Lesson 3: Newton's Second Law of Motion

Newton's Second Law

Newton's first law of motion predicts the behavior of objects for which all existing forces are balanced. The first law - sometimes referred to as the "law of inertia" - states that if the forces acting upon an object are balanced, then the acceleration of that object will be 0 m/s/s. Objects at equilibrium (the condition in which all forces balance) will not accelerate. According to Newton, an object will only accelerate if there is a net or unbalanced force acting upon it. The presence of an unbalanced force will accelerate an object - changing either its speed, its direction, or both its speed and direction.

Newton's second law of motion pertains to the behavior of objects for which all existing forces are not balanced. The second law states that the acceleration of an object is dependent upon two variables - the net force acting upon the object and the mass of the object. As learned in the "The Rocket Simulation" Lab, the acceleration of an object depends directly upon the net force acting upon the object, and inversely upon the mass of the object. As the force of propulsion acting upon the rocket-chair increased, the acceleration of the rocket-chair increased. As the mass of the rocket-chair increased, the acceleration of the rocket-chair decreased.

Newton's second law of motion can be formally stated as follows:

The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.

In terms of an equation, the net force is equated to the product of the mass times the acceleration.

In this entire discussion, the emphasis has been on the "net force." The acceleration is directly proportional to the "net force;" the "net force" equals mass times acceleration; the acceleration in the same direction as the "net force;" an acceleration is produced by a "net force." The NET FORCE. It is important to remember this distinction. Do not use the value of merely "any 'ole force" in the above equation; it is the net force which is related to acceleration. As discussed in an earlier lesson, the net force is the vector sum of all the forces. If all the individual forces acting upon an object are known, then the net force can be determined. If necessary, review this principle by returning to the practice questions in Lesson 2.

The above equation also indicates that a unit of force is equal to a unit of mass times a unit of acceleration. By substituting standard metric units for force, mass, and acceleration into the above equation, the following unit equivalency can be written.

The definition of the standard metric unit of force is stated by the above equation. One Newton is defined as the amount of force required to give a 1-kg mass an acceleration of 1 m/s/s.

The F_net = m a equation can also be used as a "recipe" for algebraic problem-solving. The table below can be filled by substituting into the equation and solving for the unknown quantity. Try it yourself and then use the "pop-up menus" to view the answers.

The numerical information in the table above demonstrates some important qualitative relationships between force, mass, and acceleration. Comparing the values in rows 1 and 2, it can be seen that a doubling of the net force results in a doubling of the acceleration (if mass is held constant). Similarly, comparing the values in rows 2 and 4 demonstrates that a "halving" of the net force results in a "halving" of the acceleration (if mass is held constant). Acceleration is directly proportional to net force.

Furthermore, the qualitative relationship between mass and acceleration can be seen by a comparison of the numerical values in the above table. Observe from rows 2 and 3 that a doubling of the mass results in a "halving" of the acceleration (if force is held constant). And similarly, rows 4 and 5 show that a "halving" of the mass results in a doubling of the acceleration (if force is held constant). Acceleration is inversely proportional to mass.

The analysis of the table data illustrates that an equation such as F_net = m*a can be a guide to thinking about how a variation in one quantity might effect another quantity. Whatever alteration is made of the net force, the same change will occur with the acceleration. Double, triple or quadruple the net force, and the acceleration will do the same. On the other hand, whatever alteration is made of the mass, the opposite or inverse change will occur with the acceleration. Double, triple or quadruple the mass, and the acceleration will be one-half, one-third or one-fourth its original value.

As stated above, the direction of the net force is in the same direction as the acceleration. Thus, if the direction of the acceleration is known, then the direction of the net force is also known. Consider the two ticker tape traces below for an acceleration of a car. From the trace, determine the direction of the net force which is acting upon the car. Then depress the mouse on the "pop-up menu" to view the answer. (Review acceleration from previous unit.)

Depress mouse for answer. The net force is to the right since the acceleration is to the right. An object which moves to the right and speeds up has a rightward acceleration.
Depress mouse for answer. The net force is to the left since the acceleration is to the left. An object which moves to the right and slows down has a leftward acceleration.

In conclusion, Newton's second law provides the explanation for the behavior of objects upon which the forces do not balance. The law states that unbalanced forces cause objects to accelerate with an acceleration which is directly proportional to the net force and inversely proportional to the mass.

Check Your Understanding

1. What acceleration will result when a 12-N net force applied to a 3-kg object? A 6-kg object?

Depress mouse to view answer. A 3-kg object experiences an accel'n of 4 m/s/s. A 6-kg object experiences an accel'n of 2 m/s/s.

2. A net force of 16 N causes a mass to accelerate at a rate of 5 m/s². Determine the mass.

Depress mouse to view answer. Use Fnet=m*a with Fnet=16 N and a=5 m/s/s. So (16 N) = (m)*(5 m/s/s) And m = 3.2 kg

3. An object is accelerating at 2 m/s². If the net force is tripled and the mass is doubled, then what is the new acceleration?

Depress mouse to view answer. Answer: 3 m/s/s. The original value of 2 m/s/s must be multiplied by 3 (since a and F are directly proportional) and divided by 2 (since a and m are inversely proportional)

4. An object is accelerating at 2 m/s². If the net force is tripled and the mass is halved, then what is the new acceleration?

Depress mouse to view answer. Answer: 12 m/s/s. The original value of 2 m/s/s must be multiplied by 3 (since a and F are directly proportional) and divided by 1/2 (since a and m are inversely proportional)

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