The Vector Nature of Newton’s Second Law of Motion

When a football player throws a pass, the direction of the force he applies to the ball is important. Both the force and the resulting acceleration of the ball are vector quantities, as are all forces and accelerations. The directions of these vectors can be taken into account in two dimensions by using x and y components. The net force

[&~bf~Fˆ{*N*[−6.5%0.0]|bfArarr|}&] in Newton’s second law has components

and

, while the acceleration

has components a_x and a_y. Consequently, Newton’s second law, as expressed in Equation 4.1, can be written in an equivalent form as two equations, one for the x components and one for the y components:


	(4.2a)


	(4.2b)

This procedure is similar to that employed in Chapter 3 for the equations of two-dimensional kinematics (see Table 3-1). The components in Equations 4.2a and 4.2b are scalar components and will be either positive or negative numbers, depending on whether they point along the positive or negative x or y axis. The remainder of this section deals with examples that show how these equations are used.

Example 2

Applying Newton’s Second Law Using Components

A man is stranded on a raft (mass of man and raft = 1300 kg), as shown in Figure 4-6a. By paddling, he causes an average force

of 17 N to be applied to the raft in a direction due east (the +x direction). The wind also exerts a force

on the raft. This force has a magnitude of 15 N and points 67° north of east. Ignoring any resistance from the water, find the x and y components of the raft’s acceleration.

Reasoning Since the mass of the man and the raft is known, Newton’s second law can be used to determine the acceleration components from the given forces. According to the form of the second law in Equations 4.2a and 4.2b, the acceleration component in a given direction is the component of the net force in that direction divided by the mass. As an aid in determining the components

and

of the net force, we use the free-body diagram in Figure 4-6b. In this diagram, the directions due east and due north are the +x and +y directions, respectively.

Figure 4-6 (a) A man is paddling a raft, as in Examples 2 and 3. (b) The free-body diagram shows the forces

and

that act on the raft. Forces acting on the raft in a direction perpendicular to the surface of the water play no role in the examples and are omitted for clarity. (c) The raft’s acceleration components a_x and a_y . (d) In 65 s, the components of the raft’s displacement are x = 48 m and y = 23 m.

Solution Figure 4-6b shows the force components:

Force	x Component	y Component
	+17 N	0 N
	+(15 N) cos 67° = + 6 N	+(15 N) sin 67° = + 14 N
	ΣF_x = + 17 N + 6 N = + 23 N	ΣF_y = + 14 N

The plus signs indicate that

points in the direction of the +x axis and

points in the direction of the +y axis. The x and y components of the acceleration point in the directions of

and

, respectively, and can now be calculated:


	(4.2a)


	(4.2b)

These acceleration components are shown in Figure 4-6c.

Example 3

The Displacement of a Raft

At the moment the forces

and

begin acting on the raft in Example 2, the velocity of the raft is 0.15 m/s, in a direction due east (the +x direction). Assuming that the forces are maintained for 65 s, find the x and y components of the raft’s displacement during this time interval.

Reasoning Once the net force acting on an object and the object’s mass have been used in Newton’s second law to determine the acceleration, it becomes possible to use the equations of kinematics to describe the resulting motion. We know from Example 2 that the acceleration components are

and

, and it is given here that the initial velocity components are

and

. Thus, Equation 3.5a

and Equation 3.5b

can be used with t = 65 s to determine the x and y components of the raft’s displacement.

Solution According to Equations 3.5a and 3.5b, the x and y components of the displacement are

Figure 4-6d shows the final location of the raft.

Check Your Understanding 1

All of the following, except one, cause the acceleration of an object to double. Which one is it? (a) All forces acting on the object double. (b) The net force acting on the object doubles. (c) Both the net force acting on the object and the mass of the object double. (d) The mass of the object is reduced by a factor of two. (The answer is given at the end of the book.)

Background: This problem depends on the concepts of force, net force, mass, and acceleration, because Newton’s second law of motion deals with them.

For similar questions (including calculational counterparts), consult Self-Assessment Test 4.1. This test is described at the end of Section 4.5.