Physics

Gravitation and Center of Mass

368 Questions

Gravitation and center of mass questions explore gravitational fields, planetary density, and the mechanics of celestial bodies. Test items include calculating gravitational strength on different planets and understanding the Roche Limit. This topic is essential for the physics syllabus of major competitive exams.

Gravitational fieldCenter of massPlanetary densityHill SphereSpace-time curvature

Gravitation and Center of Mass Questions

Multiple choice physics turning effects of forces centre of gravity forces - vectors and moments acceleration due to gravity

At which point is the centre of gravity situated in: A circular lamina.

  1. At the centre of radius.

  2. At the centre of semi circular lamina.

  3. At the centre of circular lamina.

  4. can not say
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Centre of gravity means a point from which the weight of a body or system may be considered to act. In uniform gravity it is the same as the centre of mass. For regular bodies centre of gravity lies at the centre of the body. Hence this will be at the centre of the circular lamina.

Multiple choice physics turning effects of forces centre of gravity forces - vectors and moments acceleration due to gravity

Which of following statements related to center of gravity is/are false?

  1. If an object is placed in a uniform gravitational field, center of gravity coincides with center of mass.

  2. The center of gravity of an object is defined as point through which its whole weight appears to act.

  3. The center of gravity is sometimes confused with center of mass.

  4. The center of gravity always lies inside object.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Center of gravity need not always lie inside object. Suppose take a ring. Its center of mass lies at its center. Hence it is not inside the ring but it is outside the body of the ring ie. at its center. All the other statements are correct.

Multiple choice physics turning effects of forces centre of gravity forces - vectors and moments acceleration due to gravity

A body weighs 250 N on the surface of the earth. How much will it weigh half way down to the centre of the earth.?

  1. 125 N

  2. 150 N

  3. 175 N

  4. 250 N.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let m be mass of a body.
$\therefore$ Weight of the body on the surface of the earth is W=mg=250N
Acceleration due to gravity at a depth d below the surface of the earth is $g^{\prime}=g\left(1-\dfrac{d}{R _{E}}\right)$
Weight of the body at depth d is
$W^{\prime}=mg^{\prime}=mg\left(1-\dfrac{d}{R _{E}}\right)$
Here, $d=\dfrac{R _{E}}{2}$
$\therefore W^{\prime}=mg\left(1-\dfrac{{R _{E}}/{2}}{R _{E}}\right)=\dfrac{mg}{2}=\dfrac{W}{2}=\dfrac{250N}{2}=125N$

Multiple choice physics turning effects of forces centre of gravity forces - vectors and moments acceleration due to gravity

Two planets have radii $r _1$ and $r _2$ and their densities are $\rho _1$ and $\rho _2$, respectively. The ratio of acceleration due to gravity on them will be.

  1. $r _1\rho _1:r _2\rho _2$
  2. $r _1\rho^2 _1:r _2\rho^2 _2$
  3. $r^2 _1\rho _1:r^2 _2\rho _2$
  4. $r _1\rho _2:r _2\rho _1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$\dfrac { { g } _{ 1 } }{ { g } _{ 2 } } =\dfrac { { GM } _{ 1 } }{ { r } _{ 1 }^{ 2 } } \div \dfrac { { GM } _{ 2 } }{ { r } _{ 2 }^{ 2 } } $
$=\dfrac { { Gp } _{ 1 }\dfrac { 4 }{ 3 } \pi { r } _{ 1 }^{ 3 } }{ { r } _{ 1 }^{ 2 } } \div \dfrac { { Gp } _{ 2 }\dfrac { 4 }{ 3 } \pi { r } _{ 2 }^{ 3 } }{ { r } _{ 2 }^{ 2 } } $
$=\dfrac { { p } _{ 1 }{ r } _{ 1 } }{ { p } _{ 2 }{ r } _{ 2 } } $
$=\dfrac { { r } _{ 1 }{ p } _{ 1 } }{ { r } _{ 2 }{ p } _{ 2 } } $
$={ r } _{ 1 }{ p } _{ 1 }:{ r } _{ 2 }{ p } _{ 2 }$
Multiple choice physics turning effects of forces centre of gravity forces - vectors and moments acceleration due to gravity

A man can jump 1.5m high on the earth. Calculate the approximate height he might be able to jump on a planet whose density is one-quarter that of the earth and whose radius is one-third of the earth's radius.

  1. $1.5\ m$
  2. $15\ m$
  3. $18\ m$
  4. $28\ m$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The height a person can jump is inversely proportional to the acceleration due to gravity g. g = G * M / R^2. Since M = density * volume = rho * (4/3) * pi * R^3, g is proportional to rho * R. If rho' = rho/4 and R' = R/3, then g' = g * (1/4) * (1/3) = g/12. The jump height h' = h * (g/g') = 1.5 * 12 = 18 m.

Multiple choice physics turning effects of forces centre of gravity forces - vectors and moments acceleration due to gravity

If the earth shrinks such that its mass does not change but radius decreases to one quarter of its original value then one complete day will take:

  1. 96 h

  2. 48 h

  3. 6 h

  4. 1.5 h
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We know that angular momentum of spin $\displaystyle =I\omega $
Bythe conservation of angular momentum
$\displaystyle \frac { 2 }{ 5 } M{ R }^{ 2 }.\frac { 2\pi  }{ T } =\frac { 2 }{ 5 } M{ \left( \frac { R }{ 4 }  \right)  }^{ 2 }.\frac { 2\pi  }{ T' } $
$\displaystyle T'=\frac { T }{ 16 } =\frac { 24 }{ 16 } =1.5h$

Multiple choice physics measurements and experimentation a few applications of linear shm simple pendulum example of simple harmonic motion

If R is the radius of the earth and g the acceleration due to gravity on the earth's surface, the mean density of the earth is

  1. 4πG/3gR

  2. 3πR/4gG

  3. 3g/4πRG

  4. πRg/12G
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We know that

$g=\cfrac{GM}{R^2}$
Also, density $=mass\times volume$
$M=density\times volume\M=P\times\cfrac{4\pi R^3}{3R^2}=P\times\cfrac{4\pi R}{3}$
Put value of m in $g=\cfrac{GM}{R^2}\P=\cfrac{3g}{4\pi RG}$

Multiple choice physics simple harmonic motion a few applications of linear shm simple pendulum example of simple harmonic motion

A body is broken into two parts of masses $m _1$ and $m _2$ These parts are then separated by a distance r ,What is the value of $m _1/m _2$ so that the gravitational force has maximum possible value?

  1. $1 : 1$
  2. $1 : 2$
  3. $2: 1$
  4. $4 : 3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the mass of the body is $m$

$\therefore \,{m _1} + {m _2} = m$
${F _G} = \frac{{G{m _1}{m _2}}}{{{r^2}}}$
${F _G} = \frac{{G{m _1}\left( {m - {m _1}} \right)}}{{{r^2}}}$
for ${F _G} \to \max \,\frac{{d\left( {{F _G}} \right)}}{{d{m _1}}} = 0$
$ = \frac{G}{{{r^2}}}\left( {m - 2{m _1}} \right) = 0$
$ = {m _1} = \frac{m}{2}$
${m _2} = \frac{m}{2}$
$\therefore {m _1}/{m _2} = 1:1$
Hence$,$ optin $(A)$ is correct$.$ 

Multiple choice physics simple harmonic motion a few applications of linear shm simple pendulum example of simple harmonic motion

A ball is in simple harmonic motion in a tunnel through center of the earth. Magnitude of gravitational force acting on the ball of radius $ y _o $ ,when it is at a distance $x$ from mean position is :

  1. $\dfrac{GMm}{R^{3}}x$
  2. $\dfrac{GMm}{\left [ (R-y _{0})^{2})+x^{2} \right ]}$
  3. $\dfrac{GMm}{R^{3}}\left [ (R-y _{0})^{2}+x^{2} \right ]^{1/2}$
  4. $\dfrac{GMmR^{2}}{\left [ (R-y _{0})^{2}+x^{2} \right ]^{2}}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a tunnel through the Earth, the gravitational force inside a uniform sphere is proportional to the distance from the center. The effective distance from the center for a point at distance x from the mean position (center) is r = sqrt((R-y_0)^2 + x^2). The force is F = (GMm/R^3) * r.

Multiple choice various types of barometer fluids physics

If the mass of the sheet at the top of the aneroid barometer used as an altimeter is increased, what will be the effect on the reading?

  1. It will reduce.

  2. It will increase.

  3. No effect.

  4. Can't be determined
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If the top sheet in an aneroid barometer is heavier, it will displace lesser. So, it gives reading corresponding to a lower pressure with a lighter thin sheet. Lower pressure is observed at more heights and hence, the reading of the altimeter will increase.

Multiple choice physics rotational motion of a rigid body and moment of inertia angular momentum (l) and conservation of angular momentum angular momentum in case of rotation about a fixed axis law of conservation of angular momentum

Two spherical bodies of equal mass (M) revolve about their centre of mass. The distance between the centre of the two masses is r. The angular momentum of each about their centre of mass is

  1. $2 \sqrt {GM^3 r}$
  2. $\frac{1}{2} \sqrt {GM^3 r}$
  3. $\frac{1}{2} \sqrt {2GM^3 r}$
  4. $\frac{1}{2} \sqrt {\frac{GM^3 r}{2}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For two masses M revolving about their center of mass at distance r/2, the orbital velocity v is found from GM^2 / r^2 = M * v^2 / (r/2). Thus v^2 = GM / 2, so v = sqrt(GM / 2). Angular momentum L = M * v * (r/2) = M * sqrt(GM / 2) * (r/2) = (1/2) * sqrt(GM^3 * r^2 / 2) = (1/2) * sqrt(GM^3 * r / 2).

Multiple choice

The strength of gravity depends on what two things?

  1. mass & distance between objects

  2. inertia & density

  3. force & acceleration of objects

  4. size & resistance
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The strength of the gravitational force between two objects is determined by their masses and the distance separating their centers.

Multiple choice

Why is gravity unable to pull small moons and asteroids into a spherical shape?

  1. they do not have enough mass

  2. they have too much mass

  3. gravity cannot act in space

  4. acceleration is increased
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Gravity pulls matter toward the center of mass. For an object to become spherical, it must have enough mass to generate sufficient gravitational force to overcome the structural strength of its material.