22 a Consider a system with j =1. Explicitly write (j=1, m|J,\j=1, m) in 3x3 matrix form.
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- A.) i. Define the plane polar unit vectors ŕ and 0 and show that they are orthonormal. Determine dr dê and de de ii. Hence show that for the vector r(t) = r(t)ŕ dr · = rî +rðê, d²r and = - dt dt² († − rė²)ŕ + (2ŕė +rë)ê . iii. If a mass m moves in the plane along r(t) = a and (t) = 5t determine the shape of the motion and find its period. B) Show that the motion of an object under the universal gravitational force μη Fg is governed by the radial and angular equations of motion and μ = 0, 2r0+ rö = 0. Setting r(t) =R for some constant R, solve these equations of motion and determine the period of the motion. C) Consider the following non-linear differential equation *+5= x +4. Find the stationary solutions. Expand around each of the stationary so- lutions that you have found, in order to determine if they correspond to stable or unstable equilibria. D) Show that 1 5 E = 2 - 4x is a conserved quantity. Hence draw the phase-space diagram and de- termine for which values of E oscilations…Consider a vector a = N i=1 ₁a¡ê; expressed in an orthonormal basis. Which of the following statements are true? Negative marking applies. Select one or more: a. (aa) = ₁ ai N i=1 b. (ala) (a|a) = 0 N (a|a) = Σ;₁a;ªz •j=1 □ d. e. f. = (ala) = (aa) = g. (aa) = 1 N ₁a²/ N i=1 * ₁ ažai ₁|a₁|² NLet vectors A⃗=(2,−4) A → = ( 2 , − 4 ) and B⃗=(−3,1) B → = ( − 3 , 1 ) . Calculate the following: What is the angle θAB θ A B between A⃗ A → and B⃗ B → ?
- Vector B→B→ has magnitude 7.30 and direction 14.0° below the +x-axis. Vector C→C→ has x-component Cx = −2.00 and y-component Cy = −6.70. a. What is the x-component of B→B→ ? b.What is the y-component of B→B→ ? c. What is the magnitude of C→C→ ? d. What angle does C→C→ make with the +x-axis in counterclockwise direction? A positive angle is counterclockwise from the +x-axis anwer in degrees e. What is the magnitude of C→+B→C→+B→ ? f. What angle does the resultant vector C→+B→C→+B→ make with the +x-axis in clockwise direction? A positive angle is counterclockwise from the +x-axis. asnwer in degrees g. What is the magnitude of C→−B→C→-B→ ? h. What angle does the vector C→−B→C→-B→ make with the +x-axis in counterclockwise direction? A positive angle is counterclockwise from the +x-axis. asnwer in degrees i. What is the x-component of C→−B→C→-B→ ? j.What is the y-component of C→−B→C→-B→ ?Calculate the magnitudeWhat is the x-component and magnitude of A→?
- find u(x,y) , v(x,y) where f(z)=u+iv for function f(z)=Z2 e2zVector B→B→ has magnitude 7.30 and direction 14.0° below the +x-axis. Vector C→C→ has x-component Cx = −2.00 and y-component Cy = −6.70. b. What is the y-component of B→B→ ? c. What is the magnitude of C→C→ ? d. What angle does C→C→ make with the +x-axis in counterclockwise direction? A positive angle is counterclockwise from the +x-axis. (answer in degreees) e. What is the magnitude of C→+B→C→+B→ ?Vector A→, which is directed along an x axis, is to be added to vector B→, which has a magnitude of 9.30 m. The sum is a third vector that is directed along the y axis, with a magnitude that is 5.10 times that of A→. What is the magnitude of A→?
- 6. Given vectors Ã= 2â, + 4ây + 10â, and B = -5âp + âo – 3âz, find (a) Ã+ B at P(0, –2,5) , with o = 90° (b) ÷Ē (c) Ã× B (d) The angle between A and B at P (e) The scalar component of A along B at PC pleaseUsing spherical polar coordinates r, 0, p to find CM of a uniform solid hemisphere of radius R, whose flat face lies in the xy plane with its center at the origin. The element of volume is in spherical polars of dV = r² dr sine de dip.