# The wire forms a loop and passes over

The wire forms a loop and passes over the small pulleys at A,B,C, and D. If the maximum resultant force that the wire can exert on each pulley is 120 N, determine the greatest force P that can be applied to the wire as shown.

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.

#### Solution:

Show me the final answer↓
Let us draw a free body diagram showing force P being applied to the wire.

We can write an equation of equilibrium for the y-axis forces to express T in terms of P.

$+\uparrow \sum \text{F}_\text{y}\,=\,0$

$T\text{cos}\,(30^0)\,+\,T\text{cos}\,(30^0)\,-\,P\,=\,0$

$2T\text{cos}\,(30^0)\,=\,P$

(isolate for T)

$T\,=\,\dfrac{P}{2\text{cos}\,(30^0)}$

(Simplify)

$T\,=\,0.577P$

From the previous question, we know that pulleys B and C resists the largest resultant force. Therefore, we will focus on either pulley B or C to figure out the largest possible value of P that can be applied to the wire.

Let us draw a free body diagram of pulley B.

The resultant force can be calculated using the Pythagorean theorem.

$F_R\,=\,\sqrt{(0.577P)^2+(0.577P)^2}$

$F_R\,=\,0.816P$

We know the maximum force each pulley can withstand is 120 N. Therefore:

$120\,=\,0.816P$

(Solve for P)

$P\,=\,147$ N

Final Answer:

Greatest force P: 147 N