5.5 Newton’s Second Law

Equation 5.3 is an important finding, but our experiment was limited to looking at an object’s response to a single applied force. Realistically, an object is likely to be subjected to several distinct forces $\vec{F}_{1}$ , $\vec{F}_{2}_{}$ , $\vec{F}_{3}_{}$ , . . . that may point in different directions. What happens then? In that case, it is found experimentally that the acceleration is determined by the net force.

A photo of a force sensor. — You may have used a *force sensor* in a physics lab to investigate how Newton’s laws apply to motion. Similar sensors are widely used in industry and robotics to monitor tensions and loads. Digital scales employ force sensors, as do the keys on an electric piano. A common miniature force sensor, also called a *strain gauge*, is a thin conducting wire deposited in a back-and-forth grid on a flexible backing. A current flowing from one end to the other is used to measure the wire’s electrical resistance. A force in the direction of the “fingers” stretches the wire and thereby increases its resistance. After calibration, the strength of the applied force can be deduced from the measured increase in resistance.

Newton was the first to recognize the connection between force and motion. This relationship is known today as Newton’s second law.

The significance of Newton’s second law cannot be overstated. There was no reason to suspect that there should be any simple relationship between force and acceleration. Yet there it is, a simple but exceedingly powerful equation relating the two. The critical idea is that an object accelerates in the direction of the net force vector $\vec{F}_{n e t} .$

We can rewrite Newton’s second law in the form

\vec{F}_{net} = m \vec{a}

(5.5)

which is how you’ll see it presented in many textbooks. Equations 5.4 and 5.5 are mathematically equivalent, but Equation 5.4 better describes the central idea of Newtonian mechanics: A force applied to an object causes the object to accelerate.

It’s also worth noting that the object responds only to the forces acting on it at this instant. The object has no memory of forces that may have been exerted at earlier times. This idea is sometimes called Newton’s zeroth law.

NOTE Be careful not to think that one force “overcomes” the others to determine the motion. Forces are not in competition with each other! It is $\vec{F}_{net}$ , the sum of all the forces, that determines the acceleration $\vec{a}$ .

As an example, FIGURE 5.18a shows a box being pulled by two ropes. The ropes exert tension forces $\vec{T}_{1}$ and $\vec{T}_{2}$ on the box. FIGURE 5.18b represents the box as a particle, shows the forces acting on the box, and adds them graphically to find the net force $\vec{F}_{net}$ . The box will accelerate in the direction of $\vec{F}_{net}$ with acceleration

\vec{a} = \frac{\vec{F}_{net}}{m} = \frac{\vec{T}_{1} + \vec{T}_{2}}{m}

The figure consists of two parts labeled ("a") and (b). Part ("a") illustrates the top view of a box which is being pulled by a rope from two different directions. Part (b) is a free-body diagram of the forces acting on the box in part ("a").

FIGURE 5.18 Full Alternative Text

NOTE The acceleration is not $(T_{1} + T_{2}) / m$ . You must add the forces as vectors, not merely add their magnitudes as scalars.