- Part Ii Long Answer Questions Instructions You Must Answer All The Questions In This Part Please Clearly Show The Wor 1 (282.98 KiB) Viewed 23 times
Part II: Long Answer Questions Instructions: You must answer all the questions in this Part. Please clearly show the working for these problems in your answer booklet. (a) Consider a particle of mass m = 2 kg that moves with a position vector given by: r(t): = - ( 5 m/s²) ² m/s²t² x + (2 m/s) tŷ – (2 m) 2, (i) What are the velocity v, momentum p, speed v and kinetic energy T of the particle as a function of t? (ii) What is the acceleration a of the particle and the net force Fnet it experiences as a function of t? (iii) What is the angular momentum & of the particle and the net torque I'net it experiences as a function of t? (Evaluate these quantities with respect to the origin of the coordinate system). (iv) What is the displacement Ar made by the particle between t = 0 and t = 1 s and what is the total work done on the particle over this time interval? = (b) A particle moves in an outward spiralling motion in the 2D plane, described using polar coordinates as r(t) = (2 m/s) t and o(t) (2 rad/s) t, where t is in seconds. (i) Make a sketch of the path the particle takes in the xy-plane over the time period from t = 0 until t = 2π s. On the sketch clearly indicate the position vector of the particle at t = 3 s. Also, indicate the unit vectors f and at t = 37 S. (ii) Obtain an expression for the velocity of the particle as a function of time in polar coordinates. Your result should be in terms of the unit vectors ↑ and 6. What is the velocity and speed of the particle at t = 3 s? (iii) Obtain an expression for the acceleration of the particle as a function of time in polar coordinates. What is the acceleration of the particle at t = (c) The equation of motion for a particle falling under the influence of gravity and a linear drag force is mÿ = mg - by, where m is the particle mass, g is the acceleration due to gravity, b is the coefficient of linear drag, and the vertical coordinate y is taken to have units of meters. (Note: the y-axis points downwards). (i) Identify whether each of the forces acting is conservative. Give reasons for your identification. (ii) Define the terminal speed Uter and obtain an expression for this in terms of the quantities appearing in the equation of motion. (iii) Show that the equation of motion can be equivalently written as dvy 1 = (Vier - Vy), dt T where we have set T = m/b (a constant with units of time).