Calculate Gravitational Force Mathematically - Physics
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Identify the SI unit of gravitational force $F$.
Identify the SI unit of gravitational force $F$.
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$\text{N}$. Force is measured in newtons in the SI system.
$\text{N}$. Force is measured in newtons in the SI system.
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What happens to $F$ if one mass doubles from $m_1$ to $2m_1$ (all else constant)?
What happens to $F$ if one mass doubles from $m_1$ to $2m_1$ (all else constant)?
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$F\to^2F$. Force is directly proportional to each mass.
$F\to^2F$. Force is directly proportional to each mass.
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State the formula for gravitational field strength $g$ at distance $r$ from mass $M$.
State the formula for gravitational field strength $g$ at distance $r$ from mass $M$.
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$g=G\frac{M}{r^2}$. Field strength is force per unit mass at that location.
$g=G\frac{M}{r^2}$. Field strength is force per unit mass at that location.
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State the relationship between weight $W$ and gravitational field strength $g$ for mass $m$.
State the relationship between weight $W$ and gravitational field strength $g$ for mass $m$.
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$W=mg$. Weight equals mass times gravitational field strength.
$W=mg$. Weight equals mass times gravitational field strength.
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What happens to $F$ if the separation distance doubles from $r$ to $2r$?
What happens to $F$ if the separation distance doubles from $r$ to $2r$?
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$F\to\frac{F}{4}$. Force varies as $\frac{1}{r^2}$, so doubling $r$ gives $\frac{1}{4}$ the force.
$F\to\frac{F}{4}$. Force varies as $\frac{1}{r^2}$, so doubling $r$ gives $\frac{1}{4}$ the force.
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Calculate weight $W$ for $m=2\ \text{kg}$ in a field $g=10\ \text{m/s}^2$.
Calculate weight $W$ for $m=2\ \text{kg}$ in a field $g=10\ \text{m/s}^2$.
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$20\ \text{N}$. Direct multiplication: $W=2\times10=20$.
$20\ \text{N}$. Direct multiplication: $W=2\times10=20$.
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What is the value of the universal gravitational constant $G$ in SI units?
What is the value of the universal gravitational constant $G$ in SI units?
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$G=6.67\times10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$. Fundamental constant determined experimentally by Cavendish.
$G=6.67\times10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$. Fundamental constant determined experimentally by Cavendish.
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What happens to $F$ if the separation distance is halved from $r$ to $\frac{r}{2}$?
What happens to $F$ if the separation distance is halved from $r$ to $\frac{r}{2}$?
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$F\to^4F$. Halving distance means $r^2\to\frac{r^2}{4}$, so force quadruples.
$F\to^4F$. Halving distance means $r^2\to\frac{r^2}{4}$, so force quadruples.
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Identify the SI unit of distance $r$ used in $F=G\frac{m_1m_2}{r^2}$.
Identify the SI unit of distance $r$ used in $F=G\frac{m_1m_2}{r^2}$.
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$\text{m}$. Distance is measured in meters in SI units.
$\text{m}$. Distance is measured in meters in SI units.
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What is the ratio $\frac{F_2}{F_1}$ if distance changes from $r$ to $3r$ (masses unchanged)?
What is the ratio $\frac{F_2}{F_1}$ if distance changes from $r$ to $3r$ (masses unchanged)?
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$\frac{1}{9}$. Since $F\propto\frac{1}{r^2}$, tripling $r$ gives $\frac{1}{9}$ the force.
$\frac{1}{9}$. Since $F\propto\frac{1}{r^2}$, tripling $r$ gives $\frac{1}{9}$ the force.
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What is the ratio $\frac{F_2}{F_1}$ if both masses double and distance stays the same?
What is the ratio $\frac{F_2}{F_1}$ if both masses double and distance stays the same?
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$4$. Doubling both masses gives $2\times^2=4$ times the force.
$4$. Doubling both masses gives $2\times^2=4$ times the force.
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State the formula for the magnitude of gravitational force between two point masses.
State the formula for the magnitude of gravitational force between two point masses.
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$F=G\frac{m_1m_2}{r^2}$. Newton's law of universal gravitation relates force to masses and inverse square of distance.
$F=G\frac{m_1m_2}{r^2}$. Newton's law of universal gravitation relates force to masses and inverse square of distance.
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Identify the SI unit of mass $m$ used in Newton’s law of gravitation.
Identify the SI unit of mass $m$ used in Newton’s law of gravitation.
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$\text{kg}$. Mass is measured in kilograms in SI units.
$\text{kg}$. Mass is measured in kilograms in SI units.
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What is the SI unit of the gravitational constant $G$?
What is the SI unit of the gravitational constant $G$?
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$\text{N}\cdot\text{m}^2\cdot\text{kg}^{-2}$. Derived from $F=ma$ and dimensional analysis of the formula.
$\text{N}\cdot\text{m}^2\cdot\text{kg}^{-2}$. Derived from $F=ma$ and dimensional analysis of the formula.
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Identify the variable in $F=G\frac{m_1m_2}{r^2}$ that represents center-to-center separation.
Identify the variable in $F=G\frac{m_1m_2}{r^2}$ that represents center-to-center separation.
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$r$. Distance between centers of the two masses.
$r$. Distance between centers of the two masses.
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What happens to $F$ if the distance $r$ between two masses is doubled?
What happens to $F$ if the distance $r$ between two masses is doubled?
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$F\text{ becomes }\frac{1}{4}\text{ as large}$. Force varies as $\frac{1}{r^2}$, so $(2r)^2=4$ in denominator.
$F\text{ becomes }\frac{1}{4}\text{ as large}$. Force varies as $\frac{1}{r^2}$, so $(2r)^2=4$ in denominator.
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What happens to $F$ if the distance $r$ between two masses is halved?
What happens to $F$ if the distance $r$ between two masses is halved?
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$F\text{ becomes }4\text{ times as large}$. Force varies as $\frac{1}{r^2}$, so $(\frac{r}{2})^2=\frac{1}{4}$ in denominator.
$F\text{ becomes }4\text{ times as large}$. Force varies as $\frac{1}{r^2}$, so $(\frac{r}{2})^2=\frac{1}{4}$ in denominator.
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What happens to $F$ if one mass (for example $m_1$) is doubled while $r$ is constant?
What happens to $F$ if one mass (for example $m_1$) is doubled while $r$ is constant?
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$F\text{ doubles}$. Force is directly proportional to each mass.
$F\text{ doubles}$. Force is directly proportional to each mass.
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State the formula for gravitational field strength due to a mass $M$ at distance $r$.
State the formula for gravitational field strength due to a mass $M$ at distance $r$.
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$g=G\frac{M}{r^2}$. Field strength is force per unit mass at distance $r$.
$g=G\frac{M}{r^2}$. Field strength is force per unit mass at distance $r$.
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State the relationship between gravitational force and field strength for a test mass $m$.
State the relationship between gravitational force and field strength for a test mass $m$.
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$F=mg$. Force equals mass times gravitational field strength.
$F=mg$. Force equals mass times gravitational field strength.
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Calculate $F$ on a $3\ \text{kg}$ mass where the gravitational field strength is $g=10\ \text{m}\cdot\text{s}^{-2}$.
Calculate $F$ on a $3\ \text{kg}$ mass where the gravitational field strength is $g=10\ \text{m}\cdot\text{s}^{-2}$.
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$F=30\ \text{N}$. $F=mg=3\times10=30$ N using weight formula.
$F=30\ \text{N}$. $F=mg=3\times10=30$ N using weight formula.
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Which option gives the correct proportionality for Newtonian gravity with distance: $F\propto r$, $F\propto \frac{1}{r}$, or $F\propto \frac{1}{r^2}$?
Which option gives the correct proportionality for Newtonian gravity with distance: $F\propto r$, $F\propto \frac{1}{r}$, or $F\propto \frac{1}{r^2}$?
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$F\propto \frac{1}{r^2}$. Inverse square law: force decreases with square of distance.
$F\propto \frac{1}{r^2}$. Inverse square law: force decreases with square of distance.
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What value of the gravitational constant should you use in SI calculations?
What value of the gravitational constant should you use in SI calculations?
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$G=6.67\times10^{-11}\ \text{N}\cdot\text{m}^2\cdot\text{kg}^{-2}$. Standard value used in SI unit calculations.
$G=6.67\times10^{-11}\ \text{N}\cdot\text{m}^2\cdot\text{kg}^{-2}$. Standard value used in SI unit calculations.
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Calculate $g$ at $r=2R$ from a planet: if $g(R)=g_0$, what is $g(2R)$?
Calculate $g$ at $r=2R$ from a planet: if $g(R)=g_0$, what is $g(2R)$?
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$g(2R)=\frac{g_0}{4}$. Field follows inverse square law: $g∝1/r^2$, so doubling $r$ gives $g/4$.
$g(2R)=\frac{g_0}{4}$. Field follows inverse square law: $g∝1/r^2$, so doubling $r$ gives $g/4$.
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Calculate $F$ for $m_1=2\ \text{kg}$, $m_2=3\ \text{kg}$, $r=1\ \text{m}$.
Calculate $F$ for $m_1=2\ \text{kg}$, $m_2=3\ \text{kg}$, $r=1\ \text{m}$.
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$F\approx^4.00\times10^{-10}\ \text{N}$. Calculate: $F=6.67×10^{-11}×2×3/1^2=4.00×10^{-10}$ N.
$F\approx^4.00\times10^{-10}\ \text{N}$. Calculate: $F=6.67×10^{-11}×2×3/1^2=4.00×10^{-10}$ N.
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Calculate the ratio $\frac{F_2}{F_1}$ if $r$ changes from $r_1$ to $r_2$ with masses constant.
Calculate the ratio $\frac{F_2}{F_1}$ if $r$ changes from $r_1$ to $r_2$ with masses constant.
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$\frac{F_2}{F_1}=\left(\frac{r_1}{r_2}\right)^2$. Ratio of forces equals inverse ratio of distances squared.
$\frac{F_2}{F_1}=\left(\frac{r_1}{r_2}\right)^2$. Ratio of forces equals inverse ratio of distances squared.
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What is the direction of the gravitational force on each mass in a two-body system?
What is the direction of the gravitational force on each mass in a two-body system?
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$\text{Along the line joining the centers, toward the other mass}$. Gravity is always attractive between masses.
$\text{Along the line joining the centers, toward the other mass}$. Gravity is always attractive between masses.
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Find $m_2$ in terms of $F$ using $F=G\frac{m_1m_2}{r^2}$.
Find $m_2$ in terms of $F$ using $F=G\frac{m_1m_2}{r^2}$.
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$m_2=\frac{Fr^2}{Gm_1}$. Isolate $m_2$ by multiplying both sides by $r^2/(Gm_1)$.
$m_2=\frac{Fr^2}{Gm_1}$. Isolate $m_2$ by multiplying both sides by $r^2/(Gm_1)$.
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Find $r$ in terms of $F$ using $F=G\frac{m_1m_2}{r^2}$.
Find $r$ in terms of $F$ using $F=G\frac{m_1m_2}{r^2}$.
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$r=\sqrt{G\frac{m_1m_2}{F}}$. Solve for $r$ by taking square root of rearranged equation.
$r=\sqrt{G\frac{m_1m_2}{F}}$. Solve for $r$ by taking square root of rearranged equation.
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Identify the sign of gravitational potential energy $U$ for two isolated masses at finite $r$.
Identify the sign of gravitational potential energy $U$ for two isolated masses at finite $r$.
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$U<0$. Attractive force means bound system has negative energy.
$U<0$. Attractive force means bound system has negative energy.
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