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Friction, Work, and Energy on the Inclined Plane Objective To check the validity of Newton’s Second Law by measuring the motion of a cart as it accelerates up an inclined plane under the action of gravitational and other forces. Theory From Newton's second law the force acting on an (inertial) object equals the product of object's mass and acceleration, F m a (1) F ma x x (2) F ma y y (3) Consider a system of masses m 1 placed on a rough, inclined plane is connected to a string that passes over a pulley and then is fastened to a hanging m 2 . Since we can neglect the masses of the string and the pulley, and the pulley is frictionless the tension at both ends of the string are the same (magnitude T ). Let us assume m 2 accelerates downward with magnitude a. Since the two masses are connected by the string the acceleration of m also has magnitude a. 1 The equation of motion for the two masses can be written as, for m1 : T m1 g sin f k m1a (4) N 1 m1g cos 0 N 1 m1g cos (5) Since f k k N 1 k m1g cos (6) Then Eq.(4) becomes T m1g sin k m1g cos m1a (7) for m 2 : m 2 g T m 2a (8) 1 Solving Eq.(7) and Eq.(8), we get m 2 g m1 g sin k m1g cos a (9) m1 m 2 Work done by friction: If an object such as a block is lifted, work is done in order to move the block. The work, W, done by a force is defined as the component of the force that produces motion parallel to the direction of the motion (F||) times the displacement of the object on which work is being done, S, in that same direction: w F S (10) A machine can be defined as any device that multiplies forces or changes the direction of forces in order to do work. Consider the machine in figure II, which shows a system of two masses connected by a pulley, where work done on m 2 is used to lift m1 up the plane. For the object with a given mass m 2 that moves downward, work is being done on the object by the force of gravity. The work done is simply the object’s weight times the distance through which it moved: w 2 m2 g S (11) Since m1 and m 2 are connected by ropes, then the vertical distance S that m 2 moves downward is the same distance along the path of the inclined plane that m1 moves. The vertical distance that m1 moves up the plane is related to this distance by y 2 y1 sin S The total work done on m1 in order to move it is due to two distinctive kinds of forces: conservative and non-conservative. A force is conservative when it does no work on an object that moves around a closed path (the object starts and finishes at the same point). The gravitational force is a conservative force; hence, any work done by or against gravity within the system of two blocks is conservative work. w g (m1 g sin )S (12) The second component of work in our system is due to non-conservative forces. A force is non-conservative (or dissipative) if the work it does on an object moving between two points depends on the path of the motion between the points. Useful work is always lost to the kinetic frictional force because it dissipates into heat, which is un-recoverable in our system to do useful work. The non-conservative work for this system is then defined 2 by the frictional force times the distance through which the block moves. The expression for the work lost due to friction, w nc is:, w nc f k S = (k m1 g cos )S (13) Apparatus: A wooden block, an inclined plane with pulley, cord, weights, a balance, and a meter stick. Procedure: 1) Place the wooden block at the bottom of the inclined plane [angle of inclination = 04] 2) Determine the time, t , that the wooden block will travel distance, S. 3) Repeat paragraph 2 many times for the same distance S. 1 4) Find the average time, t , and calculate the acceleration of the block S at 2 . 2 5) Coefficient of kinetic friction is given by: m 2 g m1 g sin (m1 m 2 )a k . m1 g cos 6) Use eq. (7) or (8) to evaluate the tension in the rope. 7) Use eqs. (11- 13) work done within the system of two blocks. 3