Set 6 Problem number 9

• Problem

• Solution

Problem

On a graph of the net force on an object vs position, with force in Newtons parallel to the x axis and position in meters along the x axis,  we find the points ( 14 , 3 ) and ( 24 , 4 ).

• What does the area under the line segment connecting these points (i.e., the area between the line segment and the horizontal axis) represent?

Solution

The area under the segment will consist of a trapezoid with altitudes 4 Newtons and 3 Newtons, and uniform width ( 24 meters - 14 meters) = 10 meters.

• The average of the altitudes represents the average of the forces 4 Newtons and 3 Newtons.

• If force is changing at a uniform rate the segment in fact represents the force vs. position precisely; otherwise it is only an approximation to the behavior of a curving graph.
• In general we therefore say that the average of the two forces is the approximate, not the exact, average force on the interval.

• The width of the trapezoid represents the displacement over which this approximate average force is sustained.
• The area of the trapezoid is the product of the average altitude, representing the approximate average force on the interval, and the width, representing the displacement.

• The product of net force and time interval is equal to the work done on the object, which will be the change in the KE of the object.
• The area therefore represents approximate average force * displacement = work done on object = change in KE of the object during the time interval.

The average altitude in the present example is ( 3 Newtons + 4 Newtons) / 2 = 3.5 Newtons.

• The area is thus 3.5 Newtons * 10 meters = 35 Newton meters = 35 Joules
• We interpret this as the approximate work done on the object, which is the approximate change in KE corresponding to the two points on the graph.