A certain quaternary star system consists of three stars, each of mass m, moving in the same circular orbit of radius r about a central star of mass M. The stars orbit in the same sense, and are positioned one-third of a revolution apart from each other. Show that the period of each of the three stars is given by the following expression. (Submit a file with a maximum size of 1 MB.) 7-25 (M+√3) Choose File No fle chosen
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- Around 2.5 centuries ago, several physicists of the time came up with the notion of a dark star. This was a star so dense, with so much gravity, that not even light could escape. The calculations used Newtonian mechanics. In class, we calculated the escape speed from the surface of the earth or the distance from the sun, and the mass of the planet or star. Here, the process is partially reversed. Calculate the dark star radius from the mass of the star and the escape speed. Answer in kilometers. c = 3*108 m/s M = 2.4*1030 kg G = 2/3 * 10-10 N*m2/kg2Find the centre of mass of the 2D shape bounded by the lines y = 10.9 between z = 0 to 2.9. Assume the density is uniform with the value: 2.9kg. m 2 Also find the centre of mass of the 3D volume created by rotating the same lines about the z-axis. The density is uniform with the value: 2.3kg. m-³ (Give all your answers rounded to 3 significant figures.) a) Enter the mass (kg) of the 2D plate: Enter the Moment (kg.m) of the 2D plate about the y-axis: Enter the z-coordinate (m) of the centre of mass of the 2D plate: b) Enter the mass (kg) of the 3D body: Enter the Moment (kg.m) of the 3D body about the y-axis: Enter the x-coordinate (m) of the centre of mass of the 3D body:Find the escape velocity ve for an object of mass m that is initially at a distance R from the center of a planet of mass M. Assume that R≥Rplanet, the radius of the planet, and ignore air resistance. Express the escape velocity in terms of R, M, m, and G, the universal gravitational constant.
- ) Several planets possess nearly circular surrounding rings, perhaps composed of material that failed to form a satellite. In addition, many galaxies contain ringlike structures. Consider a homogeneous ring of mass M and radius R. a) What gravitational attraction does it exert on a particle of mass m located a distance x from the center of the ring along its axis? b) Suppose the particle falls from rest as a result of the attraction of the ring of matter. Find an expression for the speed with which it passes through the center of the ring. (a: see notes from class, b: Use the definition of potential energy.)Around 2.5 centuries ago, several physicists of the time came up with the notion of a dark star. This was a star so dense, with so much gravity, that not even light could escape. The calculations used Newtonian mechanics. In class, we calculated the escape speed from the surface of the earth or the distance from the sun, and the mass of the planet or star. Here, the process is partially reversed. Calculate the dark star radius from the mass of the star and the escape speed. Answer in kilometers. c = 3*108 m/s M = 3.2*1030 kg G = 2/3 * 10-10 N*m2/kg2Consider the observation that the acceleration due to the gravitational force acting on a mass around a host planet decreases with the square of the separation between the objects. We can ask ourselves: why is it still accurate to consider a gravitational acceleration value of 9.8\frac{m}{s^2}9.8s2m for all of our projectile motion problems and all of our gravitational potential energy from prior modules? Let's analyze a situation and justify this analysis method: consider an object being launched from ground level to an altitude of 10,000 meters, roughly the cruising altitude of most jet liners, and far above our everyday experiences on Earth's surface. Compare the gravitational acceleration of the object at Earth's surface (the radius of Earth is about r_E=6.37\times10^6mrE=6.37×106m) to the acceleration value at the 10,000 meter altitude by determining the following ratio: g10,000m/gsurface
- when we calculate escape speeds, we usually do so with the assumption that the object from which we are calculating escape speed is isolated. This is, of course, generally not true in the solar system. Show that the escape speed at a point near a system that consists of two stationary massive spherical objects is equal to the square root of the sum of the squares of the escape speeds from each of the two objects considered individually.White dwarf stars are produced by the collapse of regular stars, such as our sun, toward the end of their normal life. Suppose a star initially has the same mass as our sun and the same radius as that of the sun. Suppose further that it collapses into a white dwarf with a radius of 5000km. If the initial period of the star is the same 27 day period of our sun, what is the resulting period of the white dwarf?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? =