Gravitational fields

Gravitational Fields and Motion in a Circle

Learning Objectives

• Define radian, angular displacement, angular speed (ω), period (T) and frequency (f) for uniform circular motion.
• Derive and use the three equivalent forms of centripetal acceleration   \(a_c = \dfrac{v^{2}}{r}= \omega^{2}r = \dfrac{4\pi^{2}r}{T^{2}}\).
• Identify and calculate the different forces that can provide the required centripetal force (tension, friction, normal reaction, gravity, banking).
• Define a gravitational field, write its strength \(g\) and potential \(V\), and sketch field‑line diagrams.
• Apply the equality \(a_c = g\) to satellites, derive orbital speed, period and the condition for a geostationary orbit.
• Spot common misconceptions and avoid typical exam pitfalls.
• Solve a range of Cambridge‑style questions involving circular motion and gravitation.

1. Uniform Circular Motion – Kinematics

1.1 Angular quantities

• Radian: the angle subtended at the centre of a circle by an arc equal in length to the radius. 1 rad = \( \dfrac{\text{arc length}}{r}\). 360° = \(2\pi\) rad.
• Angular displacement \(\theta\) (rad) – change in direction of the radius vector.
• Angular speed \(\displaystyle \omega = \frac{\Delta\theta}{\Delta t}\) (rad s⁻¹).
• Period \(T\) – time for one complete revolution (s). \(\displaystyle T = \frac{2\pi}{\omega}\).
• Frequency \(f\) – revolutions per second (Hz). \(\displaystyle f = \frac{1}{T} = \frac{\omega}{2\pi}\).

1.2 Relationship between linear and angular quantities

\[ v = \frac{2\pi r}{T}= r\omega ,\qquad a_t = \frac{dv}{dt}\;( \text{tangential}),\qquad a_c = \frac{v^{2}}{r}= \omega^{2}r = \frac{4\pi^{2}r}{T^{2}} . \]

1.3 Derivation of the three forms of centripetal acceleration

1. Start from the definition of acceleration for a change in direction: \[ a_c = \frac{v^{2}}{r}. \]
2. Substitute \(v = \omega r\): \[ a_c = (\omega r)^{2}/r = \omega^{2}r . \]
3. Replace \(\omega\) by \(2\pi/T\): \[ a_c = \left(\frac{2\pi}{T}\right)^{2}r = \frac{4\pi^{2}r}{T^{2}} . \]

All three expressions are mathematically identical; choose the one that contains the quantities given in the question.

1.4 Sources of the required centripetal force

Source	Typical example	Force expression (inward)
Tension	Ball on a string	\(T = \dfrac{mv^{2}}{r}\)
Friction	Car on a flat curve	\(F_f = \mu N = \mu mg = \dfrac{mv^{2}}{r}\)
Normal reaction (banked curve)	Road banked at angle θ	\(N\sin\theta = \dfrac{mv^{2}}{r}\) with \(N\cos\theta = mg\)
Gravity	Satellite in orbit	\(\dfrac{GMm}{r^{2}} = \dfrac{mv^{2}}{r}\)

1.5 Example – Car on a banked curve (no friction)

\[ \begin{aligned} N\sin\theta &= \frac{mv^{2}}{r},\\ N\cos\theta &= mg . \end{aligned} \] Eliminate \(N\): \[ v = \sqrt{rg\tan\theta } . \] For \(r=50\;\text{m}\) and \(\theta =30^{\circ}\): \[ v = \sqrt{50\times9.81\times\tan30^{\circ}} \approx 12.0\;\text{m s}^{-1}. \]

2. Gravitational Fields

2.1 Definition

The gravitational field \(\mathbf g\) at a point is the force per unit mass that a test mass would experience there:

\[ \mathbf g = \frac{\mathbf F_g}{m}\qquad\text{(units: m s⁻²)} . \]

2.2 Field strength for a point mass

\[ g = \frac{GM}{r^{2}},\qquad \mathbf g \text{ directed toward the centre of mass}. \]

2.3 Gravitational potential

Potential \(V\) is the work done per unit mass in bringing a test mass from infinity to a distance \(r\):

\[ V = -\frac{GM}{r}\qquad\text{(units: J kg⁻¹)} . \] The field and potential are related by \(\mathbf g = -abla V\).

2.4 Field‑line diagrams

• Lines point radially inward toward the mass.
• Density of lines ∝ field strength (closer → denser).
• For a spherical body the field is the same at any point on a sphere of radius \(r\) (symmetry).

2.5 Example – Gravitational field 400 km above Earth

\[ \begin{aligned} R_E &= 6.37\times10^{6}\;\text{m},\quad h = 4.0\times10^{5}\;\text{m},\\ r &= R_E + h = 6.77\times10^{6}\;\text{m},\\[4pt] g &= \frac{GM_E}{r^{2}} = \frac{3.986\times10^{14}}{(6.77\times10^{6})^{2}} \approx 8.7\;\text{m s}^{-2}. \end{aligned} \]

3. Satellite Motion – Using the Gravitational Field

3.1 Equality of centripetal acceleration and field strength

\[ a_c = \frac{v^{2}}{r} = g = \frac{GM}{r^{2}} . \]

3.2 Orbital speed

\[ v = \sqrt{\frac{GM}{r}} . \]

3.3 Orbital period

\[ \begin{aligned} v &= \frac{2\pi r}{T}\;\Longrightarrow\; T = \frac{2\pi r}{v} = 2\pi\sqrt{\frac{r^{3}}{GM}} . \end{aligned} \]

3.4 Geostationary orbit

A satellite is geostationary when its period equals the Earth’s rotation period (\(T=86400\;\text{s}\)). Setting \(T\) in the period formula gives the required radius: \[ r_{\text{geo}} = \left(\frac{GM_E T^{2}}{4\pi^{2}}\right)^{\!1/3} \approx 4.22\times10^{7}\;\text{m}. \] Altitude above the surface: \[ h_{\text{geo}} = r_{\text{geo}}-R_E \approx 3.58\times10^{7}\;\text{m} \;(35\,800\;\text{km}). \] Orbital speed at this radius: \[ v_{\text{geo}} = \frac{2\pi r_{\text{geo}}}{T} \approx 3.07\times10^{3}\;\text{m s}^{-1}. \]

4. Summary Comparison – Circular Motion vs. Gravitational Field

Quantity	Symbol	Expression	Source / Physical origin	Units
Centripetal acceleration	\(a_c\)	\(\dfrac{v^{2}}{r}= \omega^{2}r = \dfrac{4\pi^{2}r}{T^{2}}\)	Resultant inward acceleration of any circular motion	m s⁻²
Gravitational field strength	g	\(\dfrac{GM}{r^{2}}\)	Mass \(M\) (Earth, planet, star)	m s⁻²
Centripetal force	\(F_c\)	\(\dfrac{mv^{2}}{r}=m\omega^{2}r\)	Provided by tension, friction, normal reaction, or gravity	N
Gravitational force	\(F_g\)	\(\dfrac{GMm}{r^{2}}\)	Interaction between two masses	N
Gravitational potential	V	\(-\dfrac{GM}{r}\)	Work per unit mass from infinity to \(r\)	J kg⁻¹

5. Common Misconceptions (Cambridge Checklist)

• Centripetal force is a new, separate force. It is simply the net inward force supplied by existing interactions (tension, friction, normal reaction, or gravity).
• ‘Centrifugal force’ is a real force. It only appears in a rotating (non‑inertial) reference frame as a fictitious force.
• Gravitational field is constant with height. It follows an inverse‑square law; \(g\) decreases as \(1/r^{2}\).
• Orbital speed depends only on altitude. It also depends on the mass of the central body: \(v=\sqrt{GM/r}\).
• All satellites have the same period. Period varies with orbital radius; only the specific geostationary radius gives a 24 h period.
• Banked‑curve formula \(v=\sqrt{rg\tan\theta}\) works on flat roads. It is valid only when friction is negligible.

6. Worked Practice Questions (Cambridge‑style)

1. Centripetal acceleration of a car Speed \(v = 20\;\text{m s}^{-1}\), curve radius \(r = 50\;\text{m}\). \[ a_c = \frac{v^{2}}{r}= \frac{20^{2}}{50}=8.0\;\text{m s}^{-2}. \] Force required for a 1500 kg car: \[ F_c = ma_c = 1500\times8.0 = 1.2\times10^{4}\;\text{N}. \]
2. Gravitational field 400 km above Earth (repeat of 2.4) \(g \approx 8.7\;\text{m s}^{-2}\). Corresponding orbital speed: \[ v = \sqrt{gr}= \sqrt{8.7\times6.77\times10^{6}} \approx 7.68\times10^{3}\;\text{m s}^{-1}. \]
3. Satellite around Mars \(M_M = 6.42\times10^{23}\;\text{kg}\), orbital radius \(r = 1.0\times10^{7}\;\text{m}\). \[ v = \sqrt{\frac{GM_M}{r}} = \sqrt{\frac{6.67\times10^{-11}\times6.42\times10^{23}}{1.0\times10^{7}}} \approx 3.28\times10^{3}\;\text{m s}^{-1}. \] \[ a_c = \frac{v^{2}}{r}\approx1.08\;\text{m s}^{-2}, \qquad g = \frac{GM_M}{r^{2}} \approx 1.08\;\text{m s}^{-2}. \]
4. Design speed of a banked curve \(r = 80\;\text{m}\), banking angle \(\theta = 20^{\circ}\). \[ v = \sqrt{rg\tan\theta}= \sqrt{80\times9.81\times\tan20^{\circ}} \approx 13.2\;\text{m s}^{-1}. \]
5. Geostationary orbit radius and speed (use \(T=86400\;\text{s}\)). \[ r_{\text{geo}} = \left(\frac{GM_E T^{2}}{4\pi^{2}}\right)^{1/3} \approx 4.22\times10^{7}\;\text{m}, \] \[ v_{\text{geo}} = \frac{2\pi r_{\text{geo}}}{T} \approx 3.07\times10^{3}\;\text{m s}^{-1}. \]
6. Energy of a satellite in circular orbit Show that the total mechanical energy per unit mass is \(\displaystyle \frac{E}{m}= -\frac{GM}{2r}\). Solution: Kinetic energy per unit mass \(= \frac{v^{2}}{2}= \frac{GM}{2r}\). Potential energy per unit mass \(= -\frac{GM}{r}\). Hence \(E/m = \frac{GM}{2r} - \frac{GM}{r}= -\frac{GM}{2r}\). (This result is useful for exam questions on orbital energy.)

7. Suggested Diagrams (to be drawn by the student)

• Top‑down view of uniform circular motion showing radius \(r\), tangential velocity \(\mathbf v\), and inward centripetal acceleration \(\mathbf a_c\).
• Free‑body diagram of a ball on a string (tension providing \(F_c\)).
• Banked curve with forces \(N\), \(mg\) and the horizontal component \(N\sin\theta\) supplying \(F_c\).
• Field‑line diagram for a point mass (lines converging toward the centre, density increasing as \(1/r^{2}\)).
• Satellite in circular orbit: show \(\mathbf g\) directed toward the planet, \(\mathbf v\) tangent to the orbit, and the equality \(a_c=g\).

8. Quick Reference Sheet

Symbol	Definition	Formula
\(\theta\)	Angular displacement (rad)	—
\(\omega\)	Angular speed (rad s⁻¹)	\(\omega = \dfrac{2\pi}{T}=2\pi f\)
\(v\)	Linear (tangential) speed (m s⁻¹)	\(v = r\omega = \dfrac{2\pi r}{T}\)
\(a_c\)	Centripetal acceleration (m s⁻²)	\(\dfrac{v^{2}}{r}= \omega^{2}r = \dfrac{4\pi^{2}r}{T^{2}}\)
\(F_c\)	Centripetal force (N)	\(F_c = ma_c = \dfrac{mv^{2}}{r}\)
\(g\)	Gravitational field strength (m s⁻²)	\(g = \dfrac{GM}{r^{2}}\)
\(V\)	Gravitational potential (J kg⁻¹)	\(V = -\dfrac{GM}{r}\)
\(T\)	Period of revolution (s)	\(T = \dfrac{2\pi r}{v}=2\pi\sqrt{\dfrac{r^{3}}{GM}}\)
\(f\)	Frequency (Hz)	\(f = 1/T\)

9. Checklist for Exam Answers

• State which form of the centripetal‑acceleration equation you are using and why.
• Show clearly how the required inward force is supplied (tension, friction, normal, or gravity).
• When dealing with satellites, start from \(F_c = F_g\) and then substitute \(v = 2\pi r/T\) if the period is required.
• Include units at every step; convert where necessary.
• For field‑line diagrams, label the direction of \(\mathbf g\) and indicate that the density of lines ∝ \(1/r^{2}\).
• Check the final answer against typical magnitude expectations (e.g., \(g\approx9.8\;\text{m s}^{-2}\) at Earth’s surface, orbital speeds of a few km s⁻¹).