Solved Examples Based On Motion Of Connected Bodies Example: Determine the constant force P that will give the system of bodies shown in Fig. (a) a velocity of 2m/sec after moving 4m from rest. Coefficient of friction between the blocks and the plane is 0.2. Pulleys are smooth.
Solution: The system of forces acting on connecting bodies is shown in Figure
N1 = 300 N ∴ F1 = N1 = 0.2 × 300 = 60 N
N
2 = 1200 cosq = 1200 × 0.6 = 720 N
F2 = 0.2 N2 = 0.2 × 720 = 144 N
N3 = 600N
∴F3 = 0.2 × 600 = 120 N.
Let the constant force to be found be P. Writing work-energy equations to the motion of the system, we get
(PF1 F21200 sin 6 F3) s = W1 + W2 + W3 / 2g (v2 u2)
i.e., (P 60 144 1200 × 0.8 120)4 = 300 + 1200 + 600 / 2 × 9.81 (22 0)
∴ P = 1391.03 N
[Note: Work done is force x distance moved in the direction of force. Hence, N1, N2, N3,(i.e. 300N, 720N, 600N) forces are not contributing to the work done.]
Example: In what distance body A of Fig (a) attains a velocity of 3m/sec after starting from rest? Take u = 0.25. Pulleys are frictionless.
Solution Let θ1and θ2be the slopes of inclined planes as shown in the figure
sin θ1= 4/5 = 0.8, cos θ1= 0.6
sin q
2 = 3/6 = 0.6, cos q
2= 0.8 By observing pulley system, it may be concluded that if 1800N block moves a distance 's', 2400N block moves a distance 0.5 s. Hence, if the velocity of 1800 N block is v, that of 2400 N block is 0.5 v. Assuming 2400 N block moves up the plane and 1800 N block moves down the plane, the forces acting on 1800 N that will do work are [Ref. Fig. 14.14(b)] 1800 sin θ
1= 1440 N down the plane.
F1 = m 1800 cos θ1= 0.25 × 1800 × 0.6
= 270 N up the plane.
The forces acting on 2400 N block when it slides up are shown in Fig. (c). The forces that do work are
2400 sin θ2= 2400 × 0.6 = 1440 N down the plane
F2 = 0.2 × 2400 cos θ2= 0.2 × 2400 × 0.8
= 384 N down the plane.
Equating work done by various forces to change in the kinetic energy of the system, we get
(1440 270) s (1440 + 384) 0.5 s
= 1800/2 X 9.81 (v2 - 0) + 2400/2 X 9.81 [(0.5 v2) - 0]
Now, v = 3 m/sec
258s = 1800/2.981 X 32 + 2400/2 X 9.81 X 0.25 X 32
Note:
Since 's' is positive, the assumed direction of motion is correct. If it comes out to be negative, recalculations are to be made, since the frictional force changes the sign.