Term Test AER301F. Dynamics. 5 November The value of each question is indicated in the table opposite.

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1 U N I V E R S I T Y O F T O R O N T O Facuty of Appied Science and Engineering Term Test AER31F Dynamics 5 November 212 Student Name: Last Name First Names Student Number: Instructions: 1. Attempt a questions. 2. The vaue of each question is indicated in the tabe opposite. 3. Write the na answers ony in the boxed space provided for each question. 4. This is a cosed-book test. 5. No cacuator is permitted. For Examiner Ony Probem Vaue Mark Tota 5 6. There are 11 pages and 5 probems in this test paper.

2 A. Denitions and Statements 1(a). What is meant by an inertia frame? A frame in which the Law of Inertia hods; aternativey, a frame in which Newton s Second Law hods. 1(b). State the definition for anguar veocity. ω v = F v T ω where ω = F v F v T 1(c). State the strong version of Newton s Third Law. f v b a = f v a b and (r v a r v b) f v b a = v where r v a, r v b are positions of partices a, b. 1(d). State the principe of virtua work. At equiibrium, δw = i f v app δr v = 1(e). State Hamiton s principe. The motion of a system is given by tb δ Ldt = t a where L = T V. 2

B. Questions Provide just the answers. 2(a). Give an exampe of when a singuarity woud occur in the use of Euer anges to represent a rotation. Consider the 3-1-3 set of Euer anges.

3 B. Questions Provide just the answers. 2(a). Give an exampe of when a singuarity woud occur in the use of Euer anges to represent a rotation. Consider the set of Euer anges. A singuarity woud occur at θ 2 =. /3 2(b). In what situations woud one use the method of Lagrange s undetermined mutipiers? Name three. 1. When a nonhoonomic constraint is present. 2. When a hoonomic constraint is present yet it might be too messy to sove dependent coordinates in terms of independent ones. 3. When a constraint force is required. /3 2(c). Sketch the transfer orbit to the Moon used by Apoo. (Indicate the motion of the Moon as we.) Why was this shape of orbit used? The figure-8 shape provides for gravity braking at the Moon, to assist in orbit insertion, and at the Earth for reentry. /3 3

4 C. Probems 3. The eiptica orbit of a panet can be represented as and reca that r 2 θ = h. r = 1 + ε cos θ (a) Show that the veocity of the panet as expressed in F o is v = h ε sin θ 1 + ε cos θ (b) Determine the acceeration of the panet as expressed in F o. (c) Is the acceeration proportiona to r 2 in the negative o v 1 direction? Why? 4

5 3(a). Show that the veocity of the panet as expressed in F o is v = h ε sin θ 1 + ε cos θ The position of the panet, in F o, is r (1 + ε cos θ) 1 r = = and the anguar veocity F o with respect to inertia space is ω = θ Hence, the inertia veocity in F o is given by v = ṙ + ω r in which ṙ = ε θ sin θ(1 + ε cos θ) 2 = h 1 ε sin θ and ω r = θ(1 + ε cos θ) 1 = h 1 (1 + ε cos θ) because r 2 θ = h. This eads to the desired resut. /4 5

6 3(b). Determine the acceeration of the panet as expressed in F o. The acceeration is determined in ike manner: a = v + ω v Here v = h θ ε cos θ ε sin θ, ω v = h θ (1 + ε cos θ) ε sin θ Thus a = h θ = h2 1/r2 /4 3(c). Is the acceeration proportiona to r 2 in the negative o v 1 direction? Why? Yes. It has to be, no?! The force, according to Newton s Second Law is proportiona to the acceeration and the force of gravity is proportiona to r 2 so the acceeration must be as we. 6

7 4. A uniform rope of mass density (per unit ength) ρ is ying on the ground and is pued up by one end at constant speed v. What is the force required to do this as a function of x, the height of the rope s end off the ground? At time t, a ength x of rope is off the ground. The momentum in the vertica direction (positive upward) is At time t + dt, the momentum is p(t) = ρxv p(t + dt) = ρ(x + dx)v The equation of motion is given by p(t + dt) p(t) = fdt Now f incudes the appied force f app that is required to ift the rope as we as the force due to gravity, which is (ρx)g. Hence, That is, ρvdx = (f app ρgx)dt f app = ρv dx dt + ρgx = ρ(v2 + gx) because dx/dt = v. /1 7

8 5. Consider a partice of mass m siding without friction on a vertica cone of haf-ange α. (a) (b) (c) (d) (e) Using coordinates r, the distance from the vertex of the cone to the partice, and θ, the horizonta anguar position, determine the kinetic energy of the partice. Using the same coordinates, determine the potentia energy of the partice. Determine the equations of motion. Show that h = mr 2 θ sin 2 α is a constant. What kind of coordinate is θ? Show that the equation for r can be obtained by taking the effective potentia energy as V eff = h 2 2mr 2 sin 2 + mgr cos α α 8

9 5(a). Using coordinates r, the distance from the vertex of the cone to the partice, and θ, the horizonta anguar position, determine the kinetic energy of the partice. The two components of veocity are ṙ in the ( radia ) direction of the side of the cone and (r sin α) θ in the tangentia direction. These direction are orthogona and hence the square of the speed is v 2 = ṙ 2 + r 2 θ2 sin 2 α and the kinetic energy accordingy is T = 1 2 m(ṙ2 + r 2 sin 2 α θ 2 ) /3 5(b). Using the same coordinates, determine the potentia energy of the partice. V = mgr cos α /1 9

10 5(c). Determine the equations of motion. By Lagrange s equations (no nonconservative force), with L = T V, ( ) d L L dt ṙ r = where L ṙ = mṙ, giving And where d dt L θ = mr2 θ sin 2 α, ( ) L = m r, ṙ m r mr θ 2 sin 2 α + mg cos α = d dt ( ) L d dt θ ( ) L θ L r = mr θ 2 sin 2 α mg cos α L θ = = m sin 2 α(2rṙ θ+r 2 θ), L r = yieding m sin 2 α(2rṙ θ + r 2 θ) = /4 1

11 5(d). Show that h = mr 2 θ sin 2 α is a constant. What kind of coordinate is θ? The second of Lagrange s equation can be integrated; in fact, we can observe that because L/ θ =, ( ) d L dt θ = so h = mr 2 θ sin 2 α must be constant. (It is the anguar momentum about the axis of the cone.) The coordinate θ, because it doesn t enter into L is caed an ignorabe or cycic coordinate. /3 5(e). Show that the equation for r can be obtained by taking the effective potentia energy as h 2 V eff = 2mr 2 sin 2 + mgr cos α α Using h, we may rewrite the first of Lagrange s equations as m r = h 2 mr 3 sin 2 mg cos α α We can recognized the right-hand side as the (negative) derivative of a potentia function, namey, V eff = h 2 2mr 2 sin 2 + mgr cos α α 11

Module 22: Simple Harmonic Oscillation and Torque

Module 22: Simple Harmonic Oscillation and Torque Modue : Simpe Harmonic Osciation and Torque.1 Introduction We have aready used Newton s Second Law or Conservation of Energy to anayze systems ike the boc-spring system that osciate. We sha now use torque