Newton’s Second Law

What about cases where the total force on an object is not zero, so that Newton's first law doesn't apply? The object will have an acceleration. The way we've defined positive and negative signs of force and acceleration guarantees that positive forces produce positive accelerations, and likewise for negative values. How much acceleration will it have? It will clearly depend on both the object's mass and on the amount of force.

Experiments with any particular object show that its acceleration is directly proportional to the total force applied to it. This may seem wrong, since we know of many cases where small amounts of force fail to move an object at all, and larger forces get it going. This apparent failure of proportionality actually results from forgetting that there is a frictional force in addition to the force we apply to move the object. The object's acceleration is exactly proportional to the total force on it, not to any individual force on it. In the absence of friction, even a very tiny force can slowly change the velocity of a very massive object.

Experiments also show that the acceleration is inversely proportional to the object's mass, and combining these two proportionalities gives the following way of predicting the acceleration of any object:

Newton's second law

a = Ftotal/m , where m is an object's mass Ftotal is the sum of the forces acting on it, and a is the acceleration of the object's center of mass.

We are presently restricted to the case where the forces of interest are parallel to the direction of motion.

An accelerating bus.

A generalization

As with the first law, the second law can be easily generalized to include a much larger class of interesting situations:

Suppose an object is being acted on by two sets of forces, one set lying along the object's initial direction of motion and another set acting along a perpendicular line. If the forces perpendicular to the initial direction of motion cancel out, then the object accelerates along its original line of motion according to a = F_total/m.

The relationship between mass and weight

Mass is different from weight, but they're related. An apple's mass tells us how hard it is to change its motion. Its weight measures the strength of the gravitational attraction between the apple and the planet earth. The apple's weight is less on the moon, but its mass is the same. Astronauts assembling the International Space Station in zero gravity cannot just pitch massive modules back and forth with their bare hands; the modules are weightless, but not massless.

e / A simple double-pan balance works by comparing the weight forces exerted by the earth on the contents of the two pans. Since the two pans are at almost the same location on the earth's surface, the value of g is essentially the same for each one, and equality of weight therefore also implies equality of mass.

We have already seen the experimental evidence that when weight (the force of the earth's gravity) is the only force acting on an object, its acceleration equals the constant g, and g depends on where you are on the surface of the earth, but not on the mass of the object. Applying Newton's second law then allows us to calculate the magnitude of the gravitational force on any object in terms of its mass:

|FW| = mg .

(The equation only gives the magnitude, i.e. the absolute value, of F_W, because we're defining g as a positive number, so it equals the absolute value of a falling object's acceleration.)

→ Solved problem: Decelerating a car page 148, problem 7

Weight and mass.

Calculating terminal velocity.

Self-Check

It is important to get into the habit of interpreting equations. This may be difficult at first, but eventually you will get used to this kind of reasoning. (1) Interpret the equation vterminal = in the case of ρ=0. (2) How would the terminal velocity of a 4-cm steel ball compare to that of a 1-cm ball? (3) In addition to teasing out the mathematical meaning of an equation, we also have to be able to place it in its physical context. How generally important is this equation?

Answer

(1) The case of ρ = 0 represents an object falling in a vacuum, i.e., there is no density of air. The terminal velocity would be infinite. Physically, we know that an object falling in a vacuum would never stop speeding up, since there would be no force of air friction to cancel the force of gravity. (2) The 4-cm ball would have a mass that was greater by a factor of 4×4×4, but its cross-sectional area would be greater by a factor of 4×4. Its terminal velocity would be greater by a factor of √43/42 = 2. (3) It isn't of any general importance. It's just an example of one physical situation. You should not memorize it.

Discussion Questions

A	Show that the Newton can be reexpressed in terms of the three basic mks units as the combination kg·m/s2.
B	What is wrong with the following statements? (1) "g is the force of gravity." (2) "Mass is a measure of how much space something takes up."
C	Criticize the following incorrect statement: "If an object is at rest and the total force on it is zero, it stays at rest. There can also be cases where an object is moving and keeps on moving without having any total force on it, but that can only happen when there's no friction, like in outer space."
D	The table gives laser timing data for Ben Johnson's 100 m dash at the 1987 World Championship in Rome. (His world record was later revoked because he tested positive for steroids.) How does the total force on him change over the duration of the race?