Forces and Newton's Laws of Motion

Example

Towing a Supertanker

A supertanker of mass

is being towed by two tugboats, as in Figure 4.30a. The tensions in the towing cables apply the forces

and

at equal angles of

with respect to the tanker's axis. In addition, the tanker's engines produce a forward drive force

, whose magnitude is

. Moreover, the water applies an opposing force

, whose magnitude is

. The tanker moves forward with an acceleration that points along the tanker's axis and has a magnitude of

. Find the magnitudes of the tensions

and

Figure 4.30
(a) Four forces act on a supertanker: and are the tension forces due to the towing cables, is the forward drive force produced by the tanker's engines, and is the force with which the water opposes the tanker's motion. (b) The free-body diagram for the tanker.

Reasoning

The unknown forces

and

contribute to the net force that accelerates the tanker. To determine

and

, therefore, we analyze the net force, which we will do using components. The various force components can be found by referring to the free-body diagram for the tanker in Figure 4.30b, where the ship's axis is chosen as the x axis. We will then use Newton's second law in its component form,

and

, to obtain the magnitudes of

and

Solution

The individual force components are summarized as follows:

Force	x Component	y Component


		0
		0

MATH SKILLS

The sine and cosine functions are defined in Equations 1.1 and 1.2 as

and

, where

is the length of the side of a right triangle that is opposite the angle

is the length of the side adjacent to the angle

, and h is the length of the hypotenuse (see Figure 4.31a). When using the sine and cosine functions to determine the scalar components of a vector, we begin by identifying the angle

. Figure 4.31b indicates that

for the vector

. The components of

are

and

. Comparing the shaded triangles in Figure 4.31, we can see that

, and

. Therefore, we have

Figure 4.31

Math Skills drawing.

Since the acceleration points along the x axis, there is no y component of the acceleration

. Consequently, the sum of the y components of the forces must be zero:

This result shows that the magnitudes of the tensions in the cables are equal,

. Since the ship accelerates along the x direction, the sum of the x components of the forces is not zero. The second law indicates that

Since

, we can replace the two separate tension symbols by a single symbol T, the magnitude of the tension. Solving for T gives