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A-Level CAIE Physics 13 3 Gravitational Field Of A Point Mass Question Bank

Practice A-Level CAIE Physics 13 3 Gravitational Field Of A Point Mass questions by syllabus topic with past-paper context, marks, difficulty and question previews on Eduninja.

10 matching questions · Open interactive library

Question 1

1

9 marks

Question 1(a)

1(a)

2 marks

Question 1(a)(ii)

1(a)(ii)

Explain, with reference to gravitational field lines, why the gravitational field near the surface of the Earth is approximately constant for small changes in height.

Mediumstructured2 marks

Answer

change in height negligible compared with radius (of Earth) B1 (so) field lines are (effectively) parallel B1

Question 1(b)

1(b)

A large isolated uniform sphere has mass M and radius R. Point P lies on a straight line passing through the centre of the sphere, at a variable displacement x from the centre, as shown in Fig. 1.1. Fig. 1.2 shows the variation with x of the gravitational field g at point P due to the sphere for the values of x for which P is inside the sphere. The magnitude of the gravitational field at the surface of the sphere is Y.

structured7 marks

Question 1(b)(i)

1(b)(i)

Determine an expression for Y in terms of M and R. Identify any other symbols that you use.

Mediumstructured2 marks

Answer

\(Y=G M / R^{2}\) M1 G is the gravitational constant A1

Question 1(b)(ii)

1(b)(ii)

Explain why, at the surface of the sphere, g always has the opposite sign to x.

Mediumstructured2 marks

Answer

gravitational force is (always) attractive or gravitational force (always) acts towards the centre of the sphere B1 force is in opposite direction to displacement or at a point to the right of the centre, force acts to the left or at a point to the left of the centre, force acts to the right B1

Question 1(b)(iii)

1(b)(iii)

Complete Fig. 1.2 to show the variation of g with x for values of x, up to \(\pm 3 R\), for which point P is outside the sphere.

Mediumstructured3 marks

Answer

sketch: smooth curve with decreasing positive gradient, starting at (R,-Y) and reaching 3 R with g still negative or smooth curve with increasing positive gradient, ending at (-R, Y) and reaching -3 R with g still positive B1 both of the above curves, in correct quadrants B1 curve passing through ( \(\pm 2 R, \pm 0.25 Y\) ) and ( ± 3R, \(\pm 0.11 Y\) ) B1

Question 1

1

3 marks

Question 1(b)

1(b)

An isolated uniform conducting sphere has mass M and charge Q. The gravitational field strength at the surface of the sphere is g. The electric field strength at the surface of the sphere is E.

structured3 marks

Question 1(b)(i)

1(b)(i)

Show that where \(\alpha\) is a constant.

Mediumstructured3 marks

Answer

\(g=G M / r^{2}\) M1 \(E=Q / 4 \pi \varepsilon_{0} r^{2}\) M1 algebra showing the elimination of r leading to \(M / Q=\left(1 / 4 \pi G \varepsilon_{0}\right)(g / E)\) A1

Question 1

1

0 marks

Question 1(b)

1(b)

Artemis is a spherical planet that may be assumed to be isolated in space. The variation with distance x from the centre of Artemis of the gravitational potential \(\phi\) is shown in Fig. 1.1.

structured0 marks

Question 1(b)(i)

1(b)(i)

The radius of Artemis is 4800 km . Determine the value of \(\phi\) on the surface of Artemis.

Mediumstructured0 marks

Answer

\(-3.55 \times 10^{7} \mathrm{~J} \mathrm{~kg}^{-1}\) B1

Question 1(b)(iii)

1(b)(iii)

Calculate the gravitational field strength g on the surface of Artemis.

Mediumstructured0 marks

Answer

\(g=\frac{G M}{r^{2}} \quad\) or \(g=-\frac{\phi}{r}\) C1 \[ \begin{aligned} =\frac{6.67 \times 10^{-11} \times 2.55 \times 10^{24}}{4800000^{2}} \text { or }=\frac{3.55 \times 10^{7}}{4800000} =7.4 \mathrm{~N} \mathrm{~kg}^{-1} \end{aligned} \] A1

Question 1

1

3 marks

Question 1(a)

1(a)

2 marks

Question 1(a)(ii)

1(a)(ii)

Use Newton's law of gravitation to show that the gravitational field strength g at a distance r away from a point mass M is given by

Mediumstructured2 marks

Answer

g=F / m C1 \(F=G M m / r^{2}\) and so \[ g=\left[G M m / r^{2}\right] / m=G M / r^{2} \] A1

Question 1(b)

1(b)

The Earth has a mass of \(5.98 \times 10^{24} \mathrm{~kg}\) and a radius of \(6.37 \times 10^{6} \mathrm{~m}\). The Moon has a mass of \(7.35 \times 10^{22} \mathrm{~kg}\) and a radius of \(1.74 \times 10^{6} \mathrm{~m}\). The Earth and the Moon can both be considered as point masses at their centres. Their centres are a distance of \(3.84 \times 10^{8} \mathrm{~m}\) apart.

structured1 marks

Question 1(b)(i)

1(b)(i)

Show that the gravitational field strength at the surface of the Moon due to the mass of the Moon is \(1.62 \mathrm{~N} \mathrm{~kg}^{-1}\).

Mediumstructured1 marks

Answer

\(g=\left(6.67 \times 10^{-11} \times 7.35 \times 10^{22}\right) /\left(1.74 \times 10^{6}\right)^{2}=1.62 \mathrm{~N} \mathrm{~kg}^{-1}\) A1

Question 1

1

3 marks

Question 1(a)

1(a)

2 marks

Question 1(a)(ii)

1(a)(ii)

Use Newton's law of gravitation to show that the gravitational field strength g at a distance r away from a point mass M is given by

Mediumstructured2 marks

Answer

g=F / m C1 \(F=G M m / r^{2}\) and so \[ g=\left[G M m / r^{2}\right] / m=G M / r^{2} \] A1

Question 1(b)

1(b)

The Earth has a mass of \(5.98 \times 10^{24} \mathrm{~kg}\) and a radius of \(6.37 \times 10^{6} \mathrm{~m}\). The Moon has a mass of \(7.35 \times 10^{22} \mathrm{~kg}\) and a radius of \(1.74 \times 10^{6} \mathrm{~m}\). The Earth and the Moon can both be considered as point masses at their centres. Their centres are a distance of \(3.84 \times 10^{8} \mathrm{~m}\) apart.

structured1 marks

Question 1(b)(i)

1(b)(i)

Show that the gravitational field strength at the surface of the Moon due to the mass of the Moon is \(1.62 \mathrm{~N} \mathrm{~kg}^{-1}\).

Mediumstructured1 marks

Answer

\(g=\left(6.67 \times 10^{-11} \times 7.35 \times 10^{22}\right) /\left(1.74 \times 10^{6}\right)^{2}=1.62 \mathrm{~N} \mathrm{~kg}^{-1}\) A1

Question 1

1

An isolated spherical planet has a diameter of \(6.8 \times 10^{6} \mathrm{~m}\). Its mass of \(6.4 \times 10^{23} \mathrm{~kg}\) may be assumed to be a point mass at the centre of the planet.

structured2 marks

Question 1(a)

1(a)

Show that the gravitational field strength at the surface of the planet is \(3.7 \mathrm{Nkg}^{-1}\).

Mediumstructured2 marks

Answer

\(g=G M / R^{2} \quad \mathrm{C} 1\)

Question 1

1

An isolated spherical planet has a diameter of \(6.8 \times 10^{6} \mathrm{~m}\). Its mass of \(6.4 \times 10^{23} \mathrm{~kg}\) may be assumed to be a point mass at the centre of the planet.

structured2 marks

Question 1(a)

1(a)

Show that the gravitational field strength at the surface of the planet is \(3.7 \mathrm{Nkg}^{-1}\).

Mediumstructured2 marks

Answer

\(g=G M / R^{2} \quad \mathrm{C} 1\)

Question 1

1

4 marks

Question 1(c)

1(c)

A spherical planet may be assumed to be an isolated point mass with its mass concentrated at its centre. A small mass m is moving near to, and normal to, the surface of the planet. The mass moves away from the planet through a short distance h. State and explain why the change in gravitational potential energy \(\Delta E_{\mathrm{P}}\) of the mass is given by the expression where g is the acceleration of free fall.

Mediumstructured4 marks

Answer

either force on mass =m g (where g is the acceleration of free fall /gravitational field strength) \(g=G M / r^{2}\) B1 if r @ h, g is constant B1 \(\Delta E_{\mathrm{P}}=\) force × distance moved M1 =m g h A0 or \(\quad \Delta E_{\mathrm{p}}=m \Delta \phi\) \(=\operatorname{GMM}\left(1 / r_{1}-1 / r_{2}\right)=\operatorname{GMm}\left(r_{2}-r_{1}\right) / r_{1} r_{2}\) if \(r_{2} \approx r_{1}\), then \(\left(r_{2}-r_{1}\right)=h\) and \(r_{1} r_{2}=r^{2}\) \(g=G M / r^{2}\) \(\Delta E_{\mathrm{P}}=m g h\)

Question 1

1

gel and soft tissue,

structured2 marks

Question 1(b)

1(b)

An isolated star has radius R. The mass of the star may be considered to be a point mass at the centre of the star. The gravitational field strength at the surface of the star is \(g_{\mathrm{s}}\). On Fig. 1.1, sketch a graph to show the variation of the gravitational field strength of the star with distance from its centre. You should consider distances in the range R to 4 R.

Mediumstructured2 marks

Answer

graph: correct curvature M1 from \(\left(R, 1.0 g_{\mathrm{s}}\right)\) \& at least one other correct point A1

Question 1

1

2.

structured2 marks

Question 1(b)

1(b)

The Earth may be considered to be an isolated sphere of radius R with its mass concentrated at its centre. The variation of the gravitational potential \(\phi\) with distance x from the centre of the Earth is shown in Fig. 1.1. The radius R of the Earth is \(6.4 \times 10^{6} \mathrm{~m}\).

structured3 marks

Question 1(b)(i)

1(b)(i)

By considering the gravitational potential at the Earth's surface, determine a value for the mass of the Earth.

Mediumstructured3 marks

Answer

at \(R, \phi=6.3 \times 10^{7} \mathrm{~J} \mathrm{~kg}^{-1}\) (allow ± 0.1 × 107) B1 \(\phi=G M / R\) \(6.3 \times 10^{7}=\left(6.67 \times 10^{-11} \times M\right) /\left(6.4 \times 10^{6}\right)\) C1 \(M=6.0 \times 10^{24} \mathrm{~kg}\) (allow \(5.95 \rightarrow 6.14\) ) Maximum of 2/3 for any value chosen for \(\phi\) not at R