**13.39 CALC** Consider the ring-shaped object in Fig. E13.39. 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 plane. (a) Calculate the gravitational potential energy \( U \) of this system. Take the potential energy to be zero when the two objects are far apart.(b) Show that your answer to part (a) reduces to the expected result when \( x \) is much larger than the radius \( a \) of the ring.(c) Use \( F_x = -dU/dx \) to find the magnitude and direction of the force on the particle (see Section 7.4).(d) Show that your answer to part (c) reduces to the expected result when \( x \) is much larger than \( a \).(e) What are the values of \( U \) and \( F_x \) when \( x = 0 \)? Explain why these results make sense.**Figure E13.39 Explanation:**The diagram shows:- A ring of radius \( a \) with mass \( M \).- A small spherical particle with mass \( m \).- The particle is placed at a distance \( x \) from the center of the ring, along the central axis perpendicular to the plane of the ring.The image illustrates the spatial relationship between the ring and the particle, helping to visualize the problem setup for calculating gravitational interactions.

Given:- 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 plane. Find:- (a) Calculate the gravitational potential energy U of this system. Take the p when the two objects are far apart. (b) Show that your answer to part (a) reduces to the expected result when x is much larger than the radius a of the ring. (c) Use Fx = -du / dx to find the magnitude and direction of the force on the particle. (d) Find Fx at x>>a, so a→0 (e) Finding F and U at x=0Solution:- a) We know that the gravitation e potential energy U of this system is given as So the gravitation e potential energy U U = GmR dmlet R is the radius of the ring U = ∫ GmRdmU = GmR2+x2Mhere R = a So U = Gma2+x2M b) if the x>>>a then We can neglect the term of then potential energy is given as U = GmMx2= GmMxU = GmMx c) We know that we have given that Fx= -dUdxSo put the value of U here Fx= - ddxGMmR2+x2Fx= -GmMddxR2+x2-1/2So now Fx=GmM×12R2+x2-12-1×2Fx= GmM2R2+x232×2xFx= GmMx[R2+x2]32now put R = a then Fx= GmMx[a2+x2]32d) Fx= GmMx[a2+x2]32When x>>a, a becomes negligibleFx= GmMx[x2]32=GmMx2Fx=GmMx2e) At x≠0,U = GmMa2+x2Therefore, at x =0U = GmMa2=GmMaU =GmMaF= -∂U∂aF= GmMa2This results makes sense, because at if a mass m placed at the centerof the ring, there will be gravitaional force of attraction between them.

Science

Physics

Consider the ring-shaped object in Fig. E13.39. A particle with mass m is placed a distance x from the ing, along the line through the center of the ring and perpendicular to its plane. (a) Calculate the gravitational gy U of this system. Take the potential energy to be zero when the two objects are far apart. (b) Show that o part (a) reduces to the expected result when x is much larger than the radius a of the ring. (c) Use

Consider the ring-shaped object in Fig. E13.39. A particle with mass m is placed a distance x from the ing, along the line through the center of the ring and perpendicular to its plane. (a) Calculate the gravitational gy U of this system. Take the potential energy to be zero when the two objects are far apart. (b) Show that o part (a) reduces to the expected result when x is much larger than the radius a of the ring. (c) Use

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

See similar textbooks

Related questions

Q: density to be 5650kg/m³. (a) What must the muzzle velocity for the projectile to escape the planet's…

A: we are finding Mass first ,then we calculate the part a question :

Q: The first artificial satellite to orbit the Earth was Sputnik I, launched October 4, 1957. The mass…

Q: Part A Find the speed se of the spacecraft when it crashes into the earth.

Q: A satellite is in a circular Earth orbit of radius r.The area A enclosed by the orbit depends on r2…

Q: A satellite is in a circular orbit of radius r around the Earth. (Use the following as necessary: G,…

A: Let the mass of the Earth be ME. The distance between the Earth and the orbiting satellite is r. The…

Q: . Calculate the work (in MJ) necessary to bring a 103 kg object to a height of 1006 km above the…

A: a) R = radius of earth = 6.371 x 106 m h = height above the surface = 1006 km = 1.006 x 106 m r =…

Q: The code Orbit (Appendix J) can be used to generate orbital positions, given the mass of the central…

A: Required : Plots of the orbits.

Q: (a) Calculate the work (in MJ) necessary to bring a 104 kg object to a height of 991 km above the…

A: Given data: a). The mass, m2=104 kg The height, h=991 km

Q: A satellite of mass m is in an elliptical orbit around the Earth, which has mass ME and radius RE.…

A: a) The gravitational potential of the satellite at a point r from the center of the earth is…

Q: Two Earth satellites, A and B, each of mass m, are to be launched into circular orbits about Earth's…

A: Given:- Satellite A is to orbit at an altitude of 5730 km. satellite B is to orbit at an altitude of…

Q: Very far from earth (at R=∞), a spacecraft has run out of fuel and its kinetic energy is zero. If…

A: Solution:Explanation:Mechanical energy includes both kinetic energy (energy due to motion) and…

Q: calculate the gravitational potential due to a thin rod of length l and mass m.

Q: A projectile is launched vertically upward from a planet of mass M and radius R; its initial speed…

A: The initial velocity of the projectile vi is 2 ve, where ve is the escape velocity. Since the…

Q: A planet is in an elliptical orbit around a distant star. At periastron (the point of closest…

Q: The star Sirus A has a mass of 2.06 MO and a radius of 1.71 RO, where M0 is the mass of the Sun…

Q: 3/276 The Mars orbiter for the Viking mission was de- signed to make one complete trip around the…

Q: Zero, a hypothetical planet, has a mass of 5.0 * 102^3 kg, a radius of 3.0 * 10^6 m, and no…

Q: speed vpi in the opposite direction. See the diagram below: Although the satellite never touches…

Q: A space capsule of mass 445 kg is at rest 2.50 ✕ 107 m from the center of the Earth. When it has…

A: Mass of space capsule (m) = 445 kg is at rest 2.50×107 m from the centre of Earth. It has fallen…

Q: On an asteroid like Bennu, where NASA just landed a probe on October 20, a pressurized nitrogen…

Q: (a) Imagine that a space probe could be fired as a projectile from the Earth's surface with an…

A: (a) In particular, we're using the principle of conservation of mechanical energy, which states that…

Q: Consider an infinitesimally thin circular disk of radius R and mass M centered at the origin sitting…

A: (b) Consider a small element on the disk. The mass element be calculated as,…

Q: Find the z component of the gravitational field due to these five point masses at a point P on the z…

Q: (a) State Newton's law of gravitation in word form. (b) Use the law to show that that the escape…

Q: (a) Imagine that a space probe could be fired as a projectile from the Earth's surface with an…

Q: 3/224 Just after launch from the earth, the space-shuttle orbiter is in the 37 × 137-mi orbit shown.…

Q: A bullet is fired vertically upwards with velocity v from the surface of a spherical planet. When it…

A: Let h represent the bullet's climb in height. Gravitational acceleration changes with height as…

Q: Consider a spacecraft in an elliptical orbit around theearth. At the low point, or perigee, of its…

A: In astrophysics, Kepler’s third law of planetary motion has described the relationship between the…

Q: Three uniform spheres of masses m, = 3.50 kg, m, = 4.00 kg, and m, = 6.50 kg are placed at the…

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: a) Derive the escape speed vII (sometimes called the second cosmic speed) from the surface of body…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: A spaceship with m = 1.00 ✕ 104 kg is in a circular orbit around the Earth, h = 800 km above its…

A: Given: Mass of spaceship, m = 1 X104 kg Height of spaceship = 800 km Apogee , d = 3 X 104 km Mass of…

Concept explainers

Rigid Body

A rigid body is an object which does not change its shape or undergo any significant deformation due to an external force or movement. Mathematically speaking, the distance between any two points inside the body doesn't change in any situation.

Rigid Body Dynamics

Rigid bodies are defined as inelastic shapes with negligible deformation, giving them an unchanging center of mass. It is also generally assumed that the mass of a rigid body is uniformly distributed. This property of rigid bodies comes in handy when we deal with concepts like momentum, angular momentum, force and torque. The study of these properties – viz., force, torque, momentum, and angular momentum – of a rigid body, is collectively known as rigid body dynamics (RBD).

Question

**13.39 CALC** Consider the ring-shaped object in Fig. E13.39. 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 plane.

(a) Calculate the gravitational potential energy \( U \) of this system. Take the potential energy to be zero when the two objects are far apart.

(b) Show that your answer to part (a) reduces to the expected result when \( x \) is much larger than the radius \( a \) of the ring.

(c) Use \( F_x = -dU/dx \) to find the magnitude and direction of the force on the particle (see Section 7.4).

(d) Show that your answer to part (c) reduces to the expected result when \( x \) is much larger than \( a \).

(e) What are the values of \( U \) and \( F_x \) when \( x = 0 \)? Explain why these results make sense.

**Figure E13.39 Explanation:**

The diagram shows:
- A ring of radius \( a \) with mass \( M \).
- A small spherical particle with mass \( m \).
- The particle is placed at a distance \( x \) from the center of the ring, along the central axis perpendicular to the plane of the ring.

The image illustrates the spatial relationship between the ring and the particle, helping to visualize the problem setup for calculating gravitational interactions.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Trending now

This is a popular solution!

Step by step

Solved in 4 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.

Similar questions

A space capsule of mass 505 kg is at rest 8.00 x 10' m from the center of the Earth. When it has fallen 5.00 x 10° m closer to the Earth, finu le (a) What is the change in the system's gravitational potential energy? (b) Find the speed of the satellite at that point. m/s
The orbit of Earth around the Sun is almost circular:The closest and farthest distances are 1.47 * 10^8 km and 1.52 * 10^8 kmrespectively. Determine the corresponding variations in (a) totalenergy, (b) gravitational potential energy, (c) kinetic energy, and(d) orbital speed. (Hint: Use conservation of energy and conservationof angular momentum.)
A uniform solid sphere of radius R = 3.5 km produces a gravitational acceleration of ag on its surface. At what distance from the sphere's center are there points (a) inside and (b) outside the sphere where the gravitational acceleration is ag/7? (a) Number i (b) Number Units Units
A communications satellite is launched into circular orbit around the Earth, at an altitude h. The satellite's mass is m, and it was launched from Earth's surface. (Use the following as necessary: Refor the radius of the Earth, Me for the mass of the Earth, G for the gravitational constant, m, and h.) (a) How long does the satellite take to complete one circular orbit? T= (b) What is the satellite's speed? V satellite (c) What is the minimum energy input necessary to place this satellite in orbit? Ignore air resistance but include the effect of the planet's daily rotation. (Use the following as necessary: RE, ME, G, m, h, and Te for the period of Earth's rotation.) AE min
PRO BLEM 10: A PARTICLE IN A PARABOLIC BOWL := kr Suppose a particle of mass m lives inside a parabolic bowl. The height z of the bowl is given, as a function of it's radius as z = kr², where k is a number. On a particular day the particle finds itself at a height zo as measured from the bottom of the parabola. At that instant it has a horizontal speed vo. Ignoring friction calculate (1) for a given value of horizontal velocity, that is we will let vo → Vh, the particle will move in a perfect circle. What is this v,? Answer this part of the question by drawing a free body diagram and applying New ton's laws. (2) Now, suppose vo > vh, where, vh is the value you calculated above. What is the maximum height reached by the particle? Answer this part of the question by evoking the conservation of energy, and remember to discuss what happens to angular momentum here. (3) Now, suppose vo = 0, but the particle is still at 20. Let us assume that 20 is small and that he particle will undergo…
Two Earth satellites, A and B, each of mass m, are to be launched into circular orbits about Earth’s center. Satellite A is to orbit at an altitude of 6370 km. Satellite B is to orbit at an altitude of 19 110 km.The radius of Earth RE is 6370 km. (a) What is the ratio of the potential energy of satellite B to that of satellite A, in orbit? (b) What is the ratio of the kinetic energy of satellite B to that of satellite A, in orbit? (c) Which satellite has the greater total energy if each has a mass of 14.6 kg? (d) By how much?
(a) Calculate how much work is required to launch a spacecraft of mass m from the surface of the earth (mass mE, radius RE) and place it in a circular low earth orbit—that is, an orbit whose altitude above the earth’s surface is much less than RE. (As an example, the International Space Station is in low earth orbit at an altitude of about 400 km, much less than RE = 6370 km.) Ignore the kinetic energy that the spacecraft has on the ground due to the earth’s rotation. (b) Calculate the minimum amount of additional work required to move the spacecraft from low earth orbit to a very great distance from the earth. Ignore the gravitational effects of the sun, the moon, and the other planets. (c) Justify the statement “In terms of energy, low earth orbit is halfway to the edge of the universe.”
In introductory physics laboratories, a typical Cavendish balance for measuring the gravitational constant G uses lead spheres with masses of 1.20 kg and 12.0 g whose centers are separated by about 4.90 cm. Calculate the gravitational force between these spheres, treating each as a particle located at the center of the sphere. Need Help? Master It Read It
A mood of a mass m and raduis a is orbiting a planet of mass M and if radius b at a distance d (center to center) in a circular orbit. Derive an expression for the speed of the mood v in terms of M,d and the gravitational constant G.
IIISuppose a large spherical object, such as a planet, with radius R and mass M has a narrow tunnel passing diametrically through it. A particle of mass m is inside the tunnel at a distance x less than or equal to R from thecenter. It can be shown that the net gravitational force on the particlei sdue entirely to the sphere of mass with radius r ≤ ;x there is no netgravitational force from the mass in the spherical shell with r > .x a. Find an expression for the gravitational force on the particle,assuming the object has uniform density. Your expression willbe in terms of ,x R, m, M, and any necessary constants.b. You should have found that the gravitational force is a linear restoring force. Consequently, in the absence ofair resistance, objects in the tunnel will oscillate with SHM. Suppose an in- trepid astronaut exploring a 150-km-diameter, 3.5 X 1 0 kgasteroid discovers a tunnel through the center. If she jumps into the hole, how long will ti take her to fall all the way…
Problem 9.9 Consider a planet of mass M and radius R. Assume the planet is spherical and has a constant density. By direct integration determine the gravitational field and the gravitational potential at all points inside and outside the planet. (Assume the potential is zero at infinity and there are no other bodies in the universe.)
Show work to find the gravitational acceleration a distance z from the center of a uniform disk with mass M and radius a.