Gravitation flash cards

1. 1.State Newton's law of universal gravitation.

   Every particle attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them: $F=\dfrac{Gm_1m_2}{r^2}$, directed along the line joining them.

   Hint: Product of masses over distance squared.

2. 2.What is the value and SI unit of the universal gravitational constant $G$?

   $G=6.67\times10^{-11}\ \text{N m}^2\,\text{kg}^{-2}$. It is a scalar and a universal constant, independent of the medium.

   Hint: Order $10^{-11}$; units N m² kg⁻².

3. 3.What are the dimensions of $G$?

   $[G]=[M^{-1}L^{3}T^{-2}]$, obtained from $G=\dfrac{Fr^2}{m_1m_2}$.

   Hint: Rearrange $F=Gm_1m_2/r^2$.

4. 4.Is gravitational force a central force and a conservative force?

   Yes. It acts along the line joining the two masses (central) and the work done by it is path-independent (conservative), so a gravitational potential energy can be defined.

   Hint: Central = along the join; conservative = path independent.

5. 5.How does gravitational force between two masses depend on the medium between them?

   It does not depend on the intervening medium. $G$ is the same everywhere; gravitation cannot be shielded or screened.

   Hint: No screening, unlike electrostatics.

6. 6.State the principle of superposition for gravitation.

   The net gravitational force on a particle due to several masses is the vector sum of the forces due to each mass taken individually: $\vec{F}=\vec{F_1}+\vec{F_2}+\dots$

   Hint: Vector-add individual pair forces.

7. 7.How does a uniform spherical shell attract a mass placed outside it?

   It attracts as if its entire mass were concentrated at its centre: $F=\dfrac{GMm}{r^2}$ with $r$ measured from the centre.

   Hint: Outside a shell → point mass at centre.

8. 8.What is the gravitational force on a point mass placed inside a uniform spherical shell?

   Zero. The field inside a uniform shell is zero everywhere, so no net gravitational force acts on a mass inside it.

   Hint: Inside a shell → zero field.

9. 9.Define acceleration due to gravity $g$ and give its relation to Earth's mass and radius.

   $g$ is the acceleration of a freely falling body near Earth's surface: $g=\dfrac{GM}{R^2}$, where $M$ and $R$ are Earth's mass and radius. Value $\approx 9.8\ \text{m s}^{-2}$.

   Hint: $g=GM/R^2$.

10. 10.How is $g$ related to the mean density $\rho$ of the Earth?

    Using $M=\dfrac{4}{3}\pi R^3\rho$, we get $g=\dfrac{4}{3}\pi G R\rho$, so $g\propto R\rho$ for a uniform-density planet.

    Hint: Substitute $M$ in terms of density.

11. 11.Distinguish between $g$ and $G$.

    $G$ is a universal constant (same everywhere, $6.67\times10^{-11}$). $g$ is the acceleration due to gravity, a vector that varies with location, altitude, depth and latitude.

    Hint: Universal vs. local; constant vs. variable.

12. 12.How does $g$ vary with altitude $h$ above Earth's surface (general formula)?

    $g_h=\dfrac{GM}{(R+h)^2}=g\left(\dfrac{R}{R+h}\right)^2$. It decreases as height increases.

    Hint: Replace $R$ by $R+h$.

13. 13.Give the approximate expression for the decrease in $g$ at small altitude $h\ll R$.

    $g_h\approx g\left(1-\dfrac{2h}{R}\right)$, so the fractional decrease is $\dfrac{\Delta g}{g}\approx\dfrac{2h}{R}$.

    Hint: Binomial approximation, factor 2h/R.

14. 14.How does $g$ vary with depth $d$ below Earth's surface (uniform density)?

    $g_d=g\left(1-\dfrac{d}{R}\right)$. It decreases linearly with depth and becomes zero at the centre.

    Hint: Only the inner sphere of radius R−d matters.

15. 15.At what point is $g$ zero, and where is $g$ maximum?

    $g=0$ at the centre of the Earth. $g$ is maximum at the surface (for a uniform-density model) and decreases both above and below the surface.

    Hint: Peak at the surface; zero at centre.

16. 16.Compare the rate of decrease of $g$ with altitude versus with depth for small distances.

    For small $h$: $g_h\approx g(1-2h/R)$; for small $d$: $g_d= g(1-d/R)$. Thus for equal small distances, $g$ falls twice as fast going up as going down.

    Hint: Factor 2 for altitude vs 1 for depth.

17. 17.How does the rotation of the Earth affect $g$ at latitude $\lambda$?

    $g_\lambda=g-R\omega^2\cos^2\lambda$, where $\omega$ is Earth's angular speed. The rotation reduces the effective $g$ (except at the poles).

    Hint: Centrifugal term $R\omega^2\cos^2\lambda$.

18. 18.Where is the effect of Earth's rotation on $g$ maximum and where is it zero?

    The reduction $R\omega^2\cos^2\lambda$ is maximum at the equator ($\lambda=0$, $\cos\lambda=1$) and zero at the poles ($\lambda=90^\circ$).

    Hint: cos²λ is 1 at equator, 0 at poles.

19. 19.Why is $g$ greater at the poles than at the equator?

    Two reasons: (1) the equatorial radius is larger than the polar radius, and (2) rotation reduces $g$ at the equator by $R\omega^2$. Both make $g_\text{pole}>g_\text{equator}$.

    Hint: Shape (oblate) + rotation.

20. 20.By roughly what fraction would $g$ at the equator change if Earth stopped rotating?

    It would increase by $R\omega^2\cos^2\lambda$; at the equator this term is $R\omega^2\approx0.034\ \text{m s}^{-2}$, i.e. $g$ increases by about $0.034\ \text{m s}^{-2}$.

    Hint: Add back $R\omega^2$ at the equator.

21. 21.At what angular speed would bodies at the equator become weightless?

    When $g=R\omega^2$, i.e. $\omega=\sqrt{g/R}\approx1.24\times10^{-3}\ \text{rad s}^{-1}$, giving a day of about $84$ minutes.

    Hint: Set effective g to zero at equator.

22. 22.Define the gravitational field (intensity) at a point.

    It is the gravitational force per unit mass placed at that point: $\vec{E}=\dfrac{\vec{F}}{m}$. Its SI unit is $\text{N kg}^{-1}$ (same as m s⁻²).

    Hint: Force per unit test mass.

23. 23.What is the gravitational field due to a point mass $M$ at distance $r$?

    $E=\dfrac{GM}{r^2}$, directed radially towards the mass. At the surface of Earth this field equals $g$.

    Hint: Same form as g = GM/r².

24. 24.Define gravitational potential $V$ at a point.

    It is the work done per unit mass in bringing a test mass from infinity to that point: $V=\dfrac{W}{m}$. SI unit $\text{J kg}^{-1}$; it is always negative (taking zero at infinity).

    Hint: Work per unit mass from infinity.

25. 25.Write the gravitational potential due to a point mass $M$ at distance $r$.

    $V=-\dfrac{GM}{r}$. It is negative and tends to zero as $r\to\infty$.

    Hint: −GM/r; scalar and negative.

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