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Proper problem-solving techniques not only help guide you through the
difficulties of a problem but also enable you to communicate your solution
effectively to others. When working out solutions to problems in this course,
please bear the following points in mind.|
Sample ProblemHalliday, Resnick, and Krane, 6-26, p. 125.
Object B weighs 94.0 lb and object A weighs 29.0 lb. Between object B and the plane the coefficient of static friction is 0.56 and coefficient of kinetic friction is 0.25. (a) Find the acceleration of the system if B is initially at rest. (b) Find the acceleration if B is moving up the plane. (c) What is the acceleration if B is moving down the plane? The plane is inclined by 42.0°.
W = 94.0 lb uk = 0.25 w = 29.0 lb us = 0.56 [[alpha]] = 42.0deg.
From the force diagram for A, we get the equation
From the force diagram for B, we get the pair of equations
If we assume that the string is inextensible, then . We further assume that the string and pulley are massless, and that the pulley is frictionless. Then the tension appearing the the first and third equations is the same.
(a) The frictional force must be less than or equal to the product of the normal force and the coefficient of static friction, us, for the object to remain at rest. Let us calculate u assuming the objects remain at rest to determine whether this is less than the given value. Substituting in the third equation gives
which simplifies to
Since u < us, the objects remain at rest.
(b) If B is initially moving up the plane, then the sense of the frictional force is opposite to that indicated in the diagram. We can solve for either case by writing the third equation as
where the upper sign corresponds to the case of motion down the plane, and the lower sign to motion up the plane. Solving for the acceleration gives
which simplifies to
This clearly has the right dimensions. Plugging in numbers and using g = 32.2 ft s-2 gives
for motion up the plane, and
for motion down the plane. These values have been rounded to 3 significant figures, in agreement with the given information. Is it reasonable that the acceleration for motion down the plane is greater? Yes, because while friction opposes the motion in both cases, gravity opposes the motion in the former case and drives it in the latter. Furthermore, based on our choice of coordinate system as indicated on the first sketch, the acceleration is down the plane, as it must be.
Copyright © 2001 Harvey Mudd College Physics Department|
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