Practical - Assignment 2 PDF

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MECH1

Problem Sheet 2

Due Friday Feb 27

Dynamics Do all problems 1. A particle in the plane is acted upon by a force F (x, y) = y2 i + j. Determine the trajectory of the particle that starts from rest at the origin. 2. A lift is moving with acceleration a = αg, with |α| < 1. Knowing that the mass of the lift (with the load) is M , find the tension in the suspension cable. 3. A particle is suspended on a rope of length l. It is set in circular motion with constant speed, so that the angle between the rope and the vertical is α. Find the speed of the particle. 4. A particle of unit mass is suspended on a massless, unstretchable rope of unit length, the upper end of the rope being fixed. By means of the rope the particle is to move in the vertical plane. (Such a mechanical system is called mathematical pendulum, see the notes). Apart from gravity, the particle experiences a drag force (due to, say resistance of the air) as it moves. The drag force acts opposite to the particle’s velocity and its magnitude equals κv for some coefficient κ > 0. Use the general acceleration formula to write the equation of motion in terms of the angle α of the pendulum with the vertical downward direction. 5. Planes fly owing to the lift force, which is perpendicular to the wing. A dispatcher asks the crew to cruise above the airport at constant altitude in a circle of radius r with constant speed v. At what angle α will the plane tilt as it is cruising? 6. A particle of mass m slides without friction down the incline of the wedge of mass M . The wedge, in turn, is free to slide along the horizontal table. What is the number of degrees of freedom in this system? Find the accelerations of the particle and the wedge. (See figure, where for your convenience forces have been identified.) Modify the problem by assuming that there is friction between the wedge and the table, with coefficient µ.

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