Worked example 4.3: Raising a platform

Question: Consider the diagram. The platform and the attached frictionless pulley weigh a total of 34N. With what force $F$ must the (light) rope be pulled in order to lift the platform at $3.2 {\rm m/s^2}$?

Answer: Let $W$ be the weight of the platform, $m=W/g$ the mass of the platform, and $T$ the tension in the rope. From Newton's third law, it is clear that $T=F$. Let us apply Newton's second law to the upward motion of the platform. The platform is subject to two vertical forces: a downward force $W$ due to its weight, and an upward force $2 T$ due to the tension in the rope (the force is $2 T$, rather than $T$, because both the leftmost and rightmost sections of the rope, emerging from the pulley, are in tension and exerting an upward force on the pulley). Thus, the upward acceleration $a$ of the platform is

$$a = \frac{2 T-W}{m}.$$

Since $T=F$ and $m=W/g$, we obtain

$$F = \frac{W (a/g+1)}{2}.$$

Finally, given that $W=34 {\rm N}$ and $a=3.2 {\rm m/s^2}$, we have

$$F = \frac{34 (3.2/9.81+1)}{2} = 22.55 {\rm N}.$$