Determine the acceleration of the body:
Two weights P and Q are associated by the arrangement shown in Figure. Neglecting friction and the inertia of the pulleys and cord, determine the acceleration of the weight Q. Suppose P = 20 kg and Q = 15 kg.
Solution
No friction is involved and hence we can assume any direction of motion. Still we can do a little more guess work. Let us suppose block P at rest, then we have 2 T = 20 × 9.81. As P is not moving, Q also shall not move, therefore we get T = 15 × 9.81 which shows that the system will have a motion, i.e. Q shall move down and P shall be moving up. The value of T shall be in between the above two values. Furthermore, it shall be clear that acceleration of P shall be half of acceleration of Q. Considering FBD for the bodies we may write for the equilibrium of Q.
We have
T + 15 a = 15 × 9.81
∴ T = 147.15 - 15 a ---------- (1)
Equilibrium equations for P shall give us
2 T = 20 × (a/2 ) + 20 × 9.81
= 10 a + 196.2
∴ T = 5 a + 98.1 ---------- (2)
Considering the above two equations, we attain
T = 147.15 - 15 a = 5 a + 98.1
∴ 20 a = 49.05
a = 49.05/20 = 2.45 m / sec^{2}