Question

Select the correct choice(s): (1) The gravitational field inside a spherical cavity, within a spherical planet (mass is uniformly distributed) must be non zero and uniform. (2) When a body is projected horizontally at an appreciable large height above the earth, with a velocity less than for a circular orbit, it will fall to the earth along a parabolic path. (3) A body of zero total mechanical energy placed in a gravitational field will escape the field (4) Earth's satellite must be in equatorial plane.

Short answer

The correct choice is (2).

Step-by-step solution

- Analyze Statement 1

The gravitational field inside a spherical cavity within a spherical planet (with the mass uniformly distributed) must be evaluated. According to the Shell Theorem, inside a uniform spherical shell, the gravitational field is zero. Therefore, the field inside the cavity of a spherical planet is also zero.

- Analyze Statement 2

When a body is projected horizontally at a large height above the earth with a velocity less than that required for a circular orbit, it will fall to the Earth. The path will be part of a parabola because the body travels in a trajectory influenced only by gravity and its initial horizontal velocity.

- Analyze Statement 3

A body of zero total mechanical energy placed in a gravitational field will not escape the field. Zero total mechanical energy indicates that kinetic energy (KE) plus potential energy (PE) equals zero, which implies the body is bound to the field but can be at the verge of escape.

- Analyze Statement 4

Earth's satellite does not need to be exclusively in the equatorial plane. Satellites can orbit in various planes including polar orbits, depending on their mission requirements.

- Select the Correct Choices

From the analysis above: (1) is incorrect since the field inside the cavity is zero. (2) is correct since the trajectory forms part of a parabola. (3) is incorrect as zero mechanical energy implies the body can escape but it is not automatic. (4) is incorrect; satellites can have various orbital inclinations.

Key concepts

Shell Theorem

The Shell Theorem is a very useful principle in gravitational physics. It states that a spherical shell with uniform mass distribution exerts no gravitational force on objects inside the shell. As a result, the gravitational field inside the shell is zero.

In the context of the exercise, this theorem helps us understand why the gravitational field inside a spherical cavity of a planet (where the mass is uniformly distributed) must be zero. The presence of a cavity does not disrupt the uniform mass distribution of the surrounding shell, maintaining the field at zero within the cavity. This demonstrates that Statement 1 is incorrect.

Parabolic Trajectory

When a body is projected horizontally above the Earth at a considerable height but with a velocity less than what is needed for a circular orbit, it forms a parabolic trajectory. This is primarily due to the influence of Earth's gravity.

Gravity acts vertically downward, while the initial horizontal velocity causes the body to follow a curved path. Since the motion is under constant gravitational acceleration, the resulting path is parabolic.

The important takeaway here is that the body will eventually fall back to Earth, which confirms that Statement 2 is correct.

Total Mechanical Energy

Total mechanical energy in a gravitational field is the sum of kinetic energy (KE) and potential energy (PE). When a body has zero total mechanical energy, it is on the brink of escaping the gravitational field but hasn’t done so yet.

Zero total mechanical energy means:
\(KE + PE = 0\)

This implies the object's kinetic energy is just enough to counterbalance its gravitational potential energy, allowing for the possibility of escape, but it doesn’t guarantee it outright. Thus, Statement 3 is incorrect, as zero total mechanical energy does not ensure that the body will escape the gravitational field.

Satellite Orbits

Satellites orbiting Earth can have various trajectories and don't need to be confined to the equatorial plane. They can orbit in different planes, such as polar orbits that pass over the poles or geostationary orbits that stay above a fixed point over the equator.

The choice of orbit depends on the satellite's mission requirements, like weather monitoring, reconnaissance, or communication.

Therefore, Statement 4 is incorrect because not all satellites orbit in the equatorial plane.