A 120−kg object and a 420−kg object are separated by 4.10 m. (a) Find the magnitude of the net gravitational force exerted by these objects on a 46.0−kg object placed midway between them. N (b) At what position (other than an infinitely remote one) can the 46.0−kg object be placed so as to experience a net force of zero from the other two objects? m from the 420 kg mass toward the 120 kg mass
Two ships of equal mass are 108 m apart. What is the acceleration of either ship due to the gravitational attraction of the other? Treat the ships as particles and assume each has a mass of 41,000 metric tons. (Give the magnitude of your answer in m/s2. ) m/s2
In introductory physics laboratories, a typical Cavendish balance for measuring the gravitational constant G uses lead spheres with masses of 1.40 kg and 14.0 g whose centers are separated by about 5.70 cm. Calculate the gravitational force between these spheres, treating each as a particle located at the center of the sphere. N
When a falling meteoroid is at a distance above the Earth's surface of 2.80 times the Earth's radius, what is its acceleration due to the Earth's gravitation? m/s2 towards earth
The sizes of cells in living organisms on Earth has been shown to be governed by the strength of the gravitational field near the Earth's surface. In animal cells, a maximum diameter of 10 μm can be attained before auxilliary scaffold-like structures are needed to maintain the structural integrity of the cell. In our solar system, Jupiter's moon Europa (mass 4.80×1022 kg, radius 1560 km ) and Saturn's moon Enceladus (mass 1.08×1020 kg, radius 252 km ) are thought to have global oceans under a thick layer of surface ice. Because of tidal heating by the planets they orbit, these moons may have all of the necessary ingredients for life. If microbial life were to have developed on Europa and Enceladus, and cell size scales linearly with the strength of the gravitational field, what is the maximum size (in μm ) that Europan and Enceladean cells can attain without needing scaffold-type support? (a) Europan cells μm (b) Enceladean cells μm
Miranda, a satellite of Uranus, is shown in part a of the figure below. It can be modeled as a sphere of radius 242 km and mass 6.68×1019 kg (a) (b) (i) (a) Find the free-fall acceleration on its surface. m/s2 (b) A cliff on Miranda is 5.00 km high. It appears on the limb at the 11 o'clock position in part a of the figure above and is magnified in part (b) of the figure above. A devotee of extreme sports runs horizontally off the top of the cliff at 8.80 m/s. For what time interval is he in flight? (Ignore the difference in g between the lip and base of the cliff. ) s (c) How far from the base of the vertical cliff does he strike the icy surface of Miranda? m (d) What is his vector impact velocity? m/s ∘ below the horizontal
Three equal masses are located on a coordinate grid so that the masses and the origin form a square with side length ℓ, as shown in the figure. Find the magnitude and direction of the gravitational field at the origin due to these masses. Find the magnitude of the gravitational field at the origin due to these masses. (Use the following as necessary: m, l, and G. ) g = Find the direction of the gravitational field. (Give your answer in degrees counterclockwise from the +x-axis. ) ∘ counterclockwise from the +x-axis
A spacecraft in the shape of a long cylinder has a length of 100 m, and its mass with occupants is 1450 kg. It has strayed too close to a black hole having a mass 91 times that of the Sun. The nose of the spacecraft points toward the black hole, and the distance between the nose and the center of the black hole is 10.0 km. Black hole (i) (a) Determine the total force on the spacecraft. N (b) What is the difference in the gravitational fields acting on the occupants in the nose of the ship and on those in the rear of the ship, farthest from the black hole? (This difference in acceleration grows rapidly as the ship approaches the black hole. It puts the body of the ship under extreme tension and eventually tears it apart.) N/kg
You are at a weekly meeting of your Astronomy Club. The club members are excited because access is now available to real-time data on light intensity from the star at the center of another planetary system. The plane of the system is parallel to the direction from the system toward the Earth, so it is possible to detect transits: a planet passes between the Earth and the star of the system, so that the light from the star dims slightly. The planets of this system are very close to their parent star and revolve very rapidly. Planet X in this system has a period of 2.00 Earth days! As you graph the incoming data from the star as it appears on a computer, you see the dip in light intensity. You make a printout of the graph of light intensity versus time and compare it to the graph from 2.00 days ago. The latest graph shows a difference. The time interval from the beginning of the dimming of the light from the star to the return to full brightness took much longer than it did during the observation 2.00 days ago, and the light became significantly dimmer than the observation 2.00 days ago. As you continue to watch the data over several days, the light data every 2.00 days is similar to that in the first graph. Then, 10.0 days after you noticed the unusual behavior, you see it again! You realize that the unusual behavior must be due to a double occultation: two planets are lined up between you and the other star: Planet X and Planet Y! You inform your club members about the data and they excitedly ask you to find out the percentage by which the semimajor axis of the Planet Y is larger that of Planet X. %
Selected astronomical data for Jupiter's moon Adrastea is given in the table. Moon Orbital Radius (km), Orbital Period (days) Adrastea 1.30 x 10^5 0.30 From these data, calculate the mass of Jupiter (in kg ). kg
Plaskett's binary system consists of two stars that revolve in a circular orbit about a center of mass midway between them. This statement implies that the masses of the two stars are equal (see figure below). Assume the orbital speed of each star is |v→| = 180 km/s and the orbital period of each is 14.3 days. Find the mass M of each star. (For comparison, the mass of our Sun is 1.99×1030 kg.) solar masses (i)
A satellite in Earth orbit has a mass of 93 kg and is at an altitude of 2.04×106 m. (Assume that U = 0 as r → ∞. ) (a) What is the potential energy of the satellite-Earth system? J (b) What is the magnitude of the gravitational force exerted by the Earth on the satellite? N (c) What force, if any, does the satellite exert on the Earth? (Enter the magnitude of the force, if there is no force enter 0.) N
How much energy is required to move a 1350 kg object from the Earth's surface to an altitude twice the Earth's radius? J
After the Sun exhausts its nuclear fuel, its ultimate fate will be to collapse to a white dwarf state. In this state, it would have approximately the same mass as it has now, but its radius would be equal to the radius of the Earth. (a) Calculate the average density of the white dwarf (in kg/m3). kg/m3 (b) Calculate the surface free-fall acceleration (in m/s2 ). m/s2 (c) Calculate the gravitational potential energy (in J) associated with a 1.82 kg object at the surface of the white dwarf. J (d) What If? The escape speed from the "surface" of the Sun, or a distance equal to its radius, is 617.5 km/s. What would be the escape speed (in km/s ) from the surface of the white dwarf? km/s
(a) Imagine that a space probe could be fired as a projectile from the Earth's surface with an initial speed of 5.28×104 m/s relative to the Sun. What would its speed be when it is very far from the Earth (in m/s )? Ignore atmospheric friction, the effects of other planets, and the rotation of the Earth. (Consider the mass of the Sun in your calculations.) m/s (b) What If? The speed provided in part (a) is very difficult to achieve technologically. Often, Jupiter is used as a "gravitational slingshot" to increase the speed of a probe to the escape speed from the solar system, which is 1.85×104 m/s from a point on Jupiter's orbit around the Sun (if Jupiter is not nearby). If the probe is launched from the Earth's surface at a speed of 4.10×104 m/s relative to the Sun, what is the increase in speed needed from the gravitational slingshot at Jupiter for the space probe to escape the solar system (in m/s )? (Assume that the Earth and the point on Jupiter's orbit lie along the same radial line from the Sun.) m/s
A 1017−kg satellite orbits the Earth at a constant altitude of 101−km. (a) How much energy must be added to the system to move the satellite into a circular orbit with altitude 208 km ? MJ (b) What is the change in the system's kinetic energy? MJ (c) What is the change in the system's potential energy? MJ
A 53.0-kg woman wearing high-heeled shoes is invited into a home in which the kitchen has vinyl floor covering. The heel on each shoe is circular and has a radius of 0.520 cm. (a) If the woman balances on one heel, what pressure does she exert on the floor? N/m2 (b) Should the homeowner be concerned? Yes No Explain your answer.
The spring of the pressure gauge shown in the figure below has a force constant of 1400 N/m, and the piston has a diameter of 1.40 cm. As the gauge is lowered into water in a lake, what change in depth causes the piston to move in by 0.620 cm? m
The small piston of a hydraulic lift (see figure below) has a cross-sectional area of 2.70 cm2, and its large piston has a cross-sectional area of 201 cm2. What downward force of magnitude F1 must be applied to the small piston for the lift to raise a load whose weight is Fg = 14.4 kN ? N
As a rain storm passes through a region, there is an associated drop in atmospheric pressure. If the height of a mercury barometer drops by 25.6 mm from the normal height, what is the atmospheric pressure (in Pa)? Normal atmospheric pressure is 1.013×105 Pa and the density of mercury is 13.6 g/cm3. Pa
Blaise Pascal duplicated Torricelli's barometer using a red Bordeaux wine, of density 950 kg/m3, as the working liquid. (i) (a) What was the height h of the wine column for normal atmospheric pressure? m (b) Would you expect the vacuum above the column to be as good as for mercury? Yes No Explain your answer.
A girl is holding a ball with a diameter of 3.90 cm and average density of 0.0837 g/cm3 under water. Determine the force (in N ) needed to hold it completely submerged. magnitude N direction
A cube of wood having an edge dimension of 19.9 cm and a density of 649 kg/m3 floats on water. (a) What is the distance from the horizontal top surface of the cube to the water level? cm (b) What mass of lead should be placed on the cube so that the top of the cube will be just level with the water surface? kg
Water flowing through a garden hose of diameter 2.70 cm fills a 21.0 L bucket in 1.20 min. (a) What is the speed of the water leaving the end of the hose? m/s (b) A nozzle is now attached to the end of the hose. If the nozzle diameter is one-third the diameter of the hose, what is the speed of the water leaving the nozzle? m/s
A large storage tank, open at the top and filled with water, develops a small hole in its side at a point 15.8 m below the water level. The rate of flow from the leak is found to be 2.60×10−3 m3 /min. (a) Determine the speed (in m/s ) at which the water leaves the hole. m/s (b) Determine the diameter of the hole (in mm). mm (c) What If? If the hole is 10.0 m above the ground and the water is projected horizontally from the hole, how far (in m ) from the base of the tank would a bucket have to be initially placed to catch the water from the leak? m
Gasoline with a density of 737 kg/m3 moves through a constricted pipe in steady, ideal flow. At the lower point shown in the figure below, the pressure is P1 = 1.85×104 Pa, and the pipe diameter is 8.00 cm. At another point y = 0.35 m higher, the pressure is P2 = 1.15×104 Pa and the pipe diameter is 4.00 cm. (a) Find the speed of flow (in m/s ) in the lower section. m/s (b) Find the speed of flow (in m/s) in the upper section. m/s (c) Find the volume flow rate (in m3/s) through the pipe. m3/s
The block of ice (temperature 0∘C) shown in the figure below is drawn over a level surface lubricated by a layer of water 0.10 mm thick. Determine the magnitude of the force F→ (in N ) needed to pull the block with a constant speed of 0.45 m/s. At 0∘C, the viscosity of water has the value η = 1.79×10−3 N⋅s/m2. N
As a fuel saving measure, commercial jets cruise at an altitude of about 10 km. While cruising at an altitude of 10 km, a small leak occurs in one of the window seals in the passenger compartment of a Boeing 747. Within the passenger compartment, the pressure and temperature are, respectively, 1.03 atm and 20∘C and the pressure outside the craft is 0.309 atm. Model the air as an ideal fluid to find the speed (in m/s) of the stream of air flowing through the leak. (Assume the density of air to be 1.20 kg/m3.) m/s
A 4.00 kg mass is placed on top of a vertical spring, which compresses a distance of 2.62 cm. Calculate the force constant (in N/m) of the spring. N/m
A vertical spring stretches 4.4 cm when a 6-g object is hung from it. The object is replaced with a block of mass 20 g that oscillates up and down in simple harmonic motion. Calculate the period of motion. s
You attach a 2.50 kg block to a horizontal spring that is fixed at one end. You pull the block until the spring is stretched by 0.100 m and release it from rest. Assume the block slides on a horizontal surface with negligible friction. The block reaches a speed of zero again 0.500 s after release (for the first time after release). What is the maximum speed of the block (in m/s )? m/s
A 300 g load attached to a horizontal spring moves in simple harmonic motion with a period of 0.390 s. The total mechanical energy of the spring-load system is 4.00 J. (a) What is the spring constant (in N/m )? N/m (b) What is the amplitude (in m ) of the motion? m (c) What If? What is the percentage change in amplitude of motion if the total energy of the system is increased to 5.00 J ? % change = %
While driving behind a car traveling at 3.45 m/s, you notice that one of the car's tires has a small hemispherical bump on its rim as shown in the following figure. (a) Explain why the bump, from your viewpoint behind the car, executes simple harmonic motion. This answer has not been graded yet. (b) If the radii of the car's tires are 0.380 m, what is the bump's period of oscillation (in s)? s (c) What If? You hang a spring with spring constant k = 100 N/m from the rear view mirror of your car. What is the mass (in kg ) that needs to be hung from this spring to produce simple harmonic motion with the same period as the bump on the tire? kg (d) What would be the maximum speed of the hanging mass in your car (in m/s) if you initially pulled the mass down 8.00 cm beyond equilibrium before releasing it? (Enter the vertical component of the speed only.) m/s
A physical pendulum in the form of a planar object moves in simple harmonic motion with a frequency of 0.490 Hz. The pendulum has a mass of 2.40 kg, and the pivot is located 0.300 m from the center of mass. Determine the moment of inertia of the pendulum about the pivot point. kg⋅m2
A pendulum's angular position is given by θ = 0.0310 cos(ωt), where θ is in radians and ω = 6.43 rad/s. Find the period (in s) and length (in m) of the pendulum. period s length m
A pendulum with a length of 1.00 m is released from an initial angle of 17.0∘. After 1000 s, its amplitude has been reduced by friction to 3.50∘. What is the value of b/2m ? s−1
Consider the following equation. −kx − bdxdt = md2x dt2 Show that it is a solution of the following provided that b2 < 4mk. x = Ae(−b/2m)tcos(ωt + θ) (Submit a file with a maximum size of 1 MB. ) Choose File No file chosen This answer has not been graded yet.
A toddler with a mass of 15.2 kg bounces up and down in her crib. The crib mattress behaves like a light spring with force constant 726 N/m. (a) The toddler bounces with at a frequency that allows her to reach a maximum amplitude with minimum effort. What is this frequency (in Hz )? Hz (b) The toddler now bounces high enough to lose contact with the mattress once each cycle, like a trampoline. What is the minimum amplitude of oscillation (in cm ) required for this to occur? cm
Damping is negligible for a 0.145−kg object hanging from a light, 6.30−N/m spring. A sinusoidal force with an amplitude of 1.70 N drives the system. At what frequency will the force make the object vibrate with an amplitude of 0.540 m? Lower frequency Hz Higher frequency Hz
In an engine, a piston oscillates with simple harmonic motion so that its position varies according to the expression, x = 7.00 cos(t + π/5) where x is in centimeters and t is in seconds. (a) At t = 0, find the position of the piston. cm (b) At t = 0, find velocity of the piston. cm/s (c) At t = 0, find acceleration of the piston. cm/s2 (d) Find the period and amplitude of the motion. period s amplitude cm
Ocean waves with a wavelength of 9.2 m and velocity v = 1.20 m/s can be modeled using the equation shown, y(x, t) = (0.79 m)sin[(0.683 m−1)(x − vt)]. Note that x and y are in meters and t is in seconds. (a) Choose the graph of y(x, t) at t = 0. (b) Choose the graph of y(x, t) at t = 2.00 s. (c) How has the wave moved between graph (a) and graph (b)? The wave has traveled 2.40 m to the left. The wave has traveled 2.00 m to the right. The wave has traveled 2.40 m to the right. The wave has traveled 2.00 m to the left.
A wave is described by y = 0.0210 sin(kx−ωt), where k = 2.08 rad/m, ω = 3.60 rad/s, x and y are in meters, and t is in seconds. (a) Determine the amplitude of the wave. m (b) Determine the wavelength of the wave. m (c) Determine the frequency of the wave. Hz (d) Determine the speed of the wave. m/s
The figure shows a traveling sinusoidal wave with period T = 4.0 s on a string held under constant tension at t = 0. The wave is traveling to the right. Choose from the following graphs to answer the questions. Figure A Figure B Figure C Figure D Figure E Which graph represents the following original waves? (a) at time T4 later Figure (b) at time T2 later Figure (c) with amplitude 3 times larger than the original amplitude Figure (d) with wavelength 2 times larger than the original wavelength Figure (e) with frequency 2.5 times larger than the original frequency Figure
A piano string having a mass per unit length equal to 5.40×10−3 kg/m is under a tension of 1450 N. Find the speed with which a wave travels on this string. m/s
A transverse wave with an amplitude of 0.200 mm and a frequency of 570 Hz moves along a tightly stretched string with a speed of 1.96×104 cm/s. (a) If the wave can be modeled as y = Asin(kx − ωt), what are A (in m), k (in rad/m), and ω (in rad/s)? A = m k = rad/m ω = rad/s (b) What is the tension in the string, if μ = 4.20 g/m ? (Give your answer in N.) N
Sinusoidal waves 5.00 cm in amplitude are to be transmitted along a string that has a linear mass density of 4.00×10−2 kg/m. The source can deliver a maximum power of 307 W, and the string is under a tension of 111 N. What is the highest frequency f at which the source can operate? Hz
(a) Find A in the scalar equation 4(9+2) = A. (b) Find A, B, and C in the vector equation 9.00 i^ + 2.00 k^ = Ai^ + Bj^ + Ck^. A = B = C = (c) Explain how you arrive at the answers to convince a student who thinks that you cannot solve a single equation for three different unknowns. ◻ This answer has not been graded yet. (d) The functional equality or identity A + Bcos(Cx + Dt + E) = 9 cos(2x + 4t + 3) is true for all values of the variables x and t, measured in meters and in seconds, respectively. Evaluate the constants A, B, C, D, and E. ( A and B are unitless. Assume x is in meters and t is in seconds. Enter your values for C, D, and E in units of m−1, s−1, and rad, respectively. ) A = B = m−1 C = rad D = s−1 E = rad (e) Explain how you arrive at your answers to part (d).
Create a mathematical model for the pressure variation as a function of position and time for a sound wave, given that the wavelength of the wave is λ = 0.145 m and the maximum pressure variation is ΔPmax = 0.280 N/m2. Assume the sound wave is sinusoidal. (Assume the speed of sound is 343 m/s. Use the following as necessary: x and t. Assume ΔP is in Pa and x and t are in m and s, respectively. Do not include units in your answer.) ΔP = Pa
Find the pressure amplitude (in Pa ) of a 1.80 kHz sound wave in air, given that the displacement amplitude is 1.70×10−8 m. [Note: The equilibrium density of air is ρ = 1.20 kg/m3 and the speed of sound in air is v = 343 m/s. Pressure variations ΔP are measured relative to atmospheric pressure, 1.013×105 Pa. ] Pa
While floating in the ocean, a dolphin emits a sound wave directed toward the hull of an ocean liner d = 322 m away. How long (in s) until the dolphin hears an echo? Assume the water temperature is 25∘C, and the speed of sound in the water is 1,533 m/s.
The intensity of a sound wave emitted by a portable generator is 3.20 μW/m2. What is the sound level (in dB )? dB
An eagle flying at 32 m/s emits a cry whose frequency is 440 Hz. A blackbird is moving in the same direction as the eagle at 11 m/s. (Assume the speed of sound is 343 m/s.) (a) What frequency does the blackbird hear (in Hz ) as the eagle approaches the blackbird? Hz (b) What frequency does the blackbird hear (in Hz ) after the eagle passes the blackbird? Hz
Two waves are traveling in the same direction along a stretched string. The waves are 45.0∘ out of phase. Each wave has an amplitude of 10.00 cm. Find the amplitude of the resultant wave. cm
Two identical loudspeakers are placed on a wall 3.00 m apart. A listener stands 5.00 m from the wall directly in front of one of the speakers. A single oscillator is driving the speakers at a frequency of 300 Hz. (a) What is the phase difference in radians between the waves from the speakers when they reach the observer? (Your answer should be between 0 and 2π.) rad (b) What is the frequency closest to 300 Hz to which the oscillator may be adjusted such that the observer hears minimal sound? Hz
Two identical loudspeakers are driven in phase by a common oscillator at 830 Hz and face each other at a distance of 1.24 m. Locate the points along the line joining the two speakers where relative minima of sound pressure amplitude would be expected. (Take the speed of sound in air to be 343 m/s. Choose one speaker as the origin and give your answers in order of increasing distance from this speaker. Enter 'none' in all unused answer boxes.) minima Distance from speaker (m)
Two sinusoidal waves traveling in opposite directions interfere to produce a standing wave with the wave function y = (2.00)sin(0.500x)cos(800t) where x and y are in meters and t is in seconds. (a) Determine the wavelength of the interfering waves. m (b) What is the frequency of the interfering wave? Hz (c) Find the speed of the interfering waves. m/s
Consider the following figure. A thread has one end tied to a wall, extends across a small fixed pulley, and the other end is tied to a hanging object. The total length of the thread is L = 10.0 m, the mass of the thread is m = 6.00 g, the mass of the hanging object is M = 8.00 kg, and the pulley is a fixed a distance d = 7.00 m from the wall. You pluck the thread between the wall and the pulley and it starts to vibrate. What is the fundamental frequency (in Hz ) of its vibration? Hz
A 52.0 cm long lightweight rope is vibrating in such a manner that it forms a standing wave with three antinodes. (The lightweight rope is fixed at both ends. ) (a) Which harmonic does this wave represent? first harmonic second harmonic third harmonic fourth harmonic none of the above (b) Determine the wavelength (in cm ) of this wave. cm (c) How many nodes are there in the wave pattern? 1 2 3 4 none of the above (d) What If? If the lightweight rope has a linear mass density of 0.00526 kg/m and is vibrating at a frequency of 261.6 Hz, determine the tension (in N) in the lightweight rope. N
A wine glass can be shattered by establishing a standing wave vibration of high frequency around the rim of the glass. What is the frequency of such a sound (in kHz ) if it establishes four nodes and four antinodes equally spaced around the 35.0 cm circumference of the rim of the glass and has a speed of 4,320 m/s ? kHz
If the fundamental frequency of a cylinder is 853 Hz, and the speed of sound is 343 m/s, determine the length of the cylinder (in m) for each of the following cases. (a) the cylinder is closed at one end m (b) the cylinder is open at both ends m
The figure below shows a cylindrical container with an outlet valve. The cylinder is filled with water until the water level is a distance L = 20 cm from the top. A vibrating tuning fork is placed near the top of the cylinder and the valve is opened so that the length L slowly increases. If the tuning fork has a frequency of f = 502 Hz, determine the next two values of L (in m ) that correspond to a resonant mode. (Assume that the speed of sound in air is 343 m/s. ) (i) first length m second length m
In certain ranges of a piano keyboard, more than one string is tuned to the same note to provide extra loudness. For example, the note at 110 Hz has two strings at this frequency. If one string slips from its normal tension of 598 N to 560.00 N, what beat frequency is heard when the hammer strikes the two strings simultaneously? beats/s
A boy is standing at a railroad crossing for a track that runs east and west. As he faces the track, east is to his right and west is to his left. Two trains on the track some distance apart are headed west, both at speeds of 8.60 m/s, and blowing their whistles (which have the same frequency). One train is approaching him from the east and the other is traveling away from him toward the west. (Assume the speed of sound is 343 m/s.) (a) If he hears a beat frequency of 6.00 Hz, determine the frequency (in Hz) emitted by the two whistles. Hz (b) What If? What would be the frequency (in Hz ) of the beats that he hears if the lead train stops at a station (and continues to blow its whistle) while the second train continues traveling toward him? Hz
An A-major chord consists of the notes called A, C#, and E. It can be played on a piano by simultaneously striking strings with fundamental frequencies of 440 Hz, 554.37 Hz, and 659.26 Hz. The rich consonance of the chord is associated with near equality of the frequencies of some of the higher harmonics of the three tones. Consider the first 5 harmonics of each string and determine which harmonics show near equality. (Select all that apply. ) fifth harmonic of C# and third harmonic of E second harmonic of C# and fourth harmonic of E fifth harmonic of A and fourth harmonic of C# third harmonic of A and second harmonic of E first harmonic of A and first harmonic of E fourth harmonic of A and first harmonic of C#