A narrow hole is drilled through the centre of earth (mass M kg and radius R m). Now consider a particle of mass m which is inside the hole at a distance r m from the centre. (a) Show that p3 M, = M- R3 where M, is the mass of a smaller sphere with radius r inside the earth. Hint: Assume the earth is a uniform sphere and that its density is the same every- where.

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Find the z-component of the gravitational field due to the ring at a point P on the z-axis at a distance 5 m from the origin. I need to find gz.

Two concentric spherical shells with uniformly distributed masses M, and M, are situated as shown in the figure below. a M1 M2 Find the magnitude of the net gravitational force on a particle of mass m, due to the shells, when the particle is located at the following radial distances. (Use any variable or symbol stated above along with the following as necessary: a, b, c, and G for the gravitational constant.) (a) r = a F = (b) r = b F = (c) r = c F =

The mean diameters of planets A and B are 9.3 × 103 km and 1.8 × 104 km, respectively. The ratio of the mass of planet A to that of planet B is 0.88. (a) What is the ratio of the mean density of A to that of B? (b) What is the ratio of the gravitational acceleration on A to that on B? (c) What is the ratio of escape speed on A to that on B?

Kepler-10c is another exoplanet orbiting another star. It has a mass that is approximately 17.0 times the mass of the Earth. It has a radius that is approximately 2.4 times the that of the Earth. Stats for the Earth (Given): MEarth = 6×1024[kg]. REarth = 6.4×106[m]. Question: What is the escape velocity of an object sitting on the surface of Kepler-10c ? Round to the nearest [km/s] 30 [km/s] 24 [km/s] 20 [km/s] 18 [km/s] 16 [km/s] 14 [km/s] 11 [km/s] None of the answers is even close to correct

Two concentric spherical shells with uniformly distributed masses M₁ and M₂ are situated as shown in the figure. Find the magnitude of the net gravitational force on a particle of mass m, due to the shells, when the particle is located at each of the radial distances shown in the figure. Fa NOTE: Give your answer in terms of the variables given and G when applicable (a) What is the magnitude of the net gravitational force if the particle is located outside both shells with a radial distance a? F = M₁ (b) What is the magnitude of the net gravitational force if the particle is located between the two shells with a radial distance b? Fc M₂ ****** - a (c) What is the magnitude of the net gravitational force if the particle is located inside both shells with a radial distance c? =

Our solar system is roughly 2.2 x 1020 m away from the center of the Milky Way galaxy, and the system is moving at roughly 231.4 km/s around the galaxy's center. Since most of the galaxy's mass is near its center (and we are on an outer arm of this spiral galaxy), let's model the galaxy has a spherical mass distribution (like a single, giant star that our system is orbiting around).  What is the mass of the galaxy (according to our rough, spherical model)? Obviously, this will be a VERY big answer, and so enter in your answer to the order of 1040 kg. In other words, calculate the answer, and then divide by 1040 and then enter in the result.  BTW - by assuming that all mass in the galaxy is made up of stars that are about the same mass as our sun, it isn't too hard to then estimate how many stars are in the galaxy!). As an another aside, some measurements and observations that we have taken in Astronomy suggests that in reality, stars only make up a fraction of the total mass

The free-fall acceleration on the surface of Jupiter is about two and one half times that on the surface of the Earth. The radius of Jupiter is about 11.0 RE (RE = Earth's radius = 6.4  106 m). Find the ratio of their average densities, ?Jupiter/?Earth.

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