Replacing an Engine

Example 12 Replacing an Engine

An automobile engine has a weight

, whose magnitude is W = 3150 N. This engine is being positioned above an engine compartment, as Figure 4.29a illustrates. To position the engine, a worker is using a rope. Find the tension

in the supporting cable and the tension

in the positioning rope.

Figure 4.29

(a)

The ring is in equilibrium because of the three forces (the tension force in the supporting cable), (the tension force in the positioning rope), and (the weight of the engine).

(b)

The free-body diagram for the ring.

Reasoning Under the influence of the forces

, and

the ring in Figure 4.29a is at rest and, therefore, in equilibrium. Consequently, the sum of the x components and the sum of the y components of these forces must each be zero; ΣF_x = 0 and ΣF_y = 0. By using these relations, we can find T₁ and T₂. Figure 4.29b shows the free-body diagram of the ring and the force components for a suitable x, y axis system.

Solution

Problem-solving insight

When an object is in equilibrium, as here in Example 12, the net force is zero,

. This does not mean that each individual force is zero. It means that the vector sum of all the forces is zero.

The free-body diagram shows the components for each of the three forces, and the components are listed in the following table:

Force	x Component	y Component
	-T₁ sin 10.0°	+T₁ cos 10.0°
	+T₂ sin 80.0°	-T₂ cos 80.0°
	0	-W

The plus signs in the table denote components that point along the positive axes, and the minus signs denote components that point along the negative axes. Setting the sum of the x components and the sum of the y components equal to zero leads to the following equations:


	(4.9a)


	(4.9b)

Solving the first of these equations for T₁ shows that

Substituting this expression for T₁ into the second equation gives

Setting W = 3150 N in this result yields

Since

and T₂ = 582 N, it follows that