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What about cases where the total force on an object is not zero, so that Newton’s first law does not apply? The object will have an acceleration. How much acceleration will it have? It will clearly depend on both the object’s mass and on the amount of force.

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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

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 parallel to 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∥/ma=F∥/m, where F∥F∥ is the sum of the forces parallel to the line.

Example: A coin sliding across a table.

Suppose a coin is sliding to the right across a table, f, and let’s choose a positive xx axis that points to the right. The coin’s velocity is positive, and we expect based on experience that it will slow down, i.e., its acceleration should be negative.

Although the coin’s motion is purely horizontal, it feels both vertical and horizontal forces. The Earth exerts a downward gravitational force F2F2 on it, and the table makes an upward force F3F3 that prevents the coin from sinking into the wood. In fact, without these vertical forces, the horizontal frictional force would not exist: surfaces do not exert friction against one another unless they are being pressed together.

Although F2F2 and F3F3 contribute to the physics, they do so only indirectly. The only thing that directly relates to the acceleration along the horizontal direction is the horizontal force: a=F1/ma=F1/m.

The relationship between mass and weight

Mass is different from weight, but they are 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 could not just pitch massive modules back and forth with their bare hands; the modules were weightless, but not massless.

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 gg, and gg depends on where you are on the surface of 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:


(The equation only gives the magnitude, i.e. the absolute value, of FWFW, because we are defining gg as a positive number, so it equals the absolute value of a falling object’s acceleration.)

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