Gravitation flash cards

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

   Every particle attracts every other particle with a force along the line joining them: $F = \dfrac{Gm_1m_2}{r^2}$, directed along the line of centres. It is always attractive and acts as an action–reaction pair.

   Hint: Force $\propto$ product of masses, $\propto 1/r^2$.

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 universal constant, independent of the medium and the nature of the masses.

   Hint: Same everywhere; dimension $[M^{-1}L^3T^{-2}]$.

3. 3.Write the gravitational force in vector form between two masses.

   $\vec{F}_{21} = -\dfrac{Gm_1m_2}{r^2}\,\hat{r}_{21}$, where $\hat{r}_{21}$ points from 1 to 2. The minus sign shows the force is attractive (opposite to the position vector from the attracting mass).

   Hint: Minus sign = attraction.

4. 4.Is the gravitational force a central force and is it conservative?

   Yes to both. It acts along the line joining the particles (central) and the work done is path-independent, depending only on endpoints (conservative), so a potential energy can be defined.

   Hint: Path-independent work $\Rightarrow$ PE exists.

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

   The net gravitational force on a particle is the vector sum of the forces due to all other particles taken one at a time: $\vec{F}_{net} = \sum_i \vec{F}_i$. Each pair interaction is unaffected by the presence of others.

   Hint: Vector-add pairwise forces.

6. 6.How does a uniform spherical shell attract a mass outside and inside it?

   For a point outside, the shell attracts as if its whole mass were concentrated at its centre. For a point inside, the net gravitational force is zero everywhere.

   Hint: Outside: point mass at centre. Inside: zero.

7. 7.What is acceleration due to gravity $g$ and its relation to $G$, $M$, $R$?

   $g$ 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. Standard $g \approx 9.8\ \text{m s}^{-2}$.

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

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

   With $M = \frac{4}{3}\pi R^3\rho$, $\;g = \dfrac{GM}{R^2} = \dfrac{4}{3}\pi G R\rho$. So $g \propto R\rho$ for a uniform-density sphere.

   Hint: Substitute $M=\frac43\pi R^3\rho$.

9. 9.How does $g$ vary with altitude $h$ above Earth's surface?

   $g_h = \dfrac{GM}{(R+h)^2} = g\left(1+\dfrac{h}{R}\right)^{-2}$. For $h\ll R$: $\;g_h \approx g\left(1-\dfrac{2h}{R}\right)$. Gravity decreases with height.

   Hint: Fractional drop $\approx 2h/R$ for small $h$.

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

    $g_d = g\left(1-\dfrac{d}{R}\right)$. Only the mass of the inner sphere of radius $(R-d)$ contributes; the outer shell exerts zero net force. At the centre $g=0$.

    Hint: $g_d = g(1-d/R)$; centre $\Rightarrow 0$.

11. 11.For small changes, which decreases $g$ faster — going up by $h$ or down by the same $d$?

    Going up: $g$ decreases by fraction $\approx 2h/R$. Going down: it decreases by fraction $d/R$. So for equal small distances, altitude reduces $g$ about twice as fast as depth.

    Hint: Altitude: $2h/R$; depth: $d/R$.

12. 12.Why does $g$ vary with latitude due to Earth's rotation?

    Rotation supplies part of gravity as centripetal force: $g_\lambda = g - \omega^2 R\cos^2\lambda$, where $\lambda$ is latitude. Effective $g$ is largest at the poles and smallest at the equator.

    Hint: $g_\lambda = g-\omega^2R\cos^2\lambda$.

13. 13.By how much is effective $g$ reduced at the equator due to rotation?

    At equator $\lambda=0$: reduction $=\omega^2 R \approx 0.034\ \text{m s}^{-2}$. At the poles ($\lambda=90^\circ$) there is no reduction, so $g_{pole} > g_{equator}$.

    Hint: Max effect at equator, zero at poles.

14. 14.Besides rotation, why is true $g$ smaller at the equator than at the poles?

    Earth is an oblate spheroid: equatorial radius is larger than polar radius. Since $g\propto 1/R^2$, the larger equatorial $R$ gives a smaller $g$, adding to the rotational effect.

    Hint: Equatorial bulge $\Rightarrow$ larger $R$.

15. 15.Define gravitational field intensity $\vec{E}_g$ at a point.

    It is the gravitational force per unit test mass placed at that point: $\vec{E}_g = \dfrac{\vec{F}}{m}$. For a point mass $M$: $E_g = \dfrac{GM}{r^2}$, directed toward $M$. Units: $\text{N kg}^{-1}$.

    Hint: Force per unit mass; $=g$ near Earth.

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

    It is the work done by an external agent per unit mass in bringing a small mass from infinity to that point (without acceleration): $V = \dfrac{W}{m}$. For a point mass: $V = -\dfrac{GM}{r}$. Units: $\text{J kg}^{-1}$.

    Hint: $V=-GM/r$; always negative, zero at infinity.

17. 17.What is the relation between gravitational field and potential?

    The field is the negative gradient of potential: $E_g = -\dfrac{dV}{dr}$. The field points in the direction of decreasing potential.

    Hint: $E=-dV/dr$.

Open the interactive deck for the other 77 cards, with self-grading so the ones you keep missing come back.