Topic 6: Circular motion and gravitation

A satellite X of mass m orbits the Earth with a period T. What will be the orbital period of satellite Y of mass 2m occupying the same orbit as X?

A. $\frac{T}{2}$

B. T

C. $\sqrt{2T}$

D. 2T

Determine the orbital period for the satellite.

Mass of Earth = 6.0 x 10²⁴ kg

An object of mass m at the end of a string of length r moves in a vertical circle at a constant angular speed ω.

What is the tension in the string when the object is at the bottom of the circle?

A.     m(ω²r + g)

B.     m(ω²r – g)

C.     mg(ω²r + 1)

D.     mg(ω²r – 1)

Explain why the electron moves at constant speed.

Explain why the electron moves on a circular path.

State the direction of the resultant force on the ball.

An object of mass m moves in a horizontal circle of radius r with a constant speed v. What is the rate at which work is done by the centripetal force?

A.     $\frac{m{v}^3}{r}$

B.     $\frac{m{v}^3}{2\pi r}$

C.     $\frac{m{v}^3}{4\pi r}$

D.     zero

State the direction of the resultant force on the ball.

Show that the speed v of an electron in the hydrogen atom is related to the radius r of the orbit by the expression

$$v=\sqrt{\frac{k{e}^2}{m_{\text{e}}r}}$$

where k is the Coulomb constant.

A mass m attached to a string of length R moves in a vertical circle with a constant speed. The tension in the string at the top of the circle is T. What is the kinetic energy of the mass at the top of the circle?

A.   $\frac{R\left(T+mg\right)}{2}$

B.   $\frac{R\left(T-mg\right)}{2}$

C.   $\frac{Rmg}{2}$

D.   $\frac{R\left(2T+mg\right)}{2}$

For this proton, calculate, in s, the time for one full revolution.

Satellite X orbits a planet with orbital radius R. Satellite Y orbits the same planet with orbital radius 2R. Satellites X and Y have the same mass.

What is the ratio $\frac{\text{centripetal acceleration of X}}{\text{centripetal acceleration of Y}}$?

A. $\frac{1}{4}$

B. $\frac{1}{2}$

C. 2

D. 4

Outline why this force does no work on the Moon.

Outline why this force does no work on Phobos.

A particle of mass 0.02 kg moves in a horizontal circle of diameter 1 m with an angular velocity of 3$\pi$ rad s^-1. 

What is the magnitude and direction of the force responsible for this motion?

Object P moves vertically with simple harmonic motion (shm). Object Q moves in a vertical circle with a uniform speed. P and Q have the same time period T. When P is at the top of its motion, Q is at the bottom of its motion.

What is the interval between successive times when the acceleration of P is equal and opposite to the acceleration of Q?

A. $\frac{T}{4}$

B. $\frac{T}{2}$

C. $\frac{3T}{4}$

D. T

An object hangs from a light string and moves in a horizontal circle of radius r.

The string makes an angle θ with the vertical. The angular speed of the object is ω. What is tan θ?

A. $\frac{{\omega }^{2}r}{g}$

B. $\frac{g}{{\omega }^{2}r}$

C. $\frac{\omega r^2}{g}$

D. $\frac{g}{\omega r^2}$

An object of mass m makes n revolutions per second around a circle of radius r at a constant speed. What is the kinetic energy of the object?

A. 0

B. $\frac{1}{2}{\pi }^{2}m{n}^2{r}^2$

C. $2{\pi }^{2}m{n}^2{r}^2$

D. $4{\pi }^{2}m{n}^2{r}^2$

Calculate the time for one complete revolution.

Explain why the kinetic energy of the proton is constant.

Mass $M$ is attached to one end of a string. The string is passed through a hollow tube and mass $m$ is attached to the other end. Friction between the tube and string is negligible.

Mass $m$ travels at constant speed $v$ in a horizontal circle of radius $r$. What is mass $M$?

A.  $\frac{m{v}^2}{r}$

B.  $m{v}^{2}rg$

C.  $\frac{mg{v}^2}{r}$

D.  $\frac{m{v}^2}{gr}$

The person must not slide down the wall. Show that the minimum angular velocity $\omega$ of the cylinder for this situation is

$\omega =\sqrt{\frac{g}{\mu R}}$

where $\mu$ is the coefficient of static friction between the person and the cylinder.

The coefficient of static friction between the person and the cylinder is $0.40$. The radius of the cylinder is $3.5 \mathrm{m}$. The cylinder makes $28$ revolutions per minute. Deduce whether the person will slide down the inner surface of the cylinder.

Calculate the value of the centripetal force.

An object moves in a circle of constant radius. Values of the centripetal force $F$ are measured for different values of angular velocity $\omega$. A graph is plotted with $\omega$ on the $x$-axis. Which quantity plotted on the $y$-axis will produce a straight-line graph?

A.  $\sqrt{F}$

B.  $F$

C.  $F^2$

D.  $\frac{1}{F}$

The player kicks the ball again. It rolls along the ground without sliding with a horizontal velocity of $1.40 {\text{m\hspace{0.05em}s}}^{-1}$. The radius of the ball is $0.11 \text{m}$. Calculate the angular velocity of the ball. State an appropriate SI unit for your answer.

A satellite is orbiting Earth in a circular path at constant speed. Three statements about the resultant force on the satellite are:

I.   It is equal to the gravitational force of attraction on the satellite.
II.  It is equal to the mass of the satellite multiplied by its acceleration.
III. It is equal to the centripetal force on the satellite.

Which combination of statements is correct?

A.  I and II only

B.  I and III only

C.  II and III only

D.  I, II and III

The fan is rotating at 120 revolutions every minute. Calculate the centripetal acceleration of the tip of a fan blade.

The radius of the pulley is 2.5 cm. Calculate the angular speed of rotation of the pulley as the load hits the floor. State your answer to an appropriate number of significant figures.

Two masses $\text{M}$ and $\text{m}$ are connected by a string that runs without friction through a stationary tube. Mass $\text{m}$ rotates at constant speed in a horizontal circle of radius 0.25 m. The weight of $\text{M}$ provides the centripetal force for the motion of $\text{m}$. The time period for the rotation of m is 0.50 s.

What is $\frac{\text{mass of M}}{\text{mass of m}}$?

A.  1

B.  2

C.  4

D.  8

The scale diagram shows the weight W of the mass at an instant when the rod is horizontal.

Draw, on the scale diagram, an arrow to represent the force exerted on the mass by the rod.

For this proton, calculate, in s, the time for one full revolution.

For this proton, calculate, in s, the time for one full revolution.

Satellite X orbits a planet with orbital radius R. Satellite Y orbits the same planet with orbital radius 2R. Satellites X and Y have the same mass.

What is the ratio $\frac{\text{centripetal acceleration of X}}{\text{centripetal acceleration of Y}}$?

A. $\frac{1}{4}$

B. $\frac{1}{2}$

C. 2

D. 4

Outline why this force does no work on the Moon.

Outline why this force does no work on the Moon.

Outline why this force does no work on Phobos.

Outline why this force does no work on Phobos.

A particle of mass 0.02 kg moves in a horizontal circle of diameter 1 m with an angular velocity of 3$\pi$ rad s^-1. 

What is the magnitude and direction of the force responsible for this motion?

Object P moves vertically with simple harmonic motion (shm). Object Q moves in a vertical circle with a uniform speed. P and Q have the same time period T. When P is at the top of its motion, Q is at the bottom of its motion.

What is the interval between successive times when the acceleration of P is equal and opposite to the acceleration of Q?

A. $\frac{T}{4}$

B. $\frac{T}{2}$

C. $\frac{3T}{4}$

D. T

An object hangs from a light string and moves in a horizontal circle of radius r.

The string makes an angle θ with the vertical. The angular speed of the object is ω. What is tan θ?

A. $\frac{{\omega }^{2}r}{g}$

B. $\frac{g}{{\omega }^{2}r}$

C. $\frac{\omega r^2}{g}$

D. $\frac{g}{\omega r^2}$

An object of mass m makes n revolutions per second around a circle of radius r at a constant speed. What is the kinetic energy of the object?

A. 0

B. $\frac{1}{2}{\pi }^{2}m{n}^2{r}^2$

C. $2{\pi }^{2}m{n}^2{r}^2$

D. $4{\pi }^{2}m{n}^2{r}^2$

Calculate the time for one complete revolution.

Explain why the kinetic energy of the proton is constant.

Calculate the time for one complete revolution.

Explain why the kinetic energy of the proton is constant.

Mass $M$ is attached to one end of a string. The string is passed through a hollow tube and mass $m$ is attached to the other end. Friction between the tube and string is negligible.

Mass $m$ travels at constant speed $v$ in a horizontal circle of radius $r$. What is mass $M$?

A.  $\frac{m{v}^2}{r}$

B.  $m{v}^{2}rg$

C.  $\frac{mg{v}^2}{r}$

D.  $\frac{m{v}^2}{gr}$

The person must not slide down the wall. Show that the minimum angular velocity $\omega$ of the cylinder for this situation is

$\omega =\sqrt{\frac{g}{\mu R}}$

where $\mu$ is the coefficient of static friction between the person and the cylinder.

The coefficient of static friction between the person and the cylinder is $0.40$. The radius of the cylinder is $3.5 \mathrm{m}$. The cylinder makes $28$ revolutions per minute. Deduce whether the person will slide down the inner surface of the cylinder.

The person must not slide down the wall. Show that the minimum angular velocity $\omega$ of the cylinder for this situation is

$\omega =\sqrt{\frac{g}{\mu R}}$

where $\mu$ is the coefficient of static friction between the person and the cylinder.

The coefficient of static friction between the person and the cylinder is $0.40$. The radius of the cylinder is $3.5 \mathrm{m}$. The cylinder makes $28$ revolutions per minute. Deduce whether the person will slide down the inner surface of the cylinder.

Calculate the value of the centripetal force.

Calculate the value of the centripetal force.

An object moves in a circle of constant radius. Values of the centripetal force $F$ are measured for different values of angular velocity $\omega$. A graph is plotted with $\omega$ on the $x$-axis. Which quantity plotted on the $y$-axis will produce a straight-line graph?

A.  $\sqrt{F}$

B.  $F$

C.  $F^2$

D.  $\frac{1}{F}$

The player kicks the ball again. It rolls along the ground without sliding with a horizontal velocity of $1.40 {\text{m\hspace{0.05em}s}}^{-1}$. The radius of the ball is $0.11 \text{m}$. Calculate the angular velocity of the ball. State an appropriate SI unit for your answer.

The player kicks the ball again. It rolls along the ground without sliding with a horizontal velocity of $1.40 {\text{m\hspace{0.05em}s}}^{-1}$. The radius of the ball is $0.11 \text{m}$. Calculate the angular velocity of the ball. State an appropriate SI unit for your answer.

A satellite is orbiting Earth in a circular path at constant speed. Three statements about the resultant force on the satellite are:

I.   It is equal to the gravitational force of attraction on the satellite.
II.  It is equal to the mass of the satellite multiplied by its acceleration.
III. It is equal to the centripetal force on the satellite.

Which combination of statements is correct?

A.  I and II only

B.  I and III only

C.  II and III only

D.  I, II and III

The fan is rotating at 120 revolutions every minute. Calculate the centripetal acceleration of the tip of a fan blade.

The fan is rotating at 120 revolutions every minute. Calculate the centripetal acceleration of the tip of a fan blade.

The radius of the pulley is 2.5 cm. Calculate the angular speed of rotation of the pulley as the load hits the floor. State your answer to an appropriate number of significant figures.

The radius of the pulley is 2.5 cm. Calculate the angular speed of rotation of the pulley as the load hits the floor. State your answer to an appropriate number of significant figures.

Two masses $\text{M}$ and $\text{m}$ are connected by a string that runs without friction through a stationary tube. Mass $\text{m}$ rotates at constant speed in a horizontal circle of radius 0.25 m. The weight of $\text{M}$ provides the centripetal force for the motion of $\text{m}$. The time period for the rotation of m is 0.50 s.

What is $\frac{\text{mass of M}}{\text{mass of m}}$?

A.  1

B.  2

C.  4

D.  8

The scale diagram shows the weight W of the mass at an instant when the rod is horizontal.

Draw, on the scale diagram, an arrow to represent the force exerted on the mass by the rod.

The scale diagram shows the weight W of the mass at an instant when the rod is horizontal.

Draw, on the scale diagram, an arrow to represent the force exerted on the mass by the rod.

A satellite X of mass m orbits the Earth with a period T. What will be the orbital period of satellite Y of mass 2m occupying the same orbit as X?

A. $\frac{T}{2}$

B. T

C. $\sqrt{2T}$

D. 2T

Determine the orbital period for the satellite.

Mass of Earth = 6.0 x 10²⁴ kg

Determine the orbital period for the satellite.

Mass of Earth = 6.0 x 10²⁴ kg

An object of mass m at the end of a string of length r moves in a vertical circle at a constant angular speed ω.

What is the tension in the string when the object is at the bottom of the circle?

A.     m(ω²r + g)

B.     m(ω²r – g)

C.     mg(ω²r + 1)

D.     mg(ω²r – 1)

Explain why the electron moves at constant speed.

Explain why the electron moves on a circular path.

Explain why the electron moves at constant speed.

Explain why the electron moves on a circular path.

State the direction of the resultant force on the ball.

State the direction of the resultant force on the ball.

An object of mass m moves in a horizontal circle of radius r with a constant speed v. What is the rate at which work is done by the centripetal force?

A.     $\frac{m{v}^3}{r}$

B.     $\frac{m{v}^3}{2\pi r}$

C.     $\frac{m{v}^3}{4\pi r}$

D.     zero

State the direction of the resultant force on the ball.

State the direction of the resultant force on the ball.

Show that the speed v of an electron in the hydrogen atom is related to the radius r of the orbit by the expression

$$v=\sqrt{\frac{k{e}^2}{m_{\text{e}}r}}$$

where k is the Coulomb constant.

Show that the speed v of an electron in the hydrogen atom is related to the radius r of the orbit by the expression

$$v=\sqrt{\frac{k{e}^2}{m_{\text{e}}r}}$$

where k is the Coulomb constant.

A mass m attached to a string of length R moves in a vertical circle with a constant speed. The tension in the string at the top of the circle is T. What is the kinetic energy of the mass at the top of the circle?

A.   $\frac{R\left(T+mg\right)}{2}$

B.   $\frac{R\left(T-mg\right)}{2}$

C.   $\frac{Rmg}{2}$

D.   $\frac{R\left(2T+mg\right)}{2}$

Determine the orbital period for the satellite.

Mass of Earth = 6.0 x 10²⁴ kg

State what is meant by gravitational field strength.

The mass of the asteroid is 6.2 × 10¹² kg. Calculate the gravitational force experienced by the planet when the asteroid is at point P.

Two isolated point particles of mass 4M and 9M are separated by a distance 1 m. A point particle of mass M is placed a distance $x$ from the particle of mass 9M. The net gravitational force on M is zero.

What is $x$?

A.   $\frac{4}{13}$m

B.   $\frac{2}{5}$m

C.   $\frac{3}{5}$m

D.   $\frac{9}{13}$m

Mars has a mass of 6.4 × 10²³ kg. Show that, for Mars, k is about 9 × 10^–13s²m^–3.

The time taken for Mars to revolve on its axis is 8.9 × 10⁴ s. Calculate, in m s^–1, the orbital speed of the satellite.

Satellite X orbits a planet with orbital radius R. Satellite Y orbits the same planet with orbital radius 2R. Satellites X and Y have the same mass.

What is the ratio $\frac{\text{centripetal acceleration of X}}{\text{centripetal acceleration of Y}}$?

A. $\frac{1}{4}$

B. $\frac{1}{2}$

C. 2

D. 4

Planet X has a gravitational field strength of $18 \mathrm{N} {\mathrm{kg}}^{-1}$ at its surface. Planet Y has the same density as X but three times the radius of X. What is the gravitational field strength at the surface of Y?

A.  $6 \mathrm{m} {\mathrm{s}}^{-2}$

B.  $18 \mathrm{m} {\mathrm{s}}^{-2}$

C.  $54 \mathrm{m} {\mathrm{s}}^{-2}$

D.  $162 \mathrm{m} {\mathrm{s}}^{-2}$

Calculate, for the surface of $\text{Io}$, the gravitational field strength g_Io due to the mass of $\text{Io}$. State an appropriate unit for your answer.

Which is the definition of gravitational field strength at a point?

A. The sum of the gravitational fields created by all masses around the point

B. The gravitational force per unit mass experienced by a small point mass at that point

C. $G\frac{M}{r^2}$, where $M$ is the mass of a planet and $r$ is the distance from the planet to the point

D. The resultant force of gravitational attraction on a mass at that point

Determine the gravitational field of the planet.

The following data are given:

Mass of planet            $=8.0\times {10}^{24}$ kg
Radius of the planet    $=9.1\times {10}^6$ m.

The gravitational field strength at the surface of a planet of radius R is $g$. A satellite is moving in a circular orbit a distance R above the surface of the planet. What is the magnitude of the acceleration of the satellite?

A.  $0$

B.  $\frac{g}{4}$

C.  $\frac{g}{2}$

D.  $g$

The orbital radius of Titan around Saturn is $R$ and the period of revolution is $T$.

Show that $T^2=\frac{4{\mathrm{\pi }}^2{R}^{ 3}}{GM}$ where $M$ is the mass of Saturn.

The orbital radius of Titan around Saturn is 1.2 × 10⁹m and the orbital period is 15.9 days. Estimate the mass of Saturn.

The orbital radius of Titan around Saturn is $R$ and the period of revolution is $T$.

Show that $T^2=\frac{4{\mathrm{\pi }}^2{R}^{ 3}}{GM}$ where $M$ is the mass of Saturn.

The orbital radius of Titan around Saturn is 1.2 × 10⁹m and the orbital period is 15.9 days. Estimate the mass of Saturn.

P and Q are two moons of equal densities orbiting a planet. The orbital radius of P is twice the orbital radius of Q. The volume of P is half that of Q. The force exerted by the planet on P is F. What is the force exerted by the planet on Q?

A.  F

B.  2F

C.  4F

D.  8F

The centre of the Earth and the Moon are a distance $D$ apart. There is a point X between them where their gravitational fields cancel out. The distance from the centre of the Earth to X is $d$. The mass of the Earth is $M_{\text{E}}$ and the mass of the Moon is $M_{\text{M}}$.

What is correct at X?

A.  $\frac{M_{\text{E}}}{d}=\frac{M_{\text{M}}}{D-d}$

B.  $\frac{M_{\text{E}}}{D-d}=\frac{M_{\text{M}}}{d}$

C.  $\frac{M_E}{d^2}=\frac{M_{\text{M}}}{{\left(D-d\right)}^2}$

D.  $\frac{M_E}{d^2}=\frac{M_{\text{M}}}{D^2-d^2}$

Satellite X orbits a planet with orbital radius R. Satellite Y orbits the same planet with orbital radius 2R. Satellites X and Y have the same mass.

What is the ratio $\frac{\text{centripetal acceleration of X}}{\text{centripetal acceleration of Y}}$?

A. $\frac{1}{4}$

B. $\frac{1}{2}$

C. 2

D. 4

Planet X has a gravitational field strength of $18 \mathrm{N} {\mathrm{kg}}^{-1}$ at its surface. Planet Y has the same density as X but three times the radius of X. What is the gravitational field strength at the surface of Y?

A.  $6 \mathrm{m} {\mathrm{s}}^{-2}$

B.  $18 \mathrm{m} {\mathrm{s}}^{-2}$

C.  $54 \mathrm{m} {\mathrm{s}}^{-2}$

D.  $162 \mathrm{m} {\mathrm{s}}^{-2}$

Calculate, for the surface of $\text{Io}$, the gravitational field strength g_Io due to the mass of $\text{Io}$. State an appropriate unit for your answer.

Calculate, for the surface of $\text{Io}$, the gravitational field strength g_Io due to the mass of $\text{Io}$. State an appropriate unit for your answer.

Which is the definition of gravitational field strength at a point?

A. The sum of the gravitational fields created by all masses around the point

B. The gravitational force per unit mass experienced by a small point mass at that point

C. $G\frac{M}{r^2}$, where $M$ is the mass of a planet and $r$ is the distance from the planet to the point

D. The resultant force of gravitational attraction on a mass at that point

Determine the gravitational field of the planet.

The following data are given:

Mass of planet            $=8.0\times {10}^{24}$ kg
Radius of the planet    $=9.1\times {10}^6$ m.

Determine the gravitational field of the planet.

The following data are given:

Mass of planet            $=8.0\times {10}^{24}$ kg
Radius of the planet    $=9.1\times {10}^6$ m.

The gravitational field strength at the surface of a planet of radius R is $g$. A satellite is moving in a circular orbit a distance R above the surface of the planet. What is the magnitude of the acceleration of the satellite?

A.  $0$

B.  $\frac{g}{4}$

C.  $\frac{g}{2}$

D.  $g$

The orbital radius of Titan around Saturn is $R$ and the period of revolution is $T$.

Show that $T^2=\frac{4{\mathrm{\pi }}^2{R}^{ 3}}{GM}$ where $M$ is the mass of Saturn.

The orbital radius of Titan around Saturn is 1.2 × 10⁹m and the orbital period is 15.9 days. Estimate the mass of Saturn.

The orbital radius of Titan around Saturn is $R$ and the period of revolution is $T$.

Show that $T^2=\frac{4{\mathrm{\pi }}^2{R}^{ 3}}{GM}$ where $M$ is the mass of Saturn.

The orbital radius of Titan around Saturn is 1.2 × 10⁹m and the orbital period is 15.9 days. Estimate the mass of Saturn.

The orbital radius of Titan around Saturn is $R$ and the period of revolution is $T$.

Show that $T^2=\frac{4{\mathrm{\pi }}^2{R}^{ 3}}{GM}$ where $M$ is the mass of Saturn.

The orbital radius of Titan around Saturn is 1.2 × 10⁹m and the orbital period is 15.9 days. Estimate the mass of Saturn.

The orbital radius of Titan around Saturn is $R$ and the period of revolution is $T$.

Show that $T^2=\frac{4{\mathrm{\pi }}^2{R}^{ 3}}{GM}$ where $M$ is the mass of Saturn.

The orbital radius of Titan around Saturn is 1.2 × 10⁹m and the orbital period is 15.9 days. Estimate the mass of Saturn.

P and Q are two moons of equal densities orbiting a planet. The orbital radius of P is twice the orbital radius of Q. The volume of P is half that of Q. The force exerted by the planet on P is F. What is the force exerted by the planet on Q?

A.  F

B.  2F

C.  4F

D.  8F

The centre of the Earth and the Moon are a distance $D$ apart. There is a point X between them where their gravitational fields cancel out. The distance from the centre of the Earth to X is $d$. The mass of the Earth is $M_{\text{E}}$ and the mass of the Moon is $M_{\text{M}}$.

What is correct at X?

A.  $\frac{M_{\text{E}}}{d}=\frac{M_{\text{M}}}{D-d}$

B.  $\frac{M_{\text{E}}}{D-d}=\frac{M_{\text{M}}}{d}$

C.  $\frac{M_E}{d^2}=\frac{M_{\text{M}}}{{\left(D-d\right)}^2}$

D.  $\frac{M_E}{d^2}=\frac{M_{\text{M}}}{D^2-d^2}$

Determine the orbital period for the satellite.

Mass of Earth = 6.0 x 10²⁴ kg

Determine the orbital period for the satellite.

Mass of Earth = 6.0 x 10²⁴ kg

State what is meant by gravitational field strength.

The mass of the asteroid is 6.2 × 10¹² kg. Calculate the gravitational force experienced by the planet when the asteroid is at point P.

State what is meant by gravitational field strength.

The mass of the asteroid is 6.2 × 10¹² kg. Calculate the gravitational force experienced by the planet when the asteroid is at point P.

Two isolated point particles of mass 4M and 9M are separated by a distance 1 m. A point particle of mass M is placed a distance $x$ from the particle of mass 9M. The net gravitational force on M is zero.

What is $x$?

A.   $\frac{4}{13}$m

B.   $\frac{2}{5}$m

C.   $\frac{3}{5}$m

D.   $\frac{9}{13}$m

Mars has a mass of 6.4 × 10²³ kg. Show that, for Mars, k is about 9 × 10^–13s²m^–3.

The time taken for Mars to revolve on its axis is 8.9 × 10⁴ s. Calculate, in m s^–1, the orbital speed of the satellite.

Mars has a mass of 6.4 × 10²³ kg. Show that, for Mars, k is about 9 × 10^–13s²m^–3.

The time taken for Mars to revolve on its axis is 8.9 × 10⁴ s. Calculate, in m s^–1, the orbital speed of the satellite.