Calculate Gravitational Force Mathematically - Physics

1. What happens to $F$ if the distance $r$ between two masses is doubled?

   $F\text{ becomes }\frac{1}{4}\text{ as large}$. Force varies as $\frac{1}{r^2}$, so $(2r)^2=4$ in denominator.

2. Find $m_2$ in terms of $F$ using $F=G\frac{m_1m_2}{r^2}$.

   $m_2=\frac{Fr^2}{Gm_1}$. Isolate $m_2$ by multiplying both sides by $r^2/(Gm_1)$.

3. Calculate weight $W$ for $m=2\ \text{kg}$ in a field $g=10\ \text{m/s}^2$.

   $20\ \text{N}$. Direct multiplication: $W=2\times10=20$.

4. State the relationship between gravitational force and field strength for a test mass $m$.

   $F=mg$. Force equals mass times gravitational field strength.

5. What happens to $F$ if the distance $r$ between two masses is halved?

   $F\text{ becomes }4\text{ times as large}$. Force varies as $\frac{1}{r^2}$, so $(\frac{r}{2})^2=\frac{1}{4}$ in denominator.

6. What happens to $F$ if the separation distance doubles from $r$ to $2r$?

   $F\to\frac{F}{4}$. Force varies as $\frac{1}{r^2}$, so doubling $r$ gives $\frac{1}{4}$ the force.

7. Identify the SI unit of gravitational force $F$.

   $\text{N}$. Force is measured in newtons in the SI system.

8. State the formula for gravitational potential energy of two masses separated by distance $r$.

   $U=-G\frac{m_1m_2}{r}$. Negative sign indicates attractive force; zero at infinite separation.

9. What happens to $F$ if the separation distance is halved from $r$ to $\frac{r}{2}$?

   $F\to^4F$. Halving distance means $r^2\to\frac{r^2}{4}$, so force quadruples.

10. What value of the gravitational constant should you use in SI calculations?

    $G=6.67\times10^{-11}\ \text{N}\cdot\text{m}^2\cdot\text{kg}^{-2}$. Standard value used in SI unit calculations.

11. State the formula for the magnitude of gravitational force between two point masses.

    $F=G\frac{m_1m_2}{r^2}$. Newton's law of universal gravitation relates force to masses and inverse square of distance.

12. Identify the meaning of $r$ in $F=G\frac{m_1m_2}{r^2}$.

    $r=\text{center-to-center distance between the masses}$. Distance measured from center of mass to center of mass.

13. Identify the SI unit of mass $m$ used in Newton's law of gravitation.

    $\text{kg}$. Mass is measured in kilograms in SI units.

14. Identify the variable in $F=G\frac{m_1m_2}{r^2}$ that represents center-to-center separation.

    $r$. Distance between centers of the two masses.

15. What is the ratio $\frac{F_2}{F_1}$ if both masses double and distance stays the same?

    $4$. Doubling both masses gives $2\times^2=4$ times the force.

16. What is the SI unit of the gravitational constant $G$?

    $\text{N}\cdot\text{m}^2\cdot\text{kg}^{-2}$. Derived from $F=ma$ and dimensional analysis of the formula.

17. Calculate $F$ for $m_1=2\ \text{kg}$, $m_2=3\ \text{kg}$, $r=1\ \text{m}$.

    $F\approx^4.00\times10^{-10}\ \text{N}$. Calculate: $F=6.67×10^{-11}×2×3/1^2=4.00×10^{-10}$ N.

18. What happens to $F$ if one mass doubles from $m_1$ to $2m_1$ (all else constant)?

    $F\to^2F$. Force is directly proportional to each mass.

19. Find $r$ in terms of $F$ using $F=G\frac{m_1m_2}{r^2}$.

    $r=\sqrt{G\frac{m_1m_2}{F}}$. Solve for $r$ by taking square root of rearranged equation.

20. Calculate $F$ for $m_1=1\ \text{kg}$, $m_2=1\ \text{kg}$, $r=2\ \text{m}$.

    $F\approx^1.67\times10^{-11}\ \text{N}$. Calculate: $F=6.67×10^{-11}×1×1/2^2=1.67×10^{-11}$ N.

21. State the formula for gravitational field strength $g$ at distance $r$ from mass $M$.

    $g=G\frac{M}{r^2}$. Field strength is force per unit mass at that location.

22. Identify the SI unit of distance $r$ used in $F=G\frac{m_1m_2}{r^2}$.

    $\text{m}$. Distance is measured in meters in SI units.

23. Calculate the ratio $\frac{F_2}{F_1}$ if $r$ changes from $r_1$ to $r_2$ with masses constant.

    $\frac{F_2}{F_1}=\left(\frac{r_1}{r_2}\right)^2$. Ratio of forces equals inverse ratio of distances squared.

24. What is the direction of the gravitational force on each mass in a two-body system?

    $\text{Along the line joining the centers, toward the other mass}$. Gravity is always attractive between masses.

25. What is the ratio $\frac{F_2}{F_1}$ if distance changes from $r$ to $3r$ (masses unchanged)?

    $\frac{1}{9}$. Since $F\propto\frac{1}{r^2}$, tripling $r$ gives $\frac{1}{9}$ the force.

26. What is the value of the universal gravitational constant $G$ in SI units?

    $G=6.67\times10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$. Fundamental constant determined experimentally by Cavendish.

27. What happens to $F$ if one mass (for example $m_1$) is doubled while $r$ is constant?

    $F\text{ doubles}$. Force is directly proportional to each mass.

28. Calculate $g$ at $r=2R$ from a planet: if $g(R)=g_0$, what is $g(2R)$?

    $g(2R)=\frac{g_0}{4}$. Field follows inverse square law: $g∝1/r^2$, so doubling $r$ gives $g/4$.

29. Calculate $F$ on a $3\ \text{kg}$ mass where the gravitational field strength is $g=10\ \text{m}\cdot\text{s}^{-2}$.

    $F=30\ \text{N}$. $F=mg=3\times10=30$ N using weight formula.

30. Which option gives the correct proportionality for Newtonian gravity with distance: $F\propto r$, $F\propto \frac{1}{r}$, or $F\propto \frac{1}{r^2}$?

    $F\propto \frac{1}{r^2}$. Inverse square law: force decreases with square of distance.

31. Identify the sign of gravitational potential energy $U$ for two isolated masses at finite $r$.

    $U<0$. Attractive force means bound system has negative energy.

32. State the formula for gravitational field strength due to a mass $M$ at distance $r$.

    $g=G\frac{M}{r^2}$. Field strength is force per unit mass at distance $r$.

33. State the relationship between weight $W$ and gravitational field strength $g$ for mass $m$.

    $W=mg$. Weight equals mass times gravitational field strength.

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