Physics Gravitational Field Quiz

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Questions and Answers

What is the value of acceleration due to gravity on the surface of the Earth?

10.0 m/s²

9.6 m/s²

9.8 m/s² (correct)

8.5 m/s²

Which of the following factors does NOT affect the acceleration due to gravity?

Mass of the falling object (correct)

Depth below the surface of the Earth

Distance from the Earth's center

Height above the surface of the Earth

How does the acceleration due to gravity change with altitude?

It decreases as altitude increases. (correct)

It increases as altitude increases.

It remains constant regardless of altitude.

It fluctuates randomly with altitude.

At what location on Earth is the acceleration due to gravity the maximum?

Poles Signup and view all the answers

What is the relationship between the radius of the Earth and gravity as expressed in the formula g = Gm / R²?

g is inversely proportional to R Signup and view all the answers

What happens to the acceleration due to gravity if the Earth were to stop rotating?

It would have no effect at poles but increase at the equator. Signup and view all the answers

What is the acceleration due to gravity on the Moon compared to that on the Earth?

It is one-sixth that on Earth. Signup and view all the answers

At what point does the acceleration due to gravity become zero?

At the center of the Earth Signup and view all the answers

What is the formula for the intensity of gravitational field at a distance r from a body of mass M?

Eg = GM / r^2 Signup and view all the answers

What is the SI unit of gravitational potential energy?

Joule (J) Signup and view all the answers

Which of the following statements about gravitational potential is true?

Gravitational potential is equal to the work done per unit mass. Signup and view all the answers

What type of satellites are designed to appear fixed at a certain position relative to Earth?

Geostationary satellites Signup and view all the answers

Which of the following is true about the gravitational potential energy formula?

U = - GMm / r Signup and view all the answers

What is the dimensional formula for gravitational potential?

[L^3R^-2] Signup and view all the answers

Which of the following formulas represents the time period of a satellite?

T = 2π √(r^3 / GM) Signup and view all the answers

What is the gravitational mass defined by Newton's law of gravitation?

Mg = Fg / g Signup and view all the answers

What is the orbital velocity required for a satellite near the Earth's surface?

7.92 km/h Signup and view all the answers

What type of orbit do geostationary satellites occupy?

Equatorial orbit Signup and view all the answers

What is the time period of a polar satellite revolving around the Earth?

84 minutes Signup and view all the answers

What happens if the actual velocity of a satellite is less than its required orbital velocity?

It will fall back to Earth. Signup and view all the answers

What is the binding energy of a satellite of mass m at height h?

<ul> <li>GMm / 2r</li> </ul> Signup and view all the answers

What defines a polar satellite's orbit?

An angle of inclination of 90° to the equatorial plane. Signup and view all the answers

What is a characteristic of geostationary satellites?

They appear stationary to observers on the ground. Signup and view all the answers

What is the angular velocity of a satellite in a geostationary orbit?

π/12 rad/h Signup and view all the answers

What is the relationship between gravitational force and the distance separating two masses?

Inversely proportional to the square of the distance Signup and view all the answers

Which of the following statements about gravitational force is true?

It is applicable to all bodies, irrespective of size. Signup and view all the answers

What is the value of the universal gravitational constant G?

6.67 x 10-11 Nm2 kg-2 Signup and view all the answers

Which force is 10^36 times stronger than the gravitational force?

Electrostatic force Signup and view all the answers

Newton's law of gravitation applies to objects at which of the following distances?

Both very large and very short distances Signup and view all the answers

In terms of the force of gravitation, what do action-reaction pairs refer to?

The attraction between two bodies being equal in magnitude but opposite in direction Signup and view all the answers

What characterizes the gravitational force in terms of energy?

It is a conservative force. Signup and view all the answers

What is the dimensional formula of the universal gravitational constant G?

[M^-1L^3T^-2] Signup and view all the answers

What is the escape velocity at Earth?

11.2 km/s Signup and view all the answers

Which of the following statements about escape velocity is true?

Escape velocity does not depend on the direction of projection. Signup and view all the answers

What happens when the velocity of projection is equal to the escape velocity?

The object escapes following a parabolic path. Signup and view all the answers

Which of the following situations does NOT lead to weightlessness?

Standing on solid ground Signup and view all the answers

According to Kepler's laws, what is the relation between the time period of revolution and the semi-major axis of a planet's elliptical orbit?

T2 is proportional to a3 Signup and view all the answers

Which statement is true about the sum of kinetic and potential energy for a missile launched below escape velocity?

It remains negative. Signup and view all the answers

What would happen if a bottle filled with water is taken to the moon and the cork is opened?

The water will boil. Signup and view all the answers

Which of the following escape velocities is the greatest?

Neutron star Signup and view all the answers

Study Notes

Intensity of Gravitational Field

The gravitational force acting per unit mass at any point in a gravitational field is called the intensity of the gravitational field at that point.
The intensity of a gravitational field at a distance r from a body of mass M is given by: Eg or I = GM / r²
Intensity of the gravitational field is a vector quantity and its direction is towards the center of gravity of the body.
The SI unit of intensity of the gravitational field is N/m and its dimensional formula is [LT^-2].
Gravitational mass (Mg) is defined by Newton’s law of gravitation: Mg = Fg / g or Mg = W / g (Weight of body / Acceleration due to gravity)
The ratio of gravitational masses of two bodies is: (M1)g / (M2)g = Fg1g2 / Fg2g1

Gravitational Potential

The gravitational potential at any point in a gravitational field is equal to the work done per unit mass in bringing a very light body from infinity to that point.
Gravitational potential is denoted by Vg and is calculated by: Vg = W / m = – GM / r
The SI unit of gravitational potential is J/kg and it is a scalar quantity.
The dimensional formula of gravitational potential is [L³T^-2].
Gravitational potential is always negative because the work done, W, is negative.

Gravitational Potential Energy

The gravitational potential energy of any object at any point in a gravitational field is equal to the work done in bringing it from infinity to that point.
Gravitational potential energy is denoted by U and is calculated by: U = – GMm / r
The negative sign indicates that gravitational potential energy decreases with an increase in distance.
The gravitational potential energy at a height h from the surface of the earth can be calculated by: Uh = – GMm / (R + h) = mgR / (1 + h/R)

Satellite

A heavenly object which revolves around a planet is called a satellite.
Natural satellites are those heavenly objects that are not man-made and revolve around the Earth.
Artificial satellites are man-made objects launched for specific purposes that revolve around Earth.

Time Period of a Satellite

The time period of a satellite, T, is given by: T = 2π √r³ / GM = 2π √(R + h)³ / g (where g = GM / R²)
Near the earth's surface, the time period of a satellite is: T = 2π √R³ / GM = √3π / Gp (where p is the average density of earth)
Alternatively, near the earth's surface, the time period of a satellite can also be calculated by: T = 2π √R / g = 5.08 * 10³ s = 84 min.

Artificial Satellites

Artificial satellites are categorized into two types: Geostationary or parking satellites and Polar satellites.

Geostationary or Parking Satellites

A geostationary or parking satellite appears to be at a fixed position at a definite height to an observer on Earth.
They revolve around Earth in the same direction and at the same rate as Earth's rotation, making them appear stationary.
Geostationary satellites are typically used for communication purposes.
Examples include INSAT 2B and INSAT 2C, which are geostationary satellites of India.

Polar Satellites

Polar satellites revolve around Earth in polar orbits, meaning they pass over both the North and South poles.
Their inclination with the equatorial plane of Earth is 90°.
Polar satellites are used for various purposes like weather forecasting, studying the upper atmosphere, mapping, etc.
Examples include the PSLV series satellites of India.

Orbital Velocity

Orbital velocity (vo) is the minimum velocity required to launch a satellite into a given orbit around a planet.
It is calculated by: vo = √GM / r = R √g / (R + h) (where M = mass of the planet, R = radius of the planet, and h = height of the satellite from the planet's surface)
If a satellite is revolving near Earth's surface, then r = (R + h) = R, making the orbital velocity: vo = √gR = 7.92 km/h

Conditions Based on Orbital Velocity

If v < vo, then the satellite will move on a parabolic path and fall back to Earth.
If V = vo, then the satellite will revolve in a circular path/orbit around Earth.
If vo < V < ve, then the satellite will revolve around Earth in an elliptical orbit.

Energy of a Satellite in Orbit

The total energy (E) of a satellite is the sum of its kinetic energy (KE) and potential energy (PE).
It is calculated by: E = KE + PE = GMm / 2r + (- GMm / r) = – GMm / 2r

Binding Energy

Binding energy (BE) is the energy required to remove a satellite from its orbit around Earth (or a planet) to infinity.
It is calculated by: BE = + GMm / 2r

Escape Velocity

Escape velocity (ve) on Earth is the minimum velocity required to launch an object vertically upwards so that it escapes Earth's gravitational field and never returns.
It is calculated by: ve = √2GM / R = √2gR = √ 8πp GR² / 3 (where M = mass of the planet, R = radius of the planet, and p is the average density of the planet)
Escape velocity does not depend on the mass, shape, or size of the body or the direction of projection.
Earth's escape velocity is 11.2 km/s.

Some Important Escape Velocities

Moon: 2.3 km/s
Mercury: 4.28 km/s
Earth: 11.2 km/s
Jupiter: 60 km/s
Sun: 618 km/s
Neutron star: 2 x 10⁵ km/s

Relation between Escape Velocity and Orbital Velocity

The relationship between escape velocity and orbital velocity is: ve = √2vo

Velocity of Projection and Trajectory

If the velocity of projection (U) is equal to the escape velocity (v = ve), then the satellite will escape following a parabolic path.
If the velocity of projection (u) of the satellite is greater than the escape velocity (v > ve), then the satellite will escape along a hyperbolic path.

Weightlessness

Weightlessness occurs when the effective weight of a body becomes zero. This can happen in the following situations:
- During free fall under gravity
- Inside a spacecraft or satellite
- At the center of the Earth
- When a body is lying in a freely falling lift.

Kepler's Laws of Planetary Motion

Kepler's laws describe the motion of planets around the Sun:
- Law of Orbits: Every planet revolves around the Sun in an elliptical orbit with the Sun at one focus of the ellipse.
- Law of Areas: The radius vector drawn from the Sun to a planet sweeps out equal areas in equal intervals of time. This means that the planet's areal velocity around the Sun is constant.
- Law of Periods: The square of the time period of revolution of a planet around the Sun is directly proportional to the cube of the semi-major axis of its elliptical orbit. This can be represented as: T² &infi; a³ or (T₁ / T₂)² = (a₁ / a₂)³ (where a = semi-major axis of the elliptical orbit)

Important Points

For a missile launched with a velocity less than the escape velocity, the sum of its kinetic energy and potential energy is negative.
The orbital speed of Jupiter is less than the orbital speed of Earth.
You cannot hear the sound of a bomb explosion on the Moon from Earth.
If a bottle filled with water at 30°C and fitted with a cork is taken to the Moon, the water will boil when the cork is opened because of the lower atmospheric pressure on the Moon.

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Test your understanding of the intensity of gravitational fields and gravitational potential. This quiz covers key concepts, formulas, and definitions related to gravitational fields, including vector quantities and SI units.