Chapter 1 — Gravitation

Universal Force Kepler’s Laws Centripetal Force Acceleration due to Gravity Free Fall Escape Velocity

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Can You Recall?

What are the effects of a force acting on an object?
What types of forces are you familiar with?
What do you know about the gravitational force?

Gravitation is a universal force: it acts between any two masses anywhere in the universe.

1) Gravitation — The Idea

Newton’s Insight

Observing a falling apple, Newton reasoned that the Earth attracts the apple towards its centre. The vertical direction at any place is along the line joining that point to Earth’s centre. He extended this idea to the Moon, planets and the Sun: the same attractive force could be guiding celestial motions.

Force & Motion

A force changes the speed and/or the direction of motion. (Recall Newton’s laws.) Celestial motion involves continuous change of direction, hence a continuous force.

2) Circular Motion & Centripetal Force

Try this: Whirl a stone tied to a string in a circle. Your hand provides a pull towards the centre—the centripetal force. Release the string and the stone flies off along the tangent.

Definition The centripetal force is the inward force required to keep an object moving in a circular path of radius $r$ with speed $v$.

$$F_c=\frac{m v^2}{r}$$

For the Moon orbiting Earth (and planets around the Sun), this needed centripetal force is provided by gravity.

3) Kepler’s Laws of Planetary Motion

First Law (Law of Orbits)

Planets move in elliptical orbits with the Sun at one focus.

Ellipse: sum of distances from any point on the curve to the two foci is constant.

Second Law (Law of Areas)

The line joining a planet and the Sun sweeps equal areas in equal times. Planets move faster near perihelion and slower near aphelion.

Third Law (Harmonic Law)

The square of the orbital period is proportional to the cube of the semi-major axis (mean distance $r$).

$$T^2 \propto r^3 \quad\Rightarrow\quad \frac{T^2}{r^3}=K$$

4) Newton’s Universal Law of Gravitation

Statement Every pair of masses $m_1$ and $m_2$, separated by distance $d$, attract each other with a force

$$F=G\,\frac{m_1 m_2}{d^2}$$

$F$ is along the line joining their centres (or centres of mass).
$G$ is the universal gravitational constant. In SI, $[G] = \mathrm{N\,m^2\,kg^{-2}}$.
For spherical bodies, take $d$ as distance between centres.

Why inverse square? Combining uniform circular motion $F_c=mv^2/r$ with Kepler’s third law $T^2\propto r^3$ implies $F\propto 1/r^2$, matching the gravitational law.

Application: Ocean Tides

High and low tides occur mainly due to the Moon’s gravity pulling Earth’s oceans. Places roughly 90° away experience low tide at the same time.

5) Acceleration due to Gravity (g)

Earth’s gravitational force on a mass $m$ at distance $r$ from Earth’s centre (mass $M$):

$$F=\frac{GMm}{r^2},\qquad F=mg \;\Rightarrow\; g=\frac{GM}{r^2}$$

On Earth’s surface ($r=R$):

$$g=\frac{GM}{R^2}\ \approx\ 9.77~\text{to}~9.83~\mathrm{m\,s^{-2}}$$

Variation of g

With latitude (on surface): Earth is oblate; $R$ is slightly smaller at poles → $g$ higher at poles (~9.832 m/s²) and lower at equator (~9.780 m/s²).
With height: $g(h)=\dfrac{GM}{(R+h)^2}$ — decreases with $h$; small change for $h\ll R$.
With depth: $g$ decreases as we go inside Earth (effective mass contributing reduces).
Other worlds: $g$ depends on $M$ and $R$. On Moon, $g\approx \tfrac{1}{6}g_{\oplus}$.

6) Mass and Weight

Mass

Amount of matter (inertia measure). Scalar; SI unit: kg.
Independent of location (same on Earth, Moon, etc.).

Weight

The gravitational force on a body: $W=mg$. Vector (towards Earth’s centre). Varies with $g$.

In everyday speech we often say “weight in kg”, but scientifically that’s mass.

7) Free Fall

When only gravity acts, an object is in free fall. Taking upward as positive, for objects dropped from rest:

$$u=0,\quad a=g,\quad v=gt,\quad s=\tfrac{1}{2}gt^2,\quad v^2=2gs$$

Note: In air, friction (drag) and buoyancy oppose motion; true free fall occurs in vacuum. Galileo’s thought-experiment shows bodies fall together regardless of mass (neglecting air).

8) Gravitational Potential Energy (Concept)

Near Earth for small heights $h$ ($h\ll R$), we often use $U\approx mgh$ with $U=0$ on ground by choice. For large distances, it’s convenient to take $U=0$ at infinity; then the gravitational potential energy of mass $m$ at distance $r=R+h$ from Earth’s centre is

$$U(r)=-\frac{GMm}{r}\qquad (U\to 0 \text{ as } r\to\infty)$$

9) Escape Velocity

Definition The minimum speed required at the surface so that a body can escape Earth’s gravity and reach infinity with zero residual speed.

$$\tfrac{1}{2}m v_{\text{esc}}^2 - \frac{GMm}{R} = 0 \quad\Rightarrow\quad v_{\text{esc}}=\sqrt{\frac{2GM}{R}}$$

For Earth, $v_{\text{esc}}\approx 11.2\,\mathrm{km\,s^{-1}}$ (air resistance neglected).

10) Weightlessness in Space

Astronauts feel “weightless” in orbit not because $g=0$ (it isn’t), but because spacecraft and everything inside are in continuous free fall around Earth (same acceleration), producing apparent weightlessness.

Worked Examples

Example A — Tiny attraction between two people

Given: $m_1=75\,\text{kg},\; m_2=80\,\text{kg},\; d=1\,\text{m},\; G=6.67\times10^{-11}\,\mathrm{N\,m^2\,kg^{-2}}$

$$F=G\frac{m_1 m_2}{d^2} =6.67\times10^{-11}\times\frac{75\times80}{1^2} =4.002\times10^{-7}\,\text{N}$$

Extremely small—overpowered by friction and other forces in daily life.

Example B — Earth’s pull on a person

Given: $M=6\times10^{24}\,\text{kg},\; R=6.4\times10^6\,\text{m},\; m=75\,\text{kg}$

$$F=G\frac{M m}{R^2} =6.67\times10^{-11}\times \frac{(6\times10^{24})\times 75}{(6.4\times10^6)^2} \approx 733\,\text{N}$$

Corresponding acceleration $a=F/m\approx 9.77\,\mathrm{m\,s^{-2}}$; velocity after 1 s from rest: $v\approx 9.77\,\mathrm{m\,s^{-1}}$.

Example C — Free fall from 125 m

Given: $u=0,\; s=125\,\text{m},\; g=10\,\mathrm{m\,s^{-2}}$

$$s=\tfrac{1}{2}gt^2 \Rightarrow 125=5t^2 \Rightarrow t=5\,\text{s},\qquad v=u+gt=50\,\mathrm{m\,s^{-1}}$$

At half the time (2.5 s): $s=\tfrac{1}{2}g t^2=31.25\,\text{m}\Rightarrow$ height above ground $=125-31.25=93.75\,\text{m}.$

Example D — Ball thrown upwards to 4.05 m

Given: Max height $s=4.05\,\text{m},\; g=10\,\mathrm{m\,s^{-2}}$

$$v^2=u^2+2as,\ v=0,\ a=-g\ \Rightarrow\ u=9\,\mathrm{m\,s^{-1}};\quad s=\tfrac{1}{2}gt^2\Rightarrow t_{\text{up}}=0.9\,\text{s},\ t_{\text{total}}=1.8\,\text{s}.$$

Example E — Weight on the Moon

If weight on Earth is 750 N (mass $m$), then using $g_{\text{Moon}}\approx \tfrac{1}{6}g_{\oplus}$:

$$W_{\text{Moon}}\approx \tfrac{1}{6}\times 750\,\text{N}\approx 125\,\text{N}\ (\approx 126.8\,\text{N})$$

Typical Values of g with Height

Place / Height (km)	Approx. $g$ (m/s²)	Remark
Surface (avg), 0	9.8	Reference value
Mount Everest, 8.8	≈ 9.8	Very small change
Max man-made balloon, 36.6	≈ 9.77	Small decrease
Low-Earth Orbit sat., 400	≈ 8.7	Significant drop
Geostationary sat., 35,700	≈ 0.225	Very weak pull

Use Your Brain Power

Is there a gravitational force between you and your friend? If yes, why don’t you move noticeably?
If Earth’s mass doubled and radius halved, what happens to $g$? (Hint: $g\propto M/R^2$).
What would happen if there were no gravity?
Why do satellites not fall to Earth though gravity pulls them inward?

Great Scientist: Sir Isaac Newton (1642–1727)

Authored Principia; formulated the laws of motion, universal gravitation, and developed calculus. Built the first reflecting telescope.

Great Scientist: Johannes Kepler (1571–1630)

Using Tycho Brahe’s observations, Kepler discovered the three laws of planetary motion—later derived by Newton from his gravitational theory.

Do you know? Gravitational waves are ripples in space-time predicted by Einstein (1916) and directly detected a century later (LIGO). They open a new window to study the universe.

Quick Formula Sheet

Universal gravitation: $F=G\dfrac{m_1m_2}{d^2}$
Centripetal force: $F_c=\dfrac{mv^2}{r}$
$g$ at distance $r$: $g=\dfrac{GM}{r^2}$, at surface: $g=\dfrac{GM}{R^2}$
Weight: $W=mg$

Free fall (from rest): $v=gt,\ s=\tfrac{1}{2}gt^2,\ v^2=2gs$
Projectile straight up: use $a=-g$
Potential energy (far-field): $U=-\dfrac{GMm}{r}$
Escape velocity: $v_{\text{esc}}=\sqrt{\dfrac{2GM}{R}}$

Chapter Wrap-Up

Gravitation is a universal attraction explained quantitatively by Newton’s inverse-square law and supported by Kepler’s observational laws. Near Earth, it gives rise to the acceleration $g$, governs free-fall and weight, and explains orbital motion, tides, and escape velocity. In orbit, apparent weightlessness is a consequence of continuous free fall.

Chapter 1 — Gravitation: Exercise Solutions

Answer Key With Steps MathJax Equations

All formulas render with MathJax • Styles are scoped (your menu bar stays untouched).

Q1) Rewrite the table by putting connected items in a single row

I (Concept)	II (Unit)	III (Key Property)
Mass	kg	Measure of inertia
Weight	N	Depends on height (varies with $g$)
Acceleration due to gravity	m/s²	Zero at the centre (of the Earth)
Gravitational constant	N·m²/kg²	Same in the entire universe

Q2) Answer the following

a) Mass vs Weight; values on Mars?

Mass ($m$): Amount of matter; scalar; SI unit kg; same everywhere; measure of inertia.
Weight ($W$): Gravitational force on the body; vector; $W=mg$; SI unit N; changes with local $g$.
On Mars: $m$ remains the same; $W$ decreases because $g_{\text{Mars}} < g_{\oplus}$.

b) Definitions

Free fall: Motion only under gravity (no other forces). From rest: $\;v=gt,\; s=\tfrac{1}{2}gt^2,\; v^2=2gs$.
Acceleration due to gravity ($g$): $g=\dfrac{GM}{r^2}$ (towards Earth’s centre). Near surface $g\approx 9.8\,\mathrm{m/s^2}$.
Escape velocity ($v_{\text{esc}}$): Minimum speed at surface to reach infinity with zero speed: $\;v_{\text{esc}}=\sqrt{\dfrac{2GM}{R}}$.
Centripetal force: Inward force to keep circular motion: $\;F_c=\dfrac{mv^2}{r}$.

c) Kepler’s Laws → Newton’s inverse-square

Law of Orbits: Planetary orbits are ellipses with Sun at one focus.
Law of Areas: Equal areas are swept in equal times (variable speed).
Harmonic Law: $T^2 \propto r^3$.

For (nearly) circular orbits: $F_c=\dfrac{mv^2}{r}=\dfrac{m(2\pi r/T)^2}{r}=\dfrac{4\pi^2 m r}{T^2}$. Using $T^2\propto r^3\Rightarrow \dfrac{1}{T^2}\propto \dfrac{1}{r^3}$, we get $F\propto \dfrac{1}{r^2}$. Hence Newton’s gravitational law $F=G\dfrac{Mm}{r^2}$.

d) Equal times up and down

Upward motion with initial speed $u$, acceleration $-g$: at top $v=0$.

$$v = u - gt \Rightarrow 0 = u - g t_{\text{up}} \;\Rightarrow\; t_{\text{up}}=\frac{u}{g}$$

Descending from rest at top: $u=0$, distance $h$, acceleration $+g$.

$$h=\tfrac12 g t_{\text{down}}^2 \quad \text{but also} \quad h=\tfrac{u^2}{2g}\Rightarrow t_{\text{down}}=\frac{u}{g}=t_{\text{up}}.$$

Hence: time to go up equals time to come down.

e) If $g$ suddenly doubles, why is pulling an object along the floor twice as difficult?

Normal reaction $N=mg$. Limiting friction $f=\mu N=\mu mg$. If $g\to 2g$, then $f\to 2\mu mg$: the resistive force doubles. Thus it becomes twice as difficult to pull the object.

Q3) Explain why the value of $g$ is zero at the centre of the Earth.

Inside a spherically symmetric Earth of uniform density, only the mass at radius $r$ contributes to gravity at that radius; outer shells cancel. The enclosed mass $M(r)\propto r^3$, and $g(r)=\dfrac{GM(r)}{r^2}\propto r$. Hence as $r\to 0$ (centre), $g\to 0$.

Q4) If a planet’s period is $T$ at distance $R$, show it becomes $8T$ at distance $2R$.

$$T^2 \propto r^3 \;\Rightarrow\; \frac{T_2^2}{T_1^2}=\frac{(2R)^3}{R^3}=8 \;\Rightarrow\; T_2=\sqrt{8}\,T_1=2\sqrt{2}\,T_1.$$

Note: If your textbook expects $8T$ (not $2\sqrt2\,T$), that would be a misprint. By Kepler’s third law, the correct factor is $2\sqrt{2}\approx 2.828$, not $8$.

Q5) Numerical Problems (Solved with steps)

a) Drop from 5 m takes 5 s → find $g$

$$s=\tfrac12 g t^2 \Rightarrow 5=\tfrac12 g (5)^2 \Rightarrow g=\frac{10}{25}=0.4~\mathrm{m/s^2}.$$

Ans: $0.4~\mathrm{m/s^2}$ ✓

b) $R_A=\tfrac12 R_B$. If $g_B=\tfrac12 g_A$, find $M_B$ in terms of $M_A$.

$$g=\frac{GM}{R^2},\quad \frac{GM_B}{R_B^2}=\frac12\frac{GM_A}{R_A^2} \Rightarrow \frac{M_B}{(2R_A)^2}=\frac12\frac{M_A}{R_A^2}\Rightarrow M_B=2M_A.$$

Ans: $2M_A$ ✓

c) On Earth: $m=5$ kg, $W=49$ N. On Moon ($g_{\text{Moon}}=\tfrac16 g_{\oplus}$)?

Mass unchanged: $5$ kg. Weight: $W_{\text{Moon}}=mg_{\text{Moon}}=5\times \tfrac{9.8}{6}\approx 8.17$ N.

Ans: $5$ kg and $8.17$ N ✓

d) Thrown up to $h=500$ m, $g=10~\mathrm{m/s^2}$: find $u$ and total time

$$v^2=u^2-2gh=0 \Rightarrow u=\sqrt{2gh}=\sqrt{2\times 10\times 500}=100~\mathrm{m/s},$$ $$t_{\text{up}}=\frac{u}{g}=10~\text{s}\;\Rightarrow\; t_{\text{total}}=20~\text{s}.$$

Ans: $u=100$ m/s and $t=20$ s ✓

e) Ball falls from table, reaches ground in $1$ s ($g=10$)

$$v=gt=10~\mathrm{m/s},\qquad s=\tfrac12 gt^2=5~\mathrm{m}.$$

Ans: $v=10$ m/s; table height $=5$ m ✓

f) Force between Earth and Moon

Data: $M_E=6\times10^{24}$ kg, $M_M=7.4\times10^{22}$ kg, $r=3.84\times10^5$ km $=3.84\times10^{8}$ m, $G=6.7\times10^{-11}$.

$$F=G\frac{M_E M_M}{r^2} \approx 6.7\times10^{-11}\times\frac{(6\times10^{24})(7.4\times10^{22})}{(3.84\times10^8)^2} \approx 2\times10^{20}\ \text{N}.$$

Ans: $2\times 10^{20}$ N ✓

g) Mass of the Sun

Data: $F=3.5\times10^{22}$ N, $r=1.5\times10^{11}$ m, $M_E=6\times10^{24}$ kg, $G=6.7\times10^{-11}$.

$$F=G\frac{M_\odot M_E}{r^2}\Rightarrow M_\odot=\frac{F r^2}{G M_E} =\frac{(3.5\times10^{22})(1.5\times10^{11})^2}{(6.7\times10^{-11})(6\times10^{24})} \approx 1.96\times10^{30}\ \text{kg}.$$

Ans: $1.96\times10^{30}$ kg ✓

Extra: Escape Velocity Examples

On the Moon

Data: $M=7.34\times10^{22}$ kg, $R=1.74\times10^6$ m, $G=6.67\times10^{-11}$.

$$v_{\text{esc}}=\sqrt{\frac{2GM}{R}} =\sqrt{\frac{2\cdot 6.67\times10^{-11}\cdot 7.34\times10^{22}}{1.74\times10^6}} \approx 2.37~\text{km/s}.$$

Ans: $2.37\ \text{km/s}$ ✓

On the Earth (reference)

$$v_{\text{esc}}=\sqrt{2gR}\approx \sqrt{2\times 9.8\times 6.4\times10^6}\approx 11.2~\text{km/s}.$$

Final Notes

Spacecraft must launch with speed (or staged boosts) exceeding local escape conditions to break free of Earth’s gravity and set course for the Moon/planets.

All computations rounded sensibly; equations are typeset with MathJax for clarity.

Chapter 1 — Gravitation

Can You Recall?

1) Gravitation — The Idea

Newton’s Insight

Force & Motion

2) Circular Motion & Centripetal Force

3) Kepler’s Laws of Planetary Motion

First Law (Law of Orbits)

Second Law (Law of Areas)

Third Law (Harmonic Law)

4) Newton’s Universal Law of Gravitation

Application: Ocean Tides

5) Acceleration due to Gravity (g)

Variation of g

6) Mass and Weight

Mass

Weight

7) Free Fall

8) Gravitational Potential Energy (Concept)

9) Escape Velocity

10) Weightlessness in Space

Worked Examples

Example A — Tiny attraction between two people

Example B — Earth’s pull on a person

Example C — Free fall from 125 m

Example D — Ball thrown upwards to 4.05 m

Example E — Weight on the Moon

Typical Values of g with Height

Use Your Brain Power

Great Scientist: Sir Isaac Newton (1642–1727)

Great Scientist: Johannes Kepler (1571–1630)

Quick Formula Sheet

Chapter Wrap-Up

Chapter 1 — Gravitation: Exercise Solutions

Q1) Rewrite the table by putting connected items in a single row

Q2) Answer the following

a) Mass vs Weight; values on Mars?

b) Definitions

c) Kepler’s Laws → Newton’s inverse-square

d) Equal times up and down

e) If \(g\) suddenly doubles, why is pulling an object along the floor twice as difficult?

Q3) Explain why the value of \(g\) is zero at the centre of the Earth.

Q4) If a planet’s period is \(T\) at distance \(R\), show it becomes \(8T\) at distance \(2R\).

Q5) Numerical Problems (Solved with steps)

a) Drop from 5 m takes 5 s → find \(g\)

b) \(R_A=\tfrac12 R_B\). If \(g_B=\tfrac12 g_A\), find \(M_B\) in terms of \(M_A\).

c) On Earth: \(m=5\) kg, \(W=49\) N. On Moon (\(g_{\text{Moon}}=\tfrac16 g_{\oplus}\))?

d) Thrown up to \(h=500\) m, \(g=10~\mathrm{m/s^2}\): find \(u\) and total time

e) Ball falls from table, reaches ground in \(1\) s (\(g=10\))

f) Force between Earth and Moon

g) Mass of the Sun

Extra: Escape Velocity Examples

On the Moon

On the Earth (reference)

Final Notes

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