Question

The center of mass of a cart having a mass of 0.62 kg starts with a velocity of v = -3.0 m/s^x along an x-axis. It starts from a position of 2.0 m and moves without any noticeable friction acting on it to a position of 0.0 m. This is in spite of the fact that a fan assembly is exerting a force on it in a positive x-direction. However, instead of being powered by batteries the fan is driven by a voltage source that is programmed to change with distance from a motion detector. This program leads to a set of changes in the force that is shown in the diagram that follows.

Answer 1

Force is varying with position.

work done = F.dx

Total workdone is equal to the area under the F,x curve

1) from 2.0 to 1.2 force is linearly changing from 0.2 to 0.05 N

W1 = (2.0-1.2)*(0.2+0.05)/2 = 0.1 J

2 ) then constant 0.05N from 1.2 to 1.0

W2 = 0.05 * 0.2 = 0.01 J

3) and increase uniformly from 0.05 to 0.25 over x=1.0 to x=0.4

W3 = (1-0.4)*(0.05+0.25)/2 = 0.09 J

4) then constant 0.25 N from x= 0.4 to x=0.0

W4 = 0.4*0.25 = 0.1 J

Total work done W = w1+w2+w3+w4 =0.1+0.01+0.09+0.1 = 0.3 J

change in KE = workd done = 0.3 J

change in velocity = sqrt(2*0.3/0.62) = 0.98 m/s

Final velocity = -3.0 + 0.98 = -2.02 m/s