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physics
mechanics

Questions and Answers of Mechanics

A uniform ball of radius R rolls without slipping between two rails such that the horizontal distance is d between the two contact points of the rails to the hall.(a) In a sketch, show that at any
When an object is rolling without slipping, the rolling friction force is much less than the friction force when the object is sliding; a silver dollar will roll on its edge much farther than it will
A demonstration gyroscope wheel is constructed by removing the tire from a bicycle wheel 0.650 m in diameter, wrapping lead wire around the rim, and taping it in place. The shaft projects 0.200 m at
A block with mass m is revolving with linear speed v, in a circle of radius r, on a frictionless horizontal surface (see Fig. 10.48). The string is slowly pulled from below until the radius of the
What is the ratio of the gravitational pull of the sun on the moon to that of the earth on the moon? (Assume the distance of the moon from the sun can be approximated by the distance of the earth
Cavendish Experiment in the Cavendish balance apparatus shown in Fig. 12.4, suppose that m, = 1.10 kg, m2 = 25.0 kg, and the rod connecting the m, pairs is 30.0 cm long. If, in each pair, m, and m2
Two uniform spheres, each with mass M and radius R, touch each other. What is the magnitude of their gravitational force of attraction?
An interplanetary spaceship passes through the point in space where the gravitational forces from the sun and the earth on the ship exactly cancel. (a) How far from the center of the earth is it? Use
(a) In Fig. 12.31 what are the magnitude and direction of the net gravitational force exerted on the 0.I00-kg uniform sphere by the other two uniform spheres? The centers of all three spheres are on
A typical adult human has a mass of about 70 kg. (a) What force does a full moon exert on such a human when it is directly overhead with its center 378,000 km away? (b) Compare this force with the
An 8.00-kg point mass and a 15.0-kg point mass are held in place 50.0 cm apart. A particle of mass m is released from a point between the two masses 20.0 cm from the 8.00-kg mass along the line
Calculate the magnitude and direction of the net gravitational force on the moon due to the earth and the sun when the moon is in each of the positions shown in Fig. 12.32. (Note that the figure is
Four identical masses of 800 kg each are placed at the corners of a square whose side length is 10.0 cm. What is the net gravitational force (magnitude and direction) on one of the masses, due to the
A particle of mass 3m is located 1.00 m from a particle of mass m. (a) Where should you put a thin mass M 80 that the net gravitational force on M due to the two masses is exactly zero? (b) Is the
The point masses m and 2m lie along the x-axis, with m at the origin and 2m at x = L. A third point mass M is moved along the x-axis. (a) At what point is the net gravitational force on M due to the
Two uniform spheres, each of mass 0.260 kg, are fixed at points A and B (Fig. 12.32). Find the magnitude and direction of the initial acceleration of a uniform sphere with mass 0.010 kg if released
Use the mass and radius of the dwarf planet Pluto given in Appendix F to calculate the acceleration due to gravity at the surface of Pluto.
At what distance above the surface of the earth is the acceleration due to the earth's gravity 0.980 m/s2 if the acceleration due to gravity at the surface has magnitude 9.80 mls2?
The mass of Venus is 81.5% that of the earth, and its radius is 94.9% that of the earth. (a) Compute the acceleration due to gravity on the surface of Venus from these data. (b) If a rock weighs 75.0
Titania, the largest moon of the planet Uranus, has 1 the radius of the earth and 1100 the mass of the earth. (a) What is the acceleration due to gravity at the surface of Titania? (b) What is the
Rhea, one of Saturn's moons, has a radius of 765 k/n and an acceleration due to gravity of 0.278 m/S' at its surface. Calculate its mass and average density.
Calculate the earth's gravity force on a 75-kg astronaut who is repairing the Hubble Space Telescope 600 k/n above the earth's surface, and then compare this value with his weight at the earth's
Neutron stars, such as the one at the center of the Crab Nebula, have about the same mass as our sun but have a much smaller diameter. If you weigh 675 N on the earth, what would you weigh at the
An experiment using the Cavendish balance to measure the gravitational constant G found that a uniform O.40O-kg sphere attracts another uniform 0.00300-kg sphere with a force of 8.00 X 10- 10 N, when
Exploring Europe there is strong evidence that Europe, a satellite of Jupiter, has a liquid ocean beneath its icy surface. Many scientists think we should land a vehicle there to search for life.
The asteroid Dactyl, discovered in 1993, has a radius of only about 700 m and a mass of about 3.6 X 1012 kg. Use the results of Example 12.5 (Section 12.3) to calculate the escape speed for an object
Mass of a Come L On July 4, 2005, the NASA spacecraft Deep Impact fired a projectile onto the surface of Comet Tempell. This comet is about 9.0 k/n across. Observations of surface debris released by
Use the results of Example 12.5 (Section 12.3) 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
Ten days after it was launched toward Mars in December 1998, the Mars Climate Orbiter spacecraft (mass 629 kg) was 2.87 X 106 k/n from the earth and traveling at 1.20 X 104 km/h restive to the earth.
For a satellite to be in a circular orbit 780 km above the surface of the earth, (a) What orbital speed must it be given, and (b) What is the period of the orbit (in hours)?
Assrune that the earth's orbit around the sun is circular use the earth's orbital radius and orbital period given in Appendix F to calculate the mass of the sun.
International Space Station the International Space Station makes 15.65 revolutions per day in its orbit around the earth. Assuming a circular orbit, how high is this satellite above the surface of
Deimos, a moon of Mars, is about 12 km in diameter with mass 2.0 X 10" kg. Suppose you are stranded alone on Deimos and want to playa one-person game of baseball. You would be the pitcher, and you
Planet Vulcan Suppose that a planet were discovered between the sun and Mercury, with a circular orbit of radius equal to 2/3 of the average orbit radius of Mercury. What would be the orbital period
The star Rho1 Canori is 57 light-years from the earth and has a mass 0.85 times that of our sun. A planet has been detected in a circular orbit around Rho1 Cancri with an orbital radius equal to 0.11
In March 2006, two small satellites were discovered orbiting Pluto, one at a distance of 48,000 km and the other at 64,000 km. Pluto already was known to have a large satellite Charon, orbiting at
(a) Use Fig. 12.19 to show that the sun-planet distance at perihelion is (1 - e) a, the sun-planet distance at aphelion is (1 + e la, and therefore the sum of these two distances is Za.(b) When the
Hot Jupiter’s in 2004 astronomers reported the discovery of a large Jupiter-sized planet orbiting very close to the star HD 179949 (hence the term ''hot Jupiter"). The orbit was just the distance
The Helios B spacecraft had a speed of 71 km/s when it was 4.3 X 107 km from the sun. (a) Prove that it was not in a circular orbit about the sun. (b) Prove that its orbit about the sun was closed
A uniform, spherical, l000.0-kg shell has a radius of 5.00 m.(a) Find the gravitational force this shell exerts on a 2.00-kg point mass placed at the following distances from the center of the shell:
A uniform, solid, l00-kg sphere has a radius of 5.00 m. (a) Find the gravitational force this sphere exerts on a 2.00-kg point mass placed at the following distances from the center of the sphere:
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. 12.34).(a) Calculate the gravitational
Consider the ring-shaped body of Fig. 12.35. A particle with mass m is placed a distance x from the center of the ring, along the line through the center of the ring and perpendicular to its
The weight of Santa Claus at the North Pole, as determined by a spring balance, is 875 N. What would this spring balance read for his weight at the equators, assuming that the earth is spherically
The acceleration due to gravity at the North Pole of Neptune is approximately 10.7 m/s2. Neptune has mass 1.0 X 16'6 kg and radius 25 X 104 km and rotates once around its axis in about 16 h. (a) What
Mini Black Holes 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
To what fraction of its current radius would the earth have to be compressed to become a black hole?
(a) Show that a black hole attracts an object of mass m with a force of mc'Rs/ (2r'), where r is the distance between the object and the center of the black hole. (b) Calculate the magnitude of the
At the Galaxy's Core Astronomers have observed a small, massive object at the center of our Milky Way galaxy (see Section 12.8). Airing of material orbits this massive object; the ring has a diameter
In 2005 astronomers announced the discovery of a large black hole in the galaxy Mark Arian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. (a) How far are
Three uniform spheres are fixed at the positions shown in Fig. 12.36.(a) What are the magnitude and direction of the force on a 0.0150-kg particle placed at P?(b) If the spheres are in deep outer
A uniform sphere with mass 60.0 kg is held with its center at the origin, and a second uniform sphere with mass 80.0 kg is held with its center at the point x = 0, Y = 3.00 m. (a) What are the
(a) Show that the gravitational force on the small star due to the two large stars in Example 12.3 (Section 121) is not directed toward the point midway between the two large masses. (b) Consider the
At a certain instant, the earth, the moon, and a stationary 1250-kg spacecraft lie at the vertices of an equilateral triangle whose sides are 3.84 X 10' km in length. (a) Find the magnitude and
An experiment is performed in deep space with two uniform spheres, one with mass 25.0 kg and the other with mass 100.0 kg. They have equal radii, r = 0.20 In. The spheres are released from rest with
Assume that the moon orbits the earth in a circular orbit. From the observed orbital period of27.3 days, calculate the distance of the moon from the center of the earth. Assume that the moon's motion
Geosynclines Jronous Satellites many satellites are moving in a circle in the earth's equatorial plane. They are at such a height above the earth's surface that they always remain above the same
A landing craft with mass 12,500 kg is in a circular orbit 5.75 X 10 m above the surface of a planet. The period of the orbit is 5800 s. The astronauts in the lander measure the diameter of the
What is the escape speed from a 300-km-diameter asteroid with a density of 2500 kg/m'?
(a) Asteroids have average densities of about 2500 kg/m' and radii from 470 km down to less than a kilometer. Assuming that the asteroid has a spherically symmetric mass distribution, estimate the
(a) Suppose you are at the earth's equator and observe a satellite passing directly overhead and moving from west to east in the sky. Exactly 12.0 hours later, you again observe this satellite to be
Planet X rotates it the same manner as the earth, around an axis through its north and south poles, and is perfectly spherical. An astronaut who weighs 943.0 N on the earth weighs 915.0 N at the
There are two equations from which a change it the gravitational potential energy U of the system of a mass m and the earth can be calculated. One is U = may (Eq. 7.2). The other is U = - GmEm/ r
Your starship, the Aimless Wanderer, lands on the mysterious planet Mongo. As chief scientist-engineer, you make the following measurements: A 2.50-kg stone thrown upward from the ground at 12.0 m/s
Calculate the percent difference between your weight in Sacramento, near sea level, and at the top of Mount Everest, which is 8800 m above sea level.
In Example 12.5 (Section 12.3) we ignored the gravitational effects of the moon on a spacecraft en route from the earth to the moon. In fact, we must include the gravitational potential energy due to
An unmanned spacecraft is it a circular orbit around the moon, observing the lunar surface from an altitude of 50.0 k/n (see Appendix F). To the dismay of scientists on earth, an electrical fault
What would be the length of a day (that is, the time required for one rotation of the earth on its axis) if the rate of rotation of the earth were such that g = 0 at the equator?
Falling Hammer A hatumer with mass m is dropped from rest from a height h above the earth's surface. This height is not necessarily small compared with the radius RE of the earth. If you ignore air
(a) Calculate how much walk is required to launch a spacecraft of mass m from the surface of the earth (mass m/E, radius REJ and place it is a circular low earth orbit-that is, an orbit whose
A spacecraft is to be launched from the surface of the earth so that it will escape from the solar system altogether. (a) Find the speed relative to the center of the earth with which the spacecraft
Gravity inside the Earth finds the gravitational force that the earth exerts on a 10.0-kg mass if it is placed at the following locations. Consult Fig. 12.9, and assume a constant density through
Kirkwood Gaps hundreds of thousands of asteroids orbit the sun within the asteroid belt, which extends from about 3 X 108 k/n to about 5 X 108 k/n from the sun. (a) Find the orbital period (it years)
If a satellite is in a sufficiently low orbit, it will encounter air drag from the earth's atmosphere. Since air drag does negative work (the force of air drag is directed opposite the motion), the
Binary Star-Equal Masses Two identical stars with mass M orbit around their center of mass. Each orbit is circular and has radius R, so that the two stars are always on opposite sides of the circle.
Binary Star-Different Masses two stars, with masses M, and M2, are in circular orbits around their center of mass. The star with mass M, has an orbit of radius R the star with mass M2 has an orbit of
Comets travel around the sun in elliptical orbits with large eccentricities. If a comet has speed 2.0 X 10 m/s when at a distance of 2.5 X 1011 m from the center of the sun, what is its speed when at
As Mars orbits the sun in its elliptical orbit, its distance of closest approach to the center of the sun (at perihelion) is 2067 X 1011 m, and its maximum distance from the center of the sun (at
Consider a spacecraft in an elliptical orbit around the earth. At the low point, or perigee, of its orbit, it is 400 km above the earth's surface; at the high point, or apogee, it is 4000 km above
The planet Uranus has a radius of 25,560 km and a surface acceleration due to gravity of11.1 rn/s2 at its poles. Its moon Miranda (discovered by Kniper in 1948) is in a circular orbit about Uranus at
A 3000-kg spacecraft is in a circular orbit 2000 km above the surface of Mars. How much work must the spacecraft engines perform to move the spacecraft to a circular orbit that is 4000 km above the
One of the brightest comets of the 20th century was Comet Hyakutake, which passed close to the sun in early 1996. The orbital period of this comet is estimated to be about 30,000 years. Find the
A uniform wire with mass M and length L is bent into a semicircle. Find the magnitude and direction of the gravitational force this wire exerts on a point with mass m placed at the center of
An object in the shape of a thin ring has radius a and mass M. A uniform sphere with mass m and radius R is placed with its center at a distance x to the right of the center of the ring, along a line
A thin, uniform rod has length L and mass M. Calculate the magnitude of the gravitational force the rod exerts on a particle with mass m that is at a point along the axis of the rod a distance x from
A shaft is drilled from the surface to the center of the earth (Fig. 12.25). As in Example 12.10 (Section 12.6), make the unrealistic assumption that the density of the earth is uniform. With this
(a) When an object is in a circular orbit of radius r around the earth (mass m, the period of the orbit is T, given by Eq. (12.12), and the orbital speed is v, given by Eq. (12.10). Show that when
Interplanetary Navigation The most efficient way to send a spacecraft from the earth to another planet is by using a Hohmann transfer orbit (Fig. 12.38). If the orbits of the departure and
Tidal Forces near a Black Hole. An astronaut inside a spacecraft, which protects her from harmful radiation, is orbiting a black hole at a distance of 120 k/n from its center. The black hole is 5.00
Mass M is distributed uniformly over a disk of radius a. Find the gravitational force (magnitude and direction) between this disk-shaped mass and a particle with mass m located a distance x above the
Mass M is distributed uniformly along a line of length 2L. A particle with mass m is at a point that is a distance a above the center of the line on its perpendicular bisector (point P in a Fig.
Planets are not uniform inside. Normally, they are densest at the center and have decreasing density outward toward the surface. Model a spherically symmetric planet, with the same radius as the
A piano string sounds middle A by vibrating primarily at 220 Hz. (a) Calculate the string's period and angular frequency. (b) Calculate the period and angular frequency for a soprano singing an A one
If an object on a horizontal, frictionless surface is attached to a spring, displaced, and then released, it will oscillate. If it is displaced 0.120 m from its equilibrium position and released with
The tip of a tuning fork goes through 440 complete vibrations in 0.500 s. Find the angu1ar frequency and the period of the motion.
The displacement of an oscillating object as a function of time is shown in Fig. 13.30. What are(a) The frequency;(b) The amplitude;(c) The period;(d) The angu1ar frequency of this motion?
A machine part is undergoing SHM with a frequency of 5.00 Hz and amplitude 1.80 cm. How long does it take the part to go from x = Otox = -1.80cm?
In a physics lab, you attach a 0.200-kg air-track glider to the end of an ideal spring of negligible mass and start it oscillating. The elapsed time from when the glider first moves through the
When a body of unknown mass is attached to an ideal spring with force constant 120 N/m, it is found to vibrate with a frequency of 6.00 Hz. Find (a) The period of the motion; (b) The angular
(a) When a 0.750-kg mass oscillates on an ideal spring, the frequency is 1.33 Hz. What will the frequency be if 0.220 kg are added to the original mass, and (b) Subtracted from the original mass? Try

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