 # A simple wheel and piston mechanism is used to generate linear moti...

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A simple wheel and piston mechanism is used to generate linear motion from circular motion, as shown in the diagram. The connecting rod AB is connected to the wheel at point A, and to a horizontal bar at the pivot B. The wheel spins anticlockwise at constant angular velocity w. At =0, the displacement / of the piston is 0. For each of the following statements, state if it is true or false. In each case, give a brief (one sentence) reason. You should not need to do any calculations. (1 mark is given for a correct true/false; 1 mark is given for a valid explanation.) A / y 4 B ß 00 x O 0 0 (a) In polar coordinates centred about o, the velocity of point A has no component in the radial direction. (b) The acceleration of point A is always towards the origin O. (c) The acceleration of point B is always towards or away from the origin o. (d) The rate of change of angle OAB with respect to time is equal to B. (e) The piston is in simple-harmonic motion (f) The net force exerted on the rod AB has no component in the vertical (y) direction. A garden sprinkler is shown in plan Angular speed w view in the diagram (Note that the left-hand arm is not completely shown). Water rises through the speed V central spindle (the circle) and makes its way out along the two arms at a constant speed v, propelling the sprinkler round at constant angular velocity. Consider a 'piece' of water making its way up r the right-hand pipe, a distance r from the spindle, as shown. (a) The sprinkler is rotating anticlockwise at a constant angular velocity w. Using polar co- ordinates centred on the spindle, write down vector expressions for the velocity and the acceleration of the piece of water in terms of w, V and r, and unit vectors er and e, in the radial and circumferential directions respectively [4 marks]. (b) On a sketch of the sprinkler system, mark on the approximate directions of the velocity and acceleration of the piece of water [4 marks]. A ball of mass m is attached to a string of length L and secured at a point o. It is swung in a circular orbit at constant angular velocity 0 and constant apex angle ß as shown in the diagram. The string is under tension T. (a) In cylindrical polar coordinates, using the origin as the fixed point o at the top of the string, write down the acceleration of the ball in O I terms of r, 0, r, 0, F, and 0. (b) Sketch a free body diagram showing all ß the forces on the ball L (c) Resolve forces in the radial (r) direction to show that T sinB=mrd2 = (d) Resolve forces in the vertical direction m to show that Tcosß=mg (e) Therefore show that h = 9/02.

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