If a wire with linear density ρ(x, y, z) lies along a space curve C, its moments of inertia about the x-, y-, and z-axes are defined as
Ix = ∫C (y2 + z2) ρ(x, y, z) ds
Iy = ∫C (x2 + z2) ρ(x, y, z) ds
Iz = ∫C (x2 + y2) ρ(x, y, z) ds
Find the moments of inertia for the wire in Exercise 35.
35. (a) Write the formulas similar to Equations 4 for the center of mass (
(b) Find the center of mass of a wire in the shape of the helix x = 2 sin t, y = 2 cos t, z = 3t, 0 ⩽ t ⩽ 2π, if the density is a constant k.
Want to see the full answer?
Check out a sample textbook solutionChapter 16 Solutions
Multivariable Calculus
- Which of the vector valued function is/ are continuous? f:R² →R, f(x.y) =(r°y,) y+x I) 1+x II) f:R→R², f(x,y)=(arc tan xy, 2xy) III) f:R/{0,0} →R’, f(x,y)=(sin,In(x-y)) O a. Il O b. None of them O c II O d.l O e. LIIarrow_forwardFind the points on the hyperboloid of one sheet x² + 3y² - 5z² = = 16 where the tangent plane is parallel to the plane with normal vector (5,3,-1) and through (1,1,1).arrow_forwardLet f(x, y, z) = cos(ryz) Find the directional derivative of f at the point (,, 7) in the (a) direction of the vector u = i+j+ v7k. (b) Duf(, }, 7)? Among all unit vectors u, what is the largest possible value ofarrow_forward
- The equation for a plane tangent to z = f(x, y) at a point (xo, Yo) is given by z = f(xo, Yo) + f_(wo, Yo)(x – æo) + fy(x0; Y0)(y – yo) If we wanted to find an equation for the plane tangent to f(x, y) = 9xy – 3y + 5a? at the point (3, – 3), we'd start by calculating these: f(ro, Yo) = | fz(xo, Yo) = fy(xo, Yo) =arrow_forwardOne particle is acted on a point ( x,y,z ) with force F(x,y,z) = (x,yz,x^2) to move along a smooth curve C which is determined by r(t) =(t,t,t^2) when t ∈ [-1,2]Find a task for moving this particle along a curve. Carrow_forwardConsider the following problem from Science: A mitochondria with velocity vector r' (t) starts out at (-4, 0, 5) at t = 0 and mitochondrias around for 5 seconds. Where is the mitochondria located at time t = 5 if Ir' (t)| dt = 0? A) (-4, 0, 5) B) (-4, 0,5) C) (3, 0, 7) 6. D) Not enough informationarrow_forward
- The pressure at the point (x, y, z) of a gas is given by P = f(x, y, z) = (x, y) = 40 + 5.x² + 2y² + 4z2 The pressure P is in pascals and x, y, and z are in cm. a) Give a vector in the direction that the pressure is increasing most rapidly at the point (2, 1, 3). b) Compute the rate of change of the pressure in the direction of the vector Ū = (1,2, 2) at the point (2, 1, 3). Give units.arrow_forwardThe length of a curve given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in the vector form is: Select one: A. B. C. D. E. Clear my choicearrow_forwardFind a vector normal to the surface 3z 3 + x 2 y - y 2 x = 1 at P =(1,-1,1) .arrow_forward
- What is the perpendicular vector to the surface f(x.y.a)-ay-3z+y-0 at (1,-12)? (3,3.0) Ob. (1,1.3) (-1,-1-3) (4-10)arrow_forwardCompute the vector assigned to the pointP = (−3, 5) by the vectorfield:(a) F(x, y) = (xy,y − x)(b) F(x, y) = (4, 8)(c) F(x, y) = 3x+y , log2(x + y)arrow_forwardLet S be the portion of the cylinder x² - 6x + z = 4, oriented by upward-slanting normal vectors, that is cut by the planes y = 1, y = 4, and z = 0. Find the flux of F(x, y, z) = (xy, 3xz², yz) across S.arrow_forward
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning