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1. A body of mass m moves along the x-axis subject to a potential V (x). Show that the period τ of small oscillations about a stable equilibrium point x0 , is given by r m . τ = 2π 00 V (x0 ) Consider the potential V (x) = x2 x4 − 2 4 . If a body of unit mass is released from the point of stable equilibrium with velocity u, for what values of u does it (i) oscillate (ii) escape to infinity? 1 For the case u = √ show that the position at time t, is given by 2 √ 1 − e− 2t √ x= 1 + e− 2t 2. . Consider the equation of motion for an undamped, forced oscillator ẍ + ω02 x = F0 cos ωt , where ω, ω0 , F0 are constants. If the body is released from the origin with zero velocity at t = 0, show that its position at later times is given by x= 2F0 − ω2 ω02 sin(ω0 + ω)t 2 sin(ω0 − ω)t 2 . Derive an expression for the time dependence of the mechanical energy E(t) exactly at resonance, ω = ω0 . M1A1 MECHANICS (2007) Page 2 of 4 3. Consider the motion of a plane pendulum of length a. Using appropriate intrinsic co-ordinates (s, ψ), derive the expression for the conservation of energy E= 1 s mṡ2 + 2mg a sin2 2 2a , where g is the acceleration due to gravity. If the body is projected from s = 0 with velocity √ ṡ = 4ga , show that at later times r ! gt . s = 2a cos−1 sech a What type of motion does this correspond to qualitatively? 4. A body of mass m moves in a central force field F (r) directed away from the origin. Given that the radial and transverse accelerations in polar co-ordinates are arad = r̈ − rθ̇2 show that where u = and atrans = rθ̈ + 2ṙθ̇ 1 d2 u +u=− 2 dθ mh2 u2 1 r 1 F u , , and h is the angular momentum per unit mass. A body is projected from the point r = a, θ = 0 with transverse velocity 1 vtrans = a2 2 and zero radial velocity (vrad ≡ ṙ = 0) , in a force field 3a −m F (r) = 2 1 + . r 2r Show that the equation for the orbit is r= M1A1 MECHANICS (2007) a 2 − cos 2θ . Page 3 of 4 5. (a) A rocket of mass m(t) moves in a straight line under the influence of an external force F . Exhaust gas is expelled at constant velocity u relative to the rocket. Show that the equation of motion for the rocket is mv̇ + uṁ = F (b) where v̇ ≡ dv dt , , etc. Next consider the same rocket but moving around a circular track of radius a. Show that the equation of motion in polar co-ordinates is b − maθ̇2b maθ̈ + uṁ θ r=F , where b r and θb are the radial and transverse unit vectors and F is the external force. M1A1 MECHANICS (2007) Page 4 of 4