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 occur. Discuss the relation between these results and the analysis in part (c) above.

The kinetic energy of a mass moving in the x-y plane is: T = mx² + my² Using the transformation = rcos (0) and y = rsin (0), derive a general expression for kinetic energy of the mass in the r, basis.

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 N

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→ ?