Quiz1_Code1.pdf

Question and Response Sheet. Quiz 1 PHY401. 28/08/2024. Time : 40 Minutes. Marks : 10 Name :............................................................. Roll number :............................................................. This paper has 2 back to back printed pages and 10 questions in tot...

Question and Response Sheet. Quiz 1 PHY401. 28/08/2024. Time : 40 Minutes. Marks : 10 Name :............................................................. Roll number :............................................................. This paper has 2 back to back printed pages and 10 questions in total. Each question carries 1 mark. Steps leading to your answer must be shown systematically in the copy provided separately. Any answer that has been entered but working has not been shown in the copy will attract penalty of 2 marks. For each question, please give only a numerical answer, and only in the corresponding answer box provided with each question. Answer to a given question written anywhere else or in a wrong box will not be considered. 1) A particle of mass 1Kg is constrained to move on the inside of a vertical frictionless cylindrical surface of radius 1m whose vertical axis passes through the origin. It is subject to a force F~ = −2~r that acts towards the origin, where r is the distance of the particle from the origin at any instant. At an instant when z = 3m, 2 what is the magnitude of ddt2z in ms−2 ? Assume g = 10 ms−2. Ans 1: 2) A thin rod of mass 1Kg and length 4m is arranged to make a constant angle 600 with the horizontal, with its bottom end sliding in a circle on a horizontal frictionless ring of radius 2m with angular speed √13 rad/sec. The angular momentum about the center of mass of the rod is L ~ A. What is the magnitude of the horizontal component of L ~ A in Kg m2 /sec ? (Assume that the moment of inertia of a thin horizontal rod of mass m and length L about a vertical axis passing through its center is mL2 /12). Ans 2: 3) A massless axle has one end attached to a wheel (a uniform disk of mass 1 Kg and radius 1m), with the other end pivoted on the ground. The axle is perpendicular to plane of the wheel. The wheel slides on fric- tionless ground, with the axle inclined at an angle 450 to the horizontal. The point of contact on the ground traces out a circle with frequency 4 rad/sec. About the point O (shown in figure) what is the horizontal component of the total angular momentum Kg m2 /sec ? Ans 3: 4) In the Atwood’s machine shown in the figure, with the string always taut, the masses hung from the pulleys are 5 Kg, 2 Kg and 1 Kg. All pulleys are small and massless and all ropes are massless. The system starts from rest. Let x1 and x3 denote the positions shown. At a given instant of time, it is found that ẋ1 = 1 m/sec. At that instant, what is the magnitude of ẋ3 ? Ans 4: 2 5) A rigid T consists of a long rod glued perpendicular to another rod of length 1 m that is pivoted at the origin. The T rotates around in a horizontal plane with constant frequency 1 rad/sec. A particle of mass 0.5 Kg is free to slide along the long rod and is connected to the intersection of the rods by a spring with spring constant 1 N/m and relaxed length zero. r(t) denotes the position of the particle along the rod. If at t = 0, r = 21 and dr dt = 0, then what is the magnitude of r at t = π3 ? Ans 5: 2 6) For a hypothetical particle of unit mass, the Hamiltonian is given as H = p2 + pq + q 2. Assume that dimensions have been taken care of. Assume that motion starts with q = 1, q̇ = 3 at t = 0. What is the numerical value of the Hamiltonian at t = 1 ? Ans 6: 7) A particle of mass m = 1Kg is free to slide along a long massless frictionless rod lying in the x direction. At t = 0, the rod coincides with the x axis and starts from rest, and moves up vertically with constant acceleration 2 ms−2. At t = 1 sec, dx dt = 2 m/sec. Assume that potential energy is measured from the −2 x-axis, also assume g = 10 ms. Then, at t = 1 sec, what is the numerical value of the Hamiltonian of the system (in Joules) ? Ans 7: 8) A dynamical system is defined by the equations ẋ = y(13 − x2 − y 2 ) , ẏ = 12 − x(13 − x2 − y 2 ). Clearly this is a Hamiltonian system. Take x = q and y = p. What is the numerical value of the Hamiltonian at x=y=1? Ans 8: 9) A dynamical system is defined by ẋ = −3 cos x − 2 cos y , ẏ = −3 cos y − 2 cos x. Consider the region in the first quadrant with x2 + y 2 ≤ π 2. In the fixed point located in this region, what is the product of the eigenvalues of the linearised system ? Ans 9: 10) Consider the equation ẍ + x − ẋ(1 − x2 ) = 0. At the fixed point, what is the product of the eigenvalues of the linearised system ? Ans 10: Cylindrical √ coordinates are (ρ, φ, z); x = ρ cos φ, y = ρ sin φ. For a 2 × 2 matrix, the eigenvalues are 1
τ ± τ 2 − 4∆ where τ is the trace and ∆ is the determinant of the matrix. 2