7.13. EXERCISES

(a) You can shield a charge from electrical forces by putting it inside a hollow conductor. Can you shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means?

This question examines whether gravitational forces can be blocked like electric forces. In electrical cases, charges can be shielded using conductive materials, which block electric fields inside them. However, gravitational forces, which act universally, cannot similarly be shielded. No matter what material or structure you use, gravity from external masses will affect objects inside.

Imagine you’re in a swimming pool. If someone outside the pool throws a ball, the water provides no barrier, just as a hollow conductor does for electrical charges. No matter how you position your body, the water will still hit you; similarly, gravity acts everywhere.

(b) An astronaut inside a small spaceship orbiting around the earth cannot detect gravity. If the space station orbiting around the earth has a large size, can he hope to detect gravity?

In orbit, both the spaceship and the astronaut are in free fall towards Earth, creating a sensation of weightlessness. In a larger station, certain factors such as size and rigidity could allow slight detection of gravity due to the structure's pull even in a free-fall state. However, true weightlessness will still prevail without external forces.

Think of being on a roller coaster at the top of a hill. You feel weightless for a brief moment. Now imagine being in a bigger amusement park; even if the park is larger, you still feel the same brief weightlessness despite its size.

(c) If you compare the gravitational force on the Earth due to the Sun to that due to the Moon, you would find that the Sun’s pull is greater than the Moon’s pull. (You can check this yourself using the data available in the succeeding exercises). However, the tidal effect of the Moon’s pull is greater than the tidal effect of the Sun. Why?

Gravity from the Sun is indeed stronger than that from the Moon because it is significantly more massive and much closer to Earth. Tidal forces depend on the difference in gravitational pull between two points (the near side and the far side of Earth). Because the Moon is much closer, its gravitational influence creates larger differential effects, leading to more significant tides.

Imagine two people standing next to each other: one is much taller (like the Sun) but further away, while the other is shorter but very close (like the Moon). The close person can easily lift and sway the other, even if the tall one has more overall strength—this illustrates the Moon's dominant tidal effect despite the Sun's stronger gravitational pull.

7.2 Choose the correct alternative : (a) Acceleration due to gravity increases/decreases with increasing altitude. (b) Acceleration due to gravity increases/decreases with increasing depth (assume the earth to be a sphere of uniform density). (c) Acceleration due to gravity is independent of mass of the earth/mass of the body. (d) The formula – GMm(1/r2 – 1/r1) is more/less accurate than the formula mg(r2 – r1) for the difference of potential energy between two points r2 and r1 distance away from the centre of the earth.

These alternatives challenge your understanding of fundamental gravitational principles. (a) In general, gravity decreases with altitude, (b) increases with depth up to the center of the Earth, (c) gravity varies with the mass of the Earth, (d) the first formula is more accurate as it accounts for distance changes in gravitational pull compared to the simplified second formula.

Consider climbing a mountain: as you go higher, you feel lighter (gravity decreases). Imagine digging a mine: the deeper it goes, the heavier things feel due to the Earth’s mass above you pressing down. This illustrates how gravity functions both above and below Earth's surface.

7.3 Suppose there existed a planet that went around the Sun twice as fast as the earth. What would be its orbital size as compared to that of the earth?

If a planet orbits the Sun faster than Earth, it must be closer to the Sun according to Kepler's laws. The square of the orbital period is proportional to the cube of the semi-major axis. Therefore, if its period is half that of Earth's, its orbital distance would be significantly smaller.

Think of a merry-go-round: the closer you are to the center, the faster you go. Similarly, planets near the Sun travel quicker because they experience a stronger gravitational pull.

7.4 Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 108 m. Show that the mass of Jupiter is about one-thousandth that of the sun.

Using Kepler's third law of planetary motion, we can compute the mass of Jupiter based on Io's orbital data. By plugging in the period and orbital radius into the formula, after calculations we can reveal that Jupiter's mass indeed is roughly one-thousandth of the Sun’s mass.

Think of measuring the weight of different fruits using a balance scale. By comparing how much heavier or lighter one fruit is than another, you can estimate their masses relative to each other, just as we do with planets.

7.5 Let us assume that our galaxy consists of 2.5 × 1011 stars each of one solar mass. How long will a star at a distance of 50,000 ly from the galactic centre take to complete one revolution? Take the diameter of the Milky Way to be 105 ly.

To estimate the revolution period of a star in our galaxy, use formulas involving gravitational forces and circular motion. Given the total mass of stars in our galaxy and the distance of the star from the center, calculations will yield the time it takes to orbit the galaxy.

Imagine being on a merry-go-round that represents our galaxy, where each person on it spins around the center. The distance you are from the center affects how long it takes to swing around; closer you are, the faster you finish a lap, and further out, the longer it takes.