Find the density of a planet with a radius of 4000 m if the gravitational acceleration for the planet, gp units for both), where G 6.67 x 101 m/(kg s) Hint: Use the expression for the gravitational force and Newton's second law. has the same magnitude as the gravitational constant, G (keep the right

Find the density of a planet with a radius of 4000 m if the gravitational acceleration for the planet, gp units for both), where G 6.67 x 101 m/(kg s) Hint: Use the expression for the gravitational force and Newton's second law. has the same magnitude as the gravitational constant, G (keep the right

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter10: Motion In A Noninertial Reference Frame

Section: Chapter Questions

Problem 10.20P: Calculate the effective gravitational field vector g at Earths surface at the poles and the equator....

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Calculate the effective gravitational field vector g at Earths surface at the poles and the equator. Take account of the difference in the equatorial (6378 km) and polar (6357 km) radius as well as the centrifugal force. How well does the result agree with the difference calculated with the result g = 9.780356[1 + 0.0052885 sin 2 0.0000059 sin2(2)]m/s2 where is the latitude?
On a planet whose radius is 1.2107m , the acceleration due to gravity is 18m/s2 . What is the mass of the planet?
Let gM represent the difference in the gravitational fields produced by the Moon at the points on the Earths surface nearest to and farthest from the Moon. Find the fraction gM/g, where g is the Earths gravitational field. (This difference is responsible for the occurrence of the lunar tides on the Earth.)
Check Your Understanding Why not use the simpler expression U=mg(y2y1) ? How significant would the error be? (Recall the previous result, in Example 13.4, that the value g at 400 km above the Earth is 8.67m/s2 .)
The first atomic bomb was detonated on July 16, 1945, at the Trinity test site about 200 mi south of Los Alamos. In 1947, the U.S. government declassified a film reel of the explosion. From this film reel, British physicist G.I. Taylor was able to determine the rate at which the radius of the fireball from the blast grew. Using dimensional analysis, he was then able to deduce the amount of energy released in the explosion, which was a closely guarded secret at the time. Because of this, Taylor did not publish his results until 1950. This problem challenges you to recreate this famous calculation. (a) Using keen physical insight developed from years of experience, Taylor decided the radius rof the fireball should depend only on time since the explosion, t, the density of the air, , and the energy of the initial explosion, E. Thus, he made the educated guess that r=kEabtcfor some dimensionless constant kand some unknown exponents a,b, and c. Given that [E]=ML2T-2 , determine the values of the exponents necessary to make this equation dimensionally consistent. (Hint: Notice the equation implies that k=rEabtcand that [k]=1 ). (b) By analyzing data from high-energy conventional explosives, Taylor found the formula he derived seemed to be valid as long as the constant khad the value 1.03. From the film reel, he was able to determine many values of rand the corresponding values of t. For example, he found that after 25.0 ms, the fireball had a radius of 130.0 m. Use these values, along with an average air density of 1.25kg/m3 , to calculate the initial energy release of the Trinity detonation in joules (J). (Hint: To get energy in joules, you need to make sure all the numbers you substitute in are expressed in terms of SI base units.) (c) The energy released in large explosions is often cited in units of “tons of TNT” (abbreviated “t TNT”), where 1 t TNT is about 4.2 GJ. Convert yow answer to (b) into kilotons of TNT (that is, kt TNT). Compare your answer with the quick-and-duty estimate of 10 kt TNT made by physicist Enrico Fermi shortly after witnessing the explosion from what was thought to be a safe distance. (Reportedly, Fermi made his estimate by dropping some shredded bits of paper right before the remnants of the shock wave hit him and looked to see how far they were carried by it.)
Schwarzschild radius RS of a black hole is the maximum distance from the black hole’s center at which light cannot escape its gravitational field. The quantity RS (with dimensions of length) is dependent on the mass of the black hole M, the speed of light c, and the gravitational constant G. Based on the dimensions of these four parameters, predict an equation for the Schwarzschild radius. Hint: G has dimensions of [L3/MT2]
(a) Compute the mass of the earth from knowledge of the earth-moon distance (3.84 * 10^8 m) and of the lunar period (27.3 days). (b) Then calculate the average density of the earth. The average radius of the earth is 6.38 * 10^6 m.
The density of the Earth can be expressed by the equation density = C{1.22 -(r/R)}, where R is the radius of the Earth and C is a constant. Find the expression for C in terms of total mass (M) and the radius (R)..
Newton's universal law of gravitation can be stated as the force F of gravitation between two objects varies jointly as the masses m, and m, of the objects, and Gm,m2 inversely as the square of the distance r between their centers, where G is the constant of proportionality. In other words, F = The weight of an object is the force F on the object due to gravity, where F = mg. State how g (the acceleration due to gravity) varies with respect to the mass m, of the spherical body on which the object lies, and the radius of that body (assume all of the mass is at its center.). The acceleration due to gravity, g, varies as the spherical body's mass, and as
ASK YOUR TEACHER PRACTICE ANOTHER Since neutron stars consist totally of neutrons, they are extremely massive and have a density that is hard to imagine. A typical radius and mass for a neutron star are 1.65 x 10 m and 2.50 x 1028 kg respectively. (a) Determine the density of a neutron star. kg/m (b) Determine the weight (in pounds) of a penny (V = 360 mm-) if it were made from this material. (Assume 1 lb = 4.448 N.) Ib Additional Materials
According to Newton's law of universal gravitation, the attraction force between two bodies is given by: where mi and m2 are the masses of the bodies, r is the distance between the bodies, and G=6.67 × 10-" N-m²/kg² is the universal gravitational constant. Determine how many times the attraction force between the sun and the Earth is larger than the attraction force between the Earth and the moon. The distance between the sun and Earth is 149.6 x 10°m, the distance 28 between the moon and Earth is 384.4 × 10°m, mEarth = 5.98 x 10“ kg, = 2.0 x 10" kg, and mn =7.36 x 102 kg.
(a) What is the average density of the sun? (b) What is the av- erage density of a neutron star that has the same mass as the sun but a radius of only 73.199 km? (express your answer in the proper SI unit and without scientific notation)