BackA uniform, spherical, shell has a radius of . Sketch a qualitative graph of the magnitude of the gravitational force this sphere exerts on a point mass m as a function of the distance of from the center of the sphere. Include the region from to .
Agent Arlene devised the following method of measuring the muzzle velocity of a rifle (Fig. 14–34). She fires a bullet into a 4.148-kg wooden block resting on a smooth surface, and attached to a spring of spring constant k = 162.7 N/m. The bullet, whose mass is 7.450 g, remains embedded in the wooden block. She measures the maximum distance that the block compresses the spring to be 9.460 cm. What is the speed υ of the bullet?
An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk (m = 0.10 g) driven back and forth in SHM at 1.0 MHz by an electromagnetic coil. What is the disk's maximum speed at this amplitude?
Astronomers have observed a small, massive object at the center of our Milky Way galaxy. A ring of material orbits this massive object; the ring has a diameter of about 15 light-years and an orbital speed of about 200 km/s. Observations of stars, as well as theories of the structure of stars, suggest that it is impossible for a single star to have a mass of more than about 50 solar masses. Can this massive object be a single, ordinary star?
The greenhouse-gas carbon dioxide molecule CO₂ strongly absorbs infrared radiation when its vibrational normal modes are excited by light at the normal-mode frequencies. CO₂ is a linear triatomic molecule, as shown in FIGURE CP15.82, with oxygen atoms of mass mo bonded to a central carbon atom of mass mc. You know from chemistry that the atomic masses of carbon and oxygen are, respectively, 12 and 16. Assume that the bond is an ideal spring with spring constant k. There are two normal modes of this system for which oscillations take place along the axis. (You can ignore additional bending modes.) In this problem, you will find the normal modes and then use experimental data to determine the bond spring constant. The symmetric stretch frequency is known to be 4.00 X 10¹³ Hz. What is the spring constant of the C - O bond? Use 1 u = 1 atomic mass unit = 1.66 X 10⁻²⁷ kg to find the atomic masses in SI units. Interestingly, the spring constant is similar to that of springs you might use in the lab.
A 950-kg car strikes a huge spring at a speed of 25 m/s (Fig. 14–43), compressing the spring 4.0 m. What is the spring stiffness constant of the spring?
A thin, uniform rod has length L and mass M. A small uniform sphere of mass m is placed a distance x from one end of the rod, along the axis of the rod (Fig. E13.34). Use Fx = -dU/dx to find the magnitude and direction of the gravitational force exerted on the sphere by the rod (see Section 7.4). Show that your answer reduces to the expected result when x is much larger than L.
Vision is blurred if the head is vibrated at 29 Hz because the vibrations are resonant with the natural frequency of the eyeball in its socket. If the mass of the eyeball is 7.5 g, a typical value, what is the effective spring constant of the musculature that holds the eyeball in the socket?
The 15 g head of a bobble-head doll oscillates in SHM at a frequency of 4.0 Hz. The amplitude of the head's oscillations decreases to 0.5 cm in 4.0 s. What is the head's damping constant?
Jupiter's moon Io has active volcanoes (in fact, it is the most volcanically active body in the solar system) that eject material as high as 500 km (or even higher) above the surface. Io has a mass of 8.93 × 1022 kg and a radius of 1821 km. For this calculation, ignore any variation in gravity over the 500-km range of the debris. How high would this material go on earth if it were ejected with the same speed as on Io?
Use the results of Example 13.5 (Section 13.3) to calculate the escape speed for a spacecraft (a) from the surface of Mars and (b) from the surface of Jupiter. Use the data in Appendix F. (c) Why is the escape speed for a spacecraft independent of the spacecraft's mass?
In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. What is the mass of this black hole, assuming circular orbits? Express your answer in kilograms and as a multiple of our sun's mass.
Nothing can escape the event horizon of a black hole, not even light. You can think of the event horizon as being the distance from a black hole at which the escape speed is the speed of light, 3.00 ✕ 10⁸ m/s, making all escape impossible. What is the radius of the event horizon for a black hole with a mass 5.0 times the mass of the sun? This distance is called the Schwarzschild radius.
Cosmologists have speculated that black holes the size of a proton could have formed during the early days of the Big Bang when the universe began. If we take the diameter of a proton to be 1.0 × 10-15 m, what would be the mass of a mini black hole?
The greenhouse-gas carbon dioxide molecule CO₂ strongly absorbs infrared radiation when its vibrational normal modes are excited by light at the normal-mode frequencies. CO₂ is a linear triatomic molecule, as shown in FIGURE CP15.82, with oxygen atoms of mass mo bonded to a central carbon atom of mass mc. You know from chemistry that the atomic masses of carbon and oxygen are, respectively, 12 and 16. Assume that the bond is an ideal spring with spring constant k. There are two normal modes of this system for which oscillations take place along the axis. (You can ignore additional bending modes.) In this problem, you will find the normal modes and then use experimental data to determine the bond spring constant. Use the frequency of the symmetric stretch to predict the frequency of the antisymmetric stretch. The measured frequency is 7.05 × 1013 Hz so your prediction is close but not perfect. The reason is that the bonds are not ideal springs but have a slight amount of anharmonicity. Nonetheless, you’ve learned a great deal about the CO₂ molecule from a simple model of oscillating masses.