Problem 2 If u is a velocity, r a length, and a time, what are the...
PROBLEM 2 (10 POINTS) A particle moves with constant velocity u along the curve r = k(1 + cosO) (a cardioid. where k is a constant. Find a f, lal, and 0.
Problem 2] The movement of robot designed to crawl along a wall can be characterized using the body angle a (dimensionless), frequency (o), distance to wall (d), its polar moment of inertia C: dimensions ML2), input moment (Mi; dimensions ML-T2), look-ahead distance (E), damping coefficient (B; dimensions MLT]), and forward speed (u), Derive the non-dimensional parameters that relate these quantities. if the functional dependence is given as G(o)- atu 1 rewrite this equation using the non-dimensional variables that you have...
30] Find th e solution of the following boundary value problem. 1<r<2, u(r, θ = 0) = 0, u(r, θ = π) =0, 1,0-0, u(r-2,0)-sin(20), 0 < θ < π. u(r Please also draw the sketch associated with this problem. You may assume that An -n2, Hn(s)sin(ns), n 1,2,3,. are the eigenpairs for the eigenvalue problem H(0) 0, H(T)0. 30] Find th e solution of the following boundary value problem. 1
In another application of the U shape loop problem above, Consider A rod of length 40 cm moving with a velocity of 30 m/s to the left in a magnetic field of 0.2 T into the page. The loop has a small resistance of 2 ohms. Find a) the emf in the loop, b) the magnitude and direction of induced current, c) the magnetic force acting on the rod and its direction, d) the appied force needed to maintain constant...
4. Rank-2 tensors A charge q moves with a constant velocity u ux. Define the antisymmetric tensor r" by r"-1 (ημ2-η"z"), where ημ įs the four velocity of the charge. Further define (a) Show that r-lauP + y2 + z2. -tu (b) Show that the electric field produced by the charge is given by En/c subscript n refers to the s,y or z component of E where the Imc (r2) TC 4. Rank-2 tensors A charge q moves with a...
Problem 6. Describe the surface r(u, u)-R cos u x + R sin u ý + uz where 0 < u < 2π and 0 < u-H. and R and H are positive constants. What is the surface element and what is the total surface area? Show that Or/au, or/àv are continuous across the "cut at 2T coS W T
ic) At what time t, 0 sts 15,is the sauil's instantancous velocity equal to its average velocity over the isterval Osrs 15? (d) An ant arrives at the sniail's starting position at time r 12 minuses and tollows the snail's path. Daring the interval 12 , % 15 minutes, the am travels in the same direction as the snal with a constant acceleration of 2 inches per minute per minute, The art catches up to the snail at time -15...
() At)x()B(f)u() Consider the following time-varying system y(t) C(f)x(t) where x) R", u(t)E R R 1 1) Derive the state transition matrix D(t,r) when A(f) = 0 0 sint 2) Assume that x(to) = x0 is given and u(f) is known in the interval [to, 4] Based on these assumptions, derive the complete solution by using the state transition matrix D(f, r). Also show that the solution is unique in the interval [to, 4]. 3) Let x(1) 0 and u(f)...
4. Rank-2 tensors A charge q moves with a constant velocity u - ux. Define the antisymmetric tensor rue by r--(ημχν-ηνΖ), w 4. Rank-2 tensors A charge q moves with a constant velocity u = uˆx. Define the antisymmetric tensor r µν by r µν ≡ 1 c (η µx ν − η νx µ ), where η µ is the four velocity of the charge. Further define r 2 = −r µνrµν/2. (a) Show that r 2 = (x−ut)...
2/2 Problem 2 Suppose that the map T: D C R2R2 (u, v) T(u, v)- (TI(u, v), T2(u, v)) defines a change of variables whose Jacobian satisfies J(T) (u, v)1 for l (u, v) E D If R C D is a region whose area is 4, then what is the area of the region T(R) T(u, v)(u, v) E R? 5 marks 2/2 Problem 2 Suppose that the map T: D C R2R2 (u, v) T(u, v)- (TI(u, v),...