+x)^n ≈ 1+nx for |x| << 1. Be careful not to use them for large h or d.
⚠️ Common Mistakes
- ✕Confusing the formulas or approximations for variation with altitude and depth.
- ✕Incorrectly applying the approximate formulas (e.g., g_h ≈ g(1-2h/R)) when h is not much smaller than R.
- ✕Errors in the sign or the Rω^2 cos^2(λ) term when calculating variation with latitude, often leading to incorrect conclusions about max/min 'g'.
- ✕Ignoring the effect of Earth's rotation when latitude variation is relevant or assuming 'g' is uniform everywhere on the surface.
📝 Practice Questions
See allQ38.Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : A simple pendulum is taken to a planet of mass and radius, 4 times and 2 times, respectively, than the Earth. The time period of the pendulum remains same on earth and the planet. Reason (R): The mass 2025 (22 Jan Shift 2) JEE Main Previous Year Paper of the pendulum remains unchanged at Earth and the other planet. In the light of the above statements, choose the correct answer from the options given below : (1) (A) is false but (R) is true (2) (A) is true but (R) is false (3) Both (A) and (R) are true and (R) is the correct (4) Both (A) and (R) are true but (R) is NOT the explanation of (A) correct explanation of (A)
Q42.A circular disk of radius R meter and mass M kg is rotating around the axis perpendicular to the disk. An external torque is applied to the disk such that θ(t) = 5t2 −8t, where θ(t) is the angular position of the rotating disc as a function of time t. How much power is delivered by the applied torque, when t = 2 s ? (1) 72MR2 (2) 8MR2 (3) 108MR2 (4) 60MR2
Q37.Earth has mass 8 times and radius 2 times that of a planet. If the escape velocity from the earth is 11.2 km/s, the escape velocity in km/s from the planet will be : (1) 2.8 (2) 11.2 (3) 5.6 (4) 8.4 → sin [ω (t −zc )] (S.I. Units). The
Q35.A small point of mass m is placed at a distance 2R from the centre ' O′ of a big uniform solid sphere of mass M and radius R . The gravitational force on ' m ' due to M is F1 . A spherical part of radius R/3 is removed from the big sphere as shown in the figure and the gravitational force on m due to remaining part of M is found to be F2 . The value of ratio F1 : F2 is (1) 12 : 11 (2) 11 : 10 (3) 12 : 9 (4) 16 : 9
Q41.If a satellite orbiting the Earth is 9 times closer to the Earth than the Moon, what is the time period of rotation of the satellite? Given rotational time period of Moon = 27 days and gravitational attraction between the satellite and the moon is neglected. (1) 27 days (2) 1 day (3) 81 days (4) 3 days
Q49.A satellite of mass M is revolving around earth in a circular orbit at a height of R from earth surface. The 2 3 angular momentum of the satellite is . The value of x is ______ , where M and R are the mass and M√GMRx radius of earth, respectively. ( G is the gravitational constant)
NCERT Chapters
- Class 11 Physics Ch 8: Gravitation