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Example_Sheet_1 - Part IB Physics B Classical Dynamics and...

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Part IB Physics B Classical Dynamics and Fluids — Examples 1 — 2009 There will be about 2 questions for each lecture. Problem grading: (A) Problems that can be answered directly by quoting the lectured material or by straightforward calculation. (B) Problems that require some algebraic formulation and manipulation as well as calculation. (C) Problems which are either harder or longer than (B) problems. You should feel a sense of achievement in completing these. Newtonian mechanics and the energy method 1. (A) A uniform solid cylinder is set spinning about its axis and is then gently placed, with the axis horizontal, on a rough horizontal table. What fraction of its initial kinetic energy is dissipated in sliding friction before the cylinder eventually rolls smoothly along the plane? 2. (B) A ladder of length l rests against a wall at and angle θ to the vertical. There is no friction between the top of the ladder and the wall and no friction between the bottom of the ladder and the ground. Write down the kinetic and potential energies of the ladder as a function of θ and ˙ θ . Using the energy method, or otherwise, derive the equation of motion of the ladder in terms of θ . In order to work out the velocity of the centre of mass, write down expressions for the height and horizontal positions of the centre of mass as a function of θ and then differentiate with respect to time. 3. (B) A small solid cylinder of radius r lies inside a large tube of internal radius R . Find the period of oscillation of the small cylinder about its equilibrium position. Hint: first show that the angular velocity of the small tube ω is related to θ , the angle between the centre of mass of the solid cylinder, the centre of the large tube and the vertical by ω = R - r r ˙ θ .
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2 Rotating frames and fictitious forces 4. (A) Paraboloidal telescope mirrors can be made by ‘spin casting’, which involves rotating the molten glass and its container about a vertical axis as the glass solidifies. By considering the equilibrium of an element of the molten surface show that d y d x = ω 2 g x where y is the height of the surface, x is the distance from the axis of rotation and ω is the angular velocity of rotation. For a mirror of focal length 2 m, what angular velocity is required? [The equation of a parabola is y = x 2 / 4 f , where f is the focal length.] 5. (B) A train at latitude λ in the northern hemisphere is moving due north with a speed v along a straight and level track. Which rail experiences the larger vertical force? Show that the ratio R of the vertical forces on the rails is given approximately by R = 1 + 8 Ωvh sin λ ga where h is the height of the centre of gravity of the train above the rails which are at a distance a apart, g is the acceleration due to gravity, and Ω is the angular velocity of the Earth’s rotation.
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