PHY41M1 Classical Mechanics Problem sheet 1.2019 - PHYSICS DEPARTMENT PHY41M1 CLASSICAL MECHANICS Tutorial sheet 1 19 April 2019 1 2 3 4 5 6 7 8 9

# PHY41M1 Classical Mechanics Problem sheet 1.2019 -...

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PHYSICS DEPARTMENT PHY41M1 CLASSICAL MECHANICS Tutorial sheet 1 19 April 2019 1. Consider the cylindrical ()z,,φρand Cartesian ()zyx,,coordinates. (a) Write the transformation equations from ()zyx,,to ()z,,(b) Express the unit vectors zφρˆ,ˆ,ˆin terms of the unit vectors zyxˆ,ˆ,(c) Find the time derivatives of the unit vectors zφρˆ,ˆ,ˆ2. Do (1) for the spherical coordinates ()φθ,,rand the unit vectors φθrˆ,ˆ,3. Obtain the relations between the following in the centre of mass (com) and labframes: (a) the linear momenta (b) the kinetic energies (c) the angular momenta 4. If the mass density is constant, find the position of the centre of mass for: (a) a semicircular plane sheet of radius r0and surface mass density (b) a semicircular wire of radius r0and line mass density (c) a semi-cylindrical rod of radius r0, length land volume mass density (d) a semi sphere lying on base of radius r0and volume mass density ρ(e) a cone with circular base of radius r0, height hand volume mass densityρ5. φ ρ ˆ . . ˆ . σ . λ . ρ . 0 . 0 .