Gravitational Concepts Quiz

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Questions and Answers

If the distance between the centers of a large sphere and a small ball is doubled, how does the gravitational force between them change?

It will remain the same.

It will be quartered. (correct)

It will double.

It will be halved.

In the Cavendish's experiment described, what directly enables us to calculate G (the gravitational constant)?

The restoring torque.

The length of the bar connecting the spheres.

The masses of the large and small spheres.

The observed angular deflection. (correct)

A point mass m is located inside a spherical shell of radius r. What is the net gravitational force exerted on the mass m by the spherical shell?

A force acting toward the shell's center.

A force acting away from the shell's center.

Zero. (correct)

A force that depends on the mass of the shell which is greater or equal to its mass.

A point mass is located within Earth at a distance 'r' from the Earth's center. What is the primary factor in calculating the gravitational force acting on this mass?

The mass of the smaller sphere of radius r from Earth center. Signup and view all the answers

In the context of the Cavendish experiment, which of these is true of the restoring torque?

It acts to counter the gravitational torque. Signup and view all the answers

What does Kepler's Law of Periods state about the relationship between a planet's orbital period and the semi-major axis of its orbit?

The square of the time period of revolution is proportional to the cube of the semi-major axis. Signup and view all the answers

According to the provided data, which of the following planets has the smallest semi-major axis?

Mercury Signup and view all the answers

Based on the information given, what is the approximate value of the quotient (T²/a³) for Saturn in units of 10⁻³⁴ y² m⁻³?

2.98 Signup and view all the answers

If a planet's speed at perihelion is $v_P$ and its distance from the Sun is $r_P$, and at aphelion, the speed is $v_A$ and distance is $r_A$, what is the relationship between these quantities according to the conservation of angular momentum?

$v_P r_P$ = $v_A r_A$ Signup and view all the answers

Given that $r_A$ (distance at aphelion) is greater than $r_P$ (distance at perihelion), how does the speed of the planet at perihelion $v_P$ compare to its speed at aphelion $v_A$?

$v_P > v_A$ Signup and view all the answers

Based on the provided context, what is the relation between the angular momentum at perihelion (Lp) and aphelion (LA)?

Lp = LA Signup and view all the answers

Why does the Law of Areas (Kepler's Second law) hold true for planetary motion?

Because gravitation is a central force. Signup and view all the answers

According to Kepler's second law, if a planet sweeps out two areas, one larger (SBAC) and one smaller (SBPC), how will the time taken to sweep each area be related?

It will take the same time to sweep both areas. Signup and view all the answers

What is the mathematical representation of the magnitude of gravitational force between two point masses according to Newton's law of gravitation?

$|F| = G \frac{m_1 m_2}{r^2}$ Signup and view all the answers

In the vector form of Newton's law of gravitation, what does the term $r̂$ represent?

The unit vector from $m_1$ to $m_2$ Signup and view all the answers

If the distance between two point masses is tripled, how does the magnitude of the gravitational force between them change?

The force becomes 1/9 as strong Signup and view all the answers

What does the negative sign in the vector form of Newton's gravitational law indicate?

The force acts in the direction opposite to the unit vector $r̂$ Signup and view all the answers

According to the provided text, what is the total force F1 on mass m1 due to masses m2, m3 and m4?

$F_1 = G \frac{m_2 m_1}{r_{21}^2}r̂_{21} + G \frac{m_3 m_1}{r_{31}^2}r̂_{31}+ G \frac{m_4 m_1}{r_{41}^2}r̂_{41}$ Signup and view all the answers

If three equal masses are placed at the vertices of an equilateral triangle, and a mass 2m is placed at the centroid, what is the net gravitational force on the mass 2m?

Zero Signup and view all the answers

In the example with three equal masses at the vertices of an equilateral triangle, if the mass at vertex A is doubled, what can be said about the net force acting on a mass at the centroid?

The force is directed towards the vertex A Signup and view all the answers

In the calculation of gravitational forces involving multiple masses, why is vector addition used rather than scalar addition?

Because gravitational force is a vector quantity implying the direction of the forces is relevant. Signup and view all the answers

What is the currently accepted value of the gravitational constant, G?

6.67 × 10⁻¹¹ N m²/kg² Signup and view all the answers

Assuming the Earth has a uniform density, how is its mass (ME) expressed?

$M_E = \frac{4\pi}{3} R_E^3 \rho$ Signup and view all the answers

What does 'r' represent in the equation $F = \frac{GmM_r}{r^2}$?

The distance of the mass m from the center of the Earth. Signup and view all the answers

If the entire Earth is assumed to have a uniform density and mass $M_E$, and a mass $m$ is located at distance ‘r’ from the Earth's center. What is the magnitude of the force on that mass?

$F = \frac{GmM_r}{r^2}$ Signup and view all the answers

What is the magnitude of gravitational force on a mass $m$ at the surface of the Earth, where $R_E$ is the radius of the Earth?

$F = \frac{GM_Em}{R_E^2}$ Signup and view all the answers

A mass ‘m’ is located at a height ‘h’ above the surface of the Earth. What is the distance in the denominator of the gravitational force equation?

$R_E + h$ Signup and view all the answers

How does the force of gravity on a mass m change when the distance from Earth's center changes from $R_E$ to $2R_E$?

The force decreases to one-fourth of the original Signup and view all the answers

If $M_r$ is the mass included within a radius $r$ and $M_E$ is the mass of Earth and $R_E$ is its radius, what is the relationship shown in the provided content?

$M_r = M_E \frac{r^3}{R_E^3}$ Signup and view all the answers

What does the variable 'a' represent in the equation for the total energy of a bound system?

The semi-major axis of the elliptical orbit. Signup and view all the answers

If a particle is located inside a uniform spherical shell, what is the gravitational force on the particle?

Zero. Signup and view all the answers

How does the gravitational acceleration, $g(h)$, at a height h above the Earth's surface, compare to the gravitational acceleration, g, at the Earth's surface?

$g(h)$ is always less than $g$. Signup and view all the answers

What happens to the total energy of a bound system?

It is negative for a closed orbit. Signup and view all the answers

What is the relationship between kinetic energy ($K$) and potential energy ($V$) for a particle in a bound system, according to the text?

$K = -\frac{1}{2}V$ Signup and view all the answers

What is the escape speed from the surface of the earth expressed in relation to $g$ and $R_E$?

$v_e = \sqrt{2gR_E}$ Signup and view all the answers

If a particle is inside a homogeneous solid sphere, where does the force on the particle act?

Toward the center of the sphere. Signup and view all the answers

A satellite with mass m orbits the earth at height h. What is its acceleration, given that the mass of the earth is $M_E$ and radius is $R_E$?

$g(h) = \frac{GM_E}{(R_E+h)^2}$ Signup and view all the answers

What does the law of areas, as described in the provided text, imply about a planet's motion?

A planet speeds up when closer to the sun and slows down when farther away, covering equal areas in equal time intervals. Signup and view all the answers

What condition must be met for a force to be considered a central force?

The force must be directed along the vector joining the planet and the Sun. Signup and view all the answers

What is the significance of angular momentum (L) in the context of planetary motion under a central force?

Angular momentum is conserved, thus making the rate at which a planet sweeps area a constant. Signup and view all the answers

What was Newton's key insight that led to the universal law of gravitation?

That the force of gravity that makes an apple fall, is related to the moon's centripetal acceleration. Signup and view all the answers

In the context of the text, what does $R_m$ represent?

The mean distance between the Earth and Moon. Signup and view all the answers

How does the centripetal acceleration of the moon ($a_m$) compare to the acceleration due to gravity on the surface of the Earth (g)?

$a_m$ is much smaller than $g$. Signup and view all the answers

What does the equation $\Delta A = \frac{1}{2} (r \times v \Delta t)$ represent?

The area swept by the planet in a time interval $\Delta t$. Signup and view all the answers

What does the text imply about the relationship between a planet's distance from the sun and its orbital speed?

Planets move faster when they are closer to the sun and slower when they are farther. Signup and view all the answers

Study Notes

Gravitation

Gravitation is the tendency of all material objects to be attracted towards the earth.
Galileo demonstrated that all bodies accelerate towards the earth with a constant acceleration.
Early models of planetary motion, proposed by Ptolemy, placed Earth at the center.
Later, Copernicus proposed a heliocentric model where the Sun is at the center.
Kepler's laws describe planetary motion:
- Planets move in elliptical orbits with the Sun at one focus.
- A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
- The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.
Newton's law of universal gravitation states that every body in the universe attracts every other body with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
The gravitational constant, G, is a fundamental constant in physics.
The acceleration due to gravity (g) varies with the distance from the center of Earth. It is maximum at the surface and decreases with increasing altitude or depth.
Gravitational potential energy is the energy an object has by virtue of its position in a gravitational field.
Escape speed is the minimum speed needed for an object to escape from a gravitational field.
Earth satellites orbit the Earth due to the gravitational force.
The total energy of an orbiting satellite is negative, indicating that it is bound to the Earth.
The gravitational potential energy associated with two particles of masses m₁ and m₂ separated by a distance r is given by V = -Gm₁m₂/r (with V=0 as r→ ∞).

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Description

Test your knowledge on gravitational forces, Kepler's laws, and Cavendish's experiment. This quiz covers essential concepts of gravity, including calculations involving distance, mass, and orbital dynamics. Challenge yourself and deepen your understanding of these fundamental physics topics!