Chapter Four Notes:Newton’s Second Law of Motion – Linear

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. 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 acting upon an object is increased, the acceleration of the object is increased. As the mass of an object is increased, the acceleration of the object is 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.This verbal statement can be expressed in equation form as follows:

a = Fnet / mThe above equation is often rearranged to a more familiar form as shown below. The net force is equated to the product of the mass times the acceleration.

Fnet = m * a

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 another 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.

  NET FORCE

Consistent with the equation on previous slide, 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 (1ms-2).

The Fnet = m • a equation is often used in 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 click the mouse to view the answers.

Net Force(N)

Mass (kg)

Acceleration(m/s/s)

1 10 2

2 20 2

3 20 4

4 10 5

5 10 10

5

10

5

2

1

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 Fnet = 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.

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.

Acceleration: An often confused quantity, acceleration has a

meaning much different than the meaning associated with it by sports announcers and other individuals. The definition of acceleration is:

Acceleration is a vector quantity which is defined as the rate at which an object changes its velocity. An object is accelerating if it is changing its velocity (Either it’s speed and/or direction).

An object is moving if its position relative to a fixed point is changing.

Speed is how fast an object is moving. You can calculate the speed of an object by dividing the distance covered by time.

Speed = distance/time SI Units: meters per second (m/s)

Instantaneous Speed◦ The speed of an object at any given instant!◦ You can tell the speed of a car at any instant by

looking at the speedometer. Average Speed

◦ Total distance covered divided by the total time: Average speed = total distance /time

interval◦ Ea: if we traveled 240 kilometers in 4 hours

◦ Average speed = 240 km/4 h = 60 km/h

◦ Simple variation of this equation gives:◦ Total distance traveled = average speed X travel

time

Speed is a description of how fast an object moves; velocity is how fast and in what direction it moves.

Constant Velocity:◦ Both speed and direction remain constant!

Changing Velocity:◦ Either the speed or the direction (or both) change.

Then you have changing velocity!

You can calculate the acceleration of an object by dividing the change in its velocity by time. Acceleration = Δ velocity / time interval

(Δ stands for “change in”) Acceleration can be either positive (increasing

speed) or negative (decreasing speed). We often call the negative acceleration, deceleration.

Acceleration also applies to a change in direction! If we change speed, direction or both, we change velocity and we accelerate!

An object moving under the influence of the gravitational force only is said to be in free fall.

The elapsed time is the time that has elapsed, or passed, since the beginning of any motion, in this case free fall.

The acceleration due to gravity on the surface of earth is given by the following value in this text:

g = 10 m/s2

Therefore, in free fall an object increases in speed by an additional 10 meters per second.

Free Fall Speeds of Objects

V = gt

Elapsed Time(seconds)

Instantaneous Speed(meters/second)

0 0

1 10

2 20

3 30

4 40

5 50

t 10t

Rising Objects:◦ An object thrown straight up, will slow down the

same way a free fall object speeds up, 10 m/s each second. If thrown up with a speed of 30 m/s, will have the following speeds at each second, assuming up is positive and down is negative.

Time (sec) Speed (m/s)

Time (sec) Speed (m/s)

3 0 3 0

2 10 4 -10

1 20 5 -20

0 30 6 -30

7 -40

8 -50

For a freely falling object, for each second of free fall, an object falls a greater distance than it did in the previous second. You need to calculate this distance by taking the average speed over the interval and multiplying it by 1 second.

During the first second: Initial speed = 0 m/s, final speed = 10 m/s, so average speed is (final speed + initial speed)/2.

[(10 m/s + 0 m/s) /2]/1sec = 5 meters during 1st sec.

During 2nd second: Initial speed = 10 m/s, final speed = 20 m/s. Therefore:

[(20 m/s + 10 m/s)/2]/1 sec = 15 meters. During 3rd second: Initial speed = 20 m/s, final

speed = 30 m/s. Therefore: [(30 m/s + 20 m/s)/2]/1 sec = 25 meters.

NOW: the total distance since the beginning

becomes: (1 sec): 0 + 5 = 5 meters (2 sec): 5 + 15 = 20 meters (3 sec): 20 + 25 = 45 meters (4 sec): 45 + 35 = 80 meters (5 sec): 80 + 45 = 125 meters During tth sec: Total Distance = ½gt2

meters.

Speed-Versus-Time◦ On a speed-versus-time graph, the slope represents speed

per time, or acceleration.

◦ Speed◦ (m/s)◦ Slope = ◦ rise/run =◦ (acceleration)

◦ ◦ Time (s)

Distance-Versus-Time◦ On a distance-versus-time graph, the slope

represents distance per time, or instantaneous speed.

◦ Distance◦ (m)◦ slope =

◦ rise/run =◦ speedinst

◦ 0 1 2 3 4 5◦ Time (s)

Air resistance noticeably slows the motion of things with large surface areas like falling feathers or pieces of paper. But air resistance less noticeably affects the motion of more compact objects like stones and baseballs.

A classic demonstration of this is called “the feather and guinea demonstration. It involves a tube with both items in it. When the air is removed, both items fall at the same speed.

Some of the confusion that occurs in analyzing the motion of falling objects comes about from mixing up “how fast” with “how far.”

When we wish to specify how fast, the equation is: v = gt

When we wish to specify how far, the equation is: d = (1/2)gt2

Acceleration is the rate at which velocity itself changes. [the rate of a rate]