IMO Practice Test — Gravitation
12 Questions • 15 min • Olympiad level
15:00
Question 1 of 12
Two solid spheres of the same material have radii in the ratio 1:2. When their surfaces touch, the gravitational force between them is proportional to:
$R^2$
$R^4$
$R^6$
$R^3$
Explanation: Mass $\propto R^3$, so $F\propto\frac{R_1^3 R_2^3}{(R_1+R_2)^2}\propto R^4$ for fixed ratio.
Question 2 of 12
The value of $g$ at a height $h=R$ above the surface is:
$\frac{g}{2}$
$\frac{g}{4}$
$\frac{g}{3}$
$\frac{g}{9}$
Explanation: $g_h=g\left(\frac{R}{R+h}\right)^2=g\left(\frac{R}{2R}\right)^2=\frac{g}{4}$.
Question 3 of 12
At what depth is the value of $g$ the same as at a height $h$ (both small)?
$d=h$
$d=2h$
$d=\frac{h}{2}$
$d=4h$
Explanation: $g_d=g(1-d/R)$ and $g_h=g(1-2h/R)$; equal when $d=2h$.
Question 4 of 12
If the Earth suddenly stopped rotating, the weight of a body at the equator would:
decrease
increase
stay the same
become zero
Explanation: Rotation reduces effective $g$ at the equator; stopping rotation removes that reduction, so weight increases.
Question 5 of 12
The escape velocity of a body from a planet of mass $9M$ and radius $9R$ (Earth $=M,R$) compared with Earth's $v_e$ is:
$v_e$
$3v_e$
$\frac{v_e}{3}$
$9v_e$
Explanation: $v_e\propto\sqrt{M/R}$; $\sqrt{9M/9R}=\sqrt{M/R}$, so unchanged.
Question 6 of 12
The ratio of the kinetic energy to the total energy of an orbiting satellite is:
$1$
$-1$
$2$
$-\frac{1}{2}$
Explanation: $KE=\frac{GMm}{2r}$ and $E=-\frac{GMm}{2r}$, so $\frac{KE}{E}=-1$.
Question 7 of 12
If the speed of a satellite in a circular orbit is increased to $\sqrt{2}$ times the orbital speed, the satellite will:
fall to Earth
stay in the same orbit
just escape the gravitational pull
move to a smaller orbit
Explanation: $\sqrt{2}\,v_o=v_e$, the escape speed at that radius, so the satellite escapes.
Question 8 of 12
A planet has the same density as Earth but twice the radius. Its escape velocity is:
the same as Earth's
twice Earth's
half Earth's
four times Earth's
Explanation: $v_e=R\sqrt{\frac{8\pi G\rho}{3}}\propto R$ at fixed density, so doubling $R$ doubles $v_e$.
Question 9 of 12
The areal velocity ($\frac{dA}{dt}$) of a planet about the Sun is:
maximum at perihelion
maximum at aphelion
constant throughout
zero at the foci
Explanation: Kepler's second law: equal areas in equal times means $\frac{dA}{dt}$ is constant.
Question 10 of 12
Two satellites orbit at radii $r$ and $4r$. The ratio of their periods $T_1:T_2$ is:
$1:2$
$1:4$
$1:16$
$1:8$
Explanation: $T\propto r^{3/2}$, so $T_2/T_1=4^{3/2}=8$; ratio $1:8$.
Question 11 of 12
The gravitational potential at the centre of a uniform solid sphere of mass $M$ and radius $R$ is:
$-\frac{GM}{R}$
$-\frac{3GM}{2R}$
$-\frac{GM}{2R}$
zero
Explanation: Inside a uniform sphere the potential at the centre is $-\frac{3GM}{2R}$.
Question 12 of 12
If the radius of Earth's orbit shrank to one-fourth keeping the Sun's mass fixed, the length of the year would become:
$\frac{1}{2}$ year
$\frac{1}{4}$ year
$\frac{1}{8}$ year
$\frac{1}{16}$ year
Explanation: $T\propto r^{3/2}$; $\left(\frac{1}{4}\right)^{3/2}=\frac{1}{8}$, so the year is $\frac{1}{8}$ of the present.