Gravitational Fields

Define gravitational potential at a point.

Show that the mass of Mars is $6.4 \times 10^{23}\ \text{kg}$.

Calculate the gravitational potential $\phi$ at the surface of Mars. Give a unit with your answer.

$\phi$ = ______ unit ______

The orbit has a period of $25$ hours.

State what can be deduced from this about the rotation of Mars on its axis.

State one other feature of this orbit.

Define gravitational potential at a point.

Show that the mass of Mars is $6.4 \times 10^{23}\ \text{kg}$.

Calculate the gravitational potential $\phi$ at the surface of Mars. Give a unit with your answer.

$\phi$ = ______ unit ______

The orbit has a period of 25 hours.

State what can be deduced from this about the rotation of Mars on its axis.

State one other feature of this orbit.

Define gravitational field.

Describe the pattern of the field lines outside the planet that represent the gravitational field due to the planet.

Explain why, for small changes in vertical height near the surface of the planet, $g$ may be assumed to be constant.

Determine a value, to three significant figures, for the radius $R$ of the Earth.

$R$ = ______ $\text{m}$

Calculate the gravitational potential at the Earth’s surface. Give a unit with your answer.

gravitational potential = ______ unit ______

Explain why the gravitational potential energy of two point masses is always negative.

Define gravitational field at a point.

By considering the force exerted by the point mass on a test mass of mass $m$ placed at P, derive an equation for the gravitational field strength $g$ at P, in terms of $M$ and $x$. Identify any other symbols you use.

On Fig. 3.1, draw an arrow to indicate the direction of the gravitational field at P.

Point Q is at distance $\frac{x}{2}$ from the point mass, on the opposite side of the mass from P, as shown in Fig. 3.2.

Compare the gravitational field at Q with that at P.

Two identical isolated uniform spheres X and Y each have radius $R$. The centres of the spheres are separated by distance $L$, as shown in Fig. 3.3.

Point P lies on the line joining the centres of X and Y, and is at a variable displacement $x$ from the centre of sphere X.

The gravitational field strength at the surface of each sphere is $g_0$.

On Fig. 3.4, sketch the variation with $x$ of the gravitational field $g$ at point P between $x = R$ and $x = L - R$.

Define gravitational field at a point.

By considering the force exerted by the point mass on a test mass of mass $m$ placed at P, derive an equation for the gravitational field strength $g$ at P, in terms of $M$ and $x$. Identify any other symbols you use.

On Fig. 3.1, draw an arrow to indicate the direction of the gravitational field at P.

Point Q is at distance $\frac{x}{2}$ from the point mass, on the opposite side of the mass from P, as shown in Fig. 3.2.

Compare the gravitational field at Q with that at P.

Two identical isolated uniform spheres X and Y each have radius $R$. The centres of the spheres are separated by distance $L$, as shown in Fig. 3.3.

Point P lies on the line joining the centres of X and Y, and is at a variable displacement $x$ from the centre of sphere X.

The gravitational field strength at the surface of each sphere is $g_0$.

On Fig. 3.4, sketch the variation with $x$ of the gravitational field $g$ at point P between $x = R$ and $x = L - R$.

Explain why the gravitational potential near to a point mass is negative.

Show that the mass of the planet is $8.8 \times 10^{25} \text{ kg}$.

The period of rotation of the planet is 0.72 Earth days.

A satellite in orbit around the planet remains above the same point on the surface of the planet.

Use the mass of the planet in (b)(i) to determine the radius $R$ of the orbit of the satellite.

$R$ = ______ $\text{m}$

The speed of the satellite in (b)(ii) is $8400 \text{ m s}^{-1}$. The mass of the satellite is $1200 \text{ kg}$.

Determine the additional energy required to move the satellite from its orbit to infinity.

energy required = ______ $\text{J}$

Define gravitational potential at a point.

Explain why the gravitational potential at X is negative.

State an expression, in terms of $\Phi$, for the gravitational potential at Y due to the planet.

gravitational potential = ______

Complete Table 1.1 by giving expressions, in terms of some or all of $M$, $R$ and $\Phi$, for the quantities indicated for each of the satellites X and Y.

Table 1.1

Define gravitational potential at a point.

Explain why the gravitational potential at X is negative.

State an expression, in terms of $\Phi$, for the gravitational potential at Y due to the planet.

gravitational potential = ______

Complete Table 1.1 by giving expressions, in terms of some or all of $M$, $R$ and $\Phi$, for the quantities indicated for each of the satellites X and Y.

Table 1.1

State Newton’s law of gravitation.

By reference to forces, explain why the orbit of the satellite is circular.

Use Newton’s law of gravitation to show that $h$ and $T$ are related by

where $G$ is the gravitational constant and $A$ and $B$ are constants that depend on the properties of the planet.

Use the gradient and intercept of the line in Fig. 1.1 to determine values for $A$ and $B$. Give units with your answers.

$A$ = ______ unit ______
$B$ = ______ unit ______

State Newton’s law of gravitation.

By reference to forces, explain why the orbit of the satellite is circular.

Use Newton’s law of gravitation to show that $h$ and $T$ are related by

where $G$ is the gravitational constant and $A$ and $B$ are constants that depend on the properties of the planet.

Use the gradient and intercept of the line in Fig. 1.1 to determine values for $A$ and $B$. Give units with your answers.

$A$ = ______ unit ______
$B$ = ______ unit ______