Equations Introduced and Used for this Topic: (All equations can be written and solved as both scalar and vector)

• [latex]F = ma[/latex]
• [latex]w = mg[/latex]

Where…

• [latex]F[/latex] is force, measured in newtons (N)
• [latex]w[/latex] is weight (it is a force), measured in newtons (N)
• [latex]m[/latex] is mass, commonly measured in kilograms (kg), grams (g) or tonnes (t)
• [latex]a[/latex] is acceleration (deceleration is negative), measured in metres per second squared (m/s²)
• [latex]g[/latex] or [latex]a_g[/latex] is the acceleration due to gravity and varies from 9.78 m/s² (equator) to 9.83 m/s² (North or South Pole). The average value of gravity is taken to be either 9.80 m/s²  or 9.81 m/s²

Newton’s Laws of Motion

Sir Isaac Newton worked in many areas of mathematics and physics. He developed the theories of gravitation in 1666, when he was only 23 years old. Some twenty years later, in 1686, he presented his three laws of motion in the “Principia Mathematica Philosophiae Naturalis.”

Newton’s first law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force. This is normally accepted as the definition of inertia. The key point here is that if there is no net force acting on an object (if all the external forces cancel each other out) then the object will maintain a constant velocity. If that velocity is zero, then the object remains at rest. If an external force is applied, the velocity will change because of the force.

Newton’s second law explains how the velocity of an object changes when it is subjected to an external force. The law defines a force to be equal to change in momentum (mass times velocity) per change in time. Newton also developed the calculus of mathematics, and the “changes” expressed in the second law are most accurately defined in differential forms. (Calculus can also be used to determine the velocity and location variations experienced by an object subjected to an external force.) For an object with a constant mass [latex]m[/latex], the second law states that the force ([latex]F[/latex]) is the product of an object’s mass and its acceleration [latex]a[/latex]:

[latex]F = m a[/latex]

Newton’s third law states that for every action (force) in nature there is an equal and opposite reaction. In other words, if object A exerts a force on object B, then object B also exerts an equal force on object A. Notice that the forces are exerted on different objects. The third law can be used to explain the generation of lift by a wing and the production of thrust by a jet engine.

6.1 Net Force… The resolution of multiple force vectors

Net Force: the resolutions of multiple forces acting on a body

Forces are Vectors … They are added, subtracted and combined according to the direction they take.

Quite often, vectors are simply left as the sum of the different vectors and not recombined.

Vectors in the same direction are added and vectors in the opposite direction are subtracted.

For instance:

10 N right + 5 N right = 15 N right

(The vectors are in the same direction so they are added.)

10 N right + 5 N left = 5 N right

(The vectors are in the opposite direction so they are subtracted.)

Vectors that are angled between two directions are first broken up into the various components for those directions before they are added or subtracted.

10 N right + 5 N down = 10 N right + 5 N down

(The vectors are in different directions so they are left alone.)

Vectors that are angled between two directions are first broken up into the various components for those directions before they are added or subtracted.

For instance:

Using right angle trigonometry this 15 N vector would be broken up into the force directed upwards and the force to the left.

To find the force upwards use the sine function:

[latex]\sin 40^{\circ} = \dfrac{\text{force up}}{15 N}[/latex]

To find the force right use the cosine function:

[latex]\cos 40^{\circ} = \dfrac{\text{force right}}{15 N}[/latex]

6.2 Newton’s Second Law

Equations Introduced or Used for this Section:

• [latex]F = ma[/latex]
• [latex]w = mg[/latex]

• Four Kinematic Equations:
  • [latex]d = \dfrac{(v_i + v_f) t}{2}[/latex]
  • [latex]2ad = v_f^2 - v_i^2[/latex]
  • [latex]\vec{v_f} = \vec{v_i} + \vec{a}t[/latex]
  • [latex]\vec{d} = \vec{v_i}t + \frac{1}{2}\vec{a}t^2[/latex]
• Newton’s second law
  • [latex]\vec{F} = m\vec{a}[/latex]

Data organized as:

• • [latex]a =[/latex] Accelerated Motion
  • [latex]v_i =[/latex]
  • [latex]v_f =[/latex]
  • [latex]d =[/latex]
  • [latex]t =[/latex]
  • [latex]F =[/latex] For Unbalanced Forces
  • [latex]m =[/latex]
  • [latex]a =[/latex]
  • [latex]w =[/latex] For Weight Questions
  • [latex]m =[/latex]
  • [latex]g =[/latex]

Newton’s Second Law: for a Balanced or Net Force = 0

From Newton’s second law: if the external forces acting on an object are balanced

([latex]F_{\text{Net}} = 0[/latex]) then the object experiences no acceleration.

This means for [latex]F_{\text{Net}} = 0[/latex], [latex]a=0\text{ m/s}^2[/latex]. Therefore, if at rest, it remains at rest ([latex]v = 0 \text{ m/s}[/latex]) or if in motion, then it remains in that constant state of motion ([latex]v = \text{ constant}[/latex]).

Newton’s Second Law: for an Unbalanced or Net Force ≠ 0

From Newton’s second law: if the external forces acting on an object are unbalanced

([latex]F_{\text{Net}} \neq 0[/latex]) then the object will experience some amount of acceleration.

This means for [latex]F_{\text{Net}} \neq 0[/latex], [latex]a \neq 0 \text{ m/s}^2[/latex]. Therefore, if originally at rest, it will accelerate and begin to move ([latex]v \neq 0 \text{ m/s}[/latex]) or if is already in motion then it will be changing its speed or velocity ([latex]v \neq \text{ constant}[/latex]).

Analyzing this change in motion resulting from an unbalanced force acting on an object requires one to use an equation derived from Newton’s second law:

[latex]F = m a[/latex]

Using Newton’s second law often requires one to share the acceleration measure with the four kinematic equations, as you will see in the following examples.

• Four Kinematic Equations:
  • [latex]d = \dfrac{(v_i + v_f) t}{2}[/latex]
  • [latex]2ad = v_f^2 - v_i^2[/latex]
  • [latex]\vec{v_f} = \vec{v_i} + \vec{a}t[/latex]
  • [latex]\vec{d} = \vec{v_i}t + \frac{1}{2}\vec{a}t^2[/latex]
• Newton’s second law
  • [latex]\vec{F} = m \vec{a}[/latex]

6.3 Weight & Mass

Equations Introduced or Used for this Section:

• [latex]F = ma[/latex]
• [latex]w = mg[/latex]

• Four Kinematic Equations:
  • [latex]d = \dfrac{(v_i + v_f) t}{2}[/latex]
  • [latex]2ad = v_f^2 - v_i^2[/latex]
  • [latex]\vec{v_f} = \vec{v_i} + \vec{a}t[/latex]
  • [latex]\vec{d} = \vec{v_i}t + \frac{1}{2}\vec{a}t^2[/latex]
• Constant Velocity Equation
  • [latex]\vec{v} = \dfrac{\vec{d}}{t}[/latex]

Weight and mass are concepts that many students confuse in their early studies.

The mass of an object refers to the measure of the amount of matter that exists in that object and the SI measure of it is in kilograms (kg)^[1]. This amount of matter remains constant anywhere in the universe and is one of the fundamental measurements in physics. At high enough speeds, relativity will increase the mass by an amount called the relativistic mass.

The weight of an object is a measure of the force of gravity that acts on the object. This can be tricky to measure as this measure is typically done by placing a scale between the object and the surface it is resting on. If the object the scale in between and the surface the scale is resting on are all in a state of free fall, then the scale can read 0 N since there is no net force holding the object to the surface.  One can generally experience this temporary weightlessness in free fall amusement park rides. As such, astronauts in space, along with their spaceship, are weightless in their orbit around the Earth as they are always in that state of free fall.

weight  = mass × gravity

Weight, like other forces, is measured in newtons.

When one measures the weight of an object in an elevator, one is required to look at weight as the measure of the sum or difference of the acceleration of gravity and of the surface the object is resting on.  Consider the following example, where the acceleration of an elevator is 4.2 m/s².

For an elevator at rest, the weight of the object is a simple measure, specifically, its weight is the product of mass × gravity.

[latex]w = m a_g[/latex]

&

[latex]w = m (9.8 \text{ m/s}^2)[/latex]

If we are now trying to measure the weight of an object on a floor that is accelerating downwards we would measure the weight to be less, which is similar to how we can feel lighter in an elevator that starts to accelerate downwards. The measured weight in this case would be the acceleration of gravity minus the acceleration of the falling floor.

[latex]w = m (a_g - a_{\text{floor}})[/latex]

&

[latex]w = m (9.8 \text{ m/s}^2 - 4.2 \text{ m/s}^2)[/latex] or [latex]w = m (5.6 \text{ m/s}^2)[/latex]

If we are now trying to measure the weight of an object on a floor that is accelerating upwards we would measure the weight to be greater which is similar to how we can feel heavier in an elevator that starts to accelerate upwards. The measured weight in this case would be the acceleration of gravity plus the acceleration of the falling floor.

[latex]w = m (a_g + a_{\text{floor}})[/latex]

&

[latex]w = m (9.8 \text{ m/s}^2 + 4.2 \text{ m/s}^2)[/latex] or [latex]w = m (14 \text{ m/s}^2)[/latex]

The following examples illustrate the difference between of weight and mass.

6.4 Newton’s Second Law & Accelerated Motion

Newton’s second law^[2], most commonly seen as [latex]F = ma[/latex] (scalar form) relates the amount of acceleration that results from an unbalanced force that acts on an object. Using this new approach to measure acceleration, we are able to expand the situations where constant acceleration occurs, and in our analysis of these problems while using the four kinematic

• Four Kinematic Equations:
  • [latex]d = \dfrac{(v_i + v_f) t}{2}[/latex]
  • [latex]2ad = v_f^2 - v_i^2[/latex]
  • [latex]\vec{v_f} = \vec{v_i} + \vec{a}t[/latex]
  • [latex]\vec{d} = \vec{v_i}t + \frac{1}{2}\vec{a}t^2[/latex]
• Newton’s second law
  • [latex]\vec{F} = m \vec{a}[/latex]

In sorting data we can use something like the following.

• • [latex]a =[/latex] accelerated motion
  • [latex]v_i =[/latex]
  • [latex]v_f =[/latex]
  • [latex]d =[/latex]
  • [latex]t =[/latex]
  • [latex]F =[/latex] for unbalanced forces
  • [latex]m =[/latex]
  • [latex]a =[/latex]

In solving these problems one often encounters situations where the measure of acceleration is shared between the kinematic equations and Newton’s second law equation. This means that you might need to find the acceleration from Newton’s second law to use in a kinematic equation, or the reverse.

Also, it is common to encounter an increase in the number of conversions required to solve these problems, specifically speed, such as km/h to SI metric m/s, acceleration in gs to its equivalent in m/s², and in being given the object’s weight, which you then need to convert to a mass in kg.

One unique feature of Newton’s second law is that you will encounter various forms of it in other parts in your studies in physics. It is also used as a quick derivation for the equations fundamental to the next chapters covered: Impulse and Momentum.

6.5 Calculating the Mass of the Earth’s Atmosphere

Exercise Answers

6.2 Newton’s Second Law

1. 7.2 kg
2. 12.5 m/s²
3. 0 N
4. − 610 N
5. 0.0010 N or 1.0 mN
6. 4.2 m/s²
7. 0.034 m/s² or 3.4 cm/s²
8. 960 m/s²

6.3 Weight & Mass

1. 1. 150 N
   2. 30 m/s²
2. 1. 5.0 m/s²
   2. 49 m/s²
3. 1. 538 N
   2. 541 N
4. 76 kg
5. 250 N
6. 730 N in both cases
7. 1.4 m/s²
8. 0.49 m/s²
9. 9960 N to 36200 N

6.4 Newton’s Second & Accelerated Motion

1. 106 m
2. 625 N
3. 390 N
4. − 4630 N
5. 5.8 s
6. 37.5 m/s
7. 4.8 s
8. 815 m

6.5 Calculating the Mass of the Earth’s Atmosphere

1. 5.3 × 10¹⁸