Gravitational potential The gravitational force between two point masses M and m is
where G is the gravitational constant.
a. Verify that this force field is conservative on any region excluding the origin.
b. Find a potential function φ for this force field such that F = –▿φ.
c. Suppose the object with mass m is moved from a point A to a point B, where A is a distance r1 from M and B is a distance r2, from M. Show that the work done in moving the object is
d. Does the work depend on the path between A and B? Explain.
Trending nowThis is a popular solution!
Chapter 17 Solutions
Calculus: Early Transcendentals (3rd Edition)
Additional Math Textbook Solutions
Calculus and Its Applications (11th Edition)
Glencoe Math Accelerated, Student Edition
Precalculus (10th Edition)
Thomas' Calculus: Early Transcendentals (14th Edition)
Calculus & Its Applications (14th Edition)
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
- Calculate the velocity and acceleration vectors, and speed for r(t) = (cos(t), sin(3t) , sin(t)) when t Velocity: Acceleration: Speed: Usage: To enter a vector, for example (x, y, z), type ""arrow_forwardCalculate the velocity and acceleration vectors, and speed for when t = 11. Velocity: Acceleration: r(t) = (cos(6t), cos(t), sin(t)) Speed: Usage: To enter a vector, for example (x, y, z), type ""arrow_forwardange Use Newton's Second Law of Motion to find the force required so that a particle of mass 2 iao position r(t) = (t², t“, sin(2t)) %3D u Select one: a. (2, 12t2,-4 sin(20)) b. (4, 24t2, -8 sin(2t;, C. 28 d. 24 e. (2t, 4t3, 2 cos(2t))arrow_forward
- 7. Properties of position vectors Let r = xi + yj + zk and let r = |r|. a. Show that Vr = r/r. b. Show that V(r") = nr"-2r. c. Find a function whose gradient equals r.arrow_forwardDetermine the moments of inertia of the Z-section about its centroidal xo- and yo-axes. -130 mm 21 mm yo 155 mm 21 mm 21 mm -130 mm Answers: (106) mm4 i (106) mm4 i lyo IIarrow_forwardDetermine the moments of inertia of the Z-section about its centroidal xo- and yo-axes. 25 mm Answers: Ixo Yo 165 mm = = i -110 mm i 25 mm 110 mm 25 mm (106) mm 4 (106) mm4arrow_forward
- The force exerted by an electric charge at the origin on a charged particle at a point (x, .) with position vector (K, y. ) is F) = Kr/ir whene K a cinstit the work done as the particle moves along a straight be from (2, 0, 0) to (2. 2, 3). 0.257karrow_forwardCould you solve it please?arrow_forwardHow do I do d) and e)? For d) I took the derivative of the vector r but I got a really weird solution which seems impossiblearrow_forward
- A sodium ion (Na+) moves in the xy-plane with a speed of 2.90 ✕ 103 m/s. If a constant magnetic field is directed along the z-axis with a magnitude of 3.25 ✕ 10−5 T, find the magnitude of the magnetic force acting on the ion and the magnitude of the ion's acceleration. HINT (a) the magnitude (in N) of the magnetic force acting on the ion N (b) the magnitude (in m/s2) of the ion's acceleration m/s2arrow_forward(5) A force F = 50 N is exerted on the automobile parking-brake lever at the position x = 250 mm. Replace the force by an equivalent force-couple system at the pivot point O. Ans: F-17.10 +47.0ĵN, Mo 17.29 N·m CCW OA = OB F 20° 15° 100 mm 10°arrow_forwardFind the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter (Use technology to sketch) x2 + y? + z? = 10, x + y = 4 x = 2 + sin (t)arrow_forward
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage