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) 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
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
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
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
terms of r, 0, r, 0, F, and 0.
(b) Sketch a free body diagram showing all
the forces on the ball
(c) Resolve forces in the radial (r) direction
to show that
T sinB=mrd2 =
(d) Resolve forces in the vertical direction
to show that
(e) Therefore show that h = 9/02.
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