5. Newton's Laws of Motion

We have already done some study of different kinds of motion (1D, 2D, circular, projectile). One can wonder what causes motion?

Force can cause a change in velocity (think of a push or pull force). We will think of this as a qualitative definition of force.

Is force a vector or a scalar quantity?

A push or pull typically is characterized by a direction of push or pull in addition to how strong (magnitude) the push or pull is. As such a force is a vector quantity.

Example: a book on a table

Suppose we have a book sitting on a table that is pulled to the right using a horizontal force \(\vec{F}\). Let us draw all the forces acting on the book in a diagram.

This kind of diagram in which we draw all the forces acting on an object is very useful in solving mechanics problems involving forces; such diagrams are called free-body diagrams.

5.1 First law

A body at rest remains at rest unless acted on by a net external force. Similarly, a body in motion remains in a constant velocity motion unless acted on by a net external force.
First law is also called the law of inertia.
Inertia is the property of an object to resist changes in its motion.

Mathematically

\(\vec{F}_{\rm net}=\) sum of all external forces acting on the body.

\[ \vec{F}_{\rm net} = \sum_i \vec{F}_i = \vec{F}_1 + \vec{F}_2 + \dots + \vec{F}_n\]

If \(\vec{F}_{\rm net} = 0\) then \(\vec{a} = 0\) and vice versa.

5.2 Second law

Newton's second law relates the net external force acting on an object to its acceleration.

Mathematically

\[ \vec{F}_{\rm net} = \sum \vec{F} = m \vec{a} \]

When only the magnitudes are considered:

\[ F_{\rm net} = | \vec{F}_{\rm net} | = m a \]

The free-body diagram of forces can then be used to write down second law equations in each of the \(x-\) and \(y-\) component directions. For the example above in which a book on a table is being pulled horizontally by a force \(\vec{F}\):

x-componenty-component

\[\sum F_x = m a_x \implies F = m a\]

As you can see, in the absence of friction between the book and the table, you can expect the book to accelerate in the \(x-\)direction with acceleration \(a=F/m\), where \(m\) is the mass of the book.

\[\sum F_y = N - m g = m a_y = 0 \implies N - mg = 0 \implies N = mg\]

The normal force, \(N\) is the force on the book from the table; in this case, the normal force equals the weight \(mg\) of the book. In general, you will have to find the normal force on an object by using Newton's laws as it may not always equal the weight of the object.

5.3 Third law

Froces between two bodies \(A\) and \(B\) (due to some kind of interaction between them) come in paris.

Mathematically

\[ \vec{F}_{\rm AB} = - \vec{F}_{BA} \]

where \(\vec{F}_{\rm AB}\) is the force on object \(A\) due to object \(B\).

Example

Two blocks \(A\) and \(B\) are on a frictionless surface. Since the motion is constrained in the horizontal direction, let us not draw the forces acting on the blocks in the vertical direction in the diagram below.

Using Second law for \(A\) gives: \(F - F_{AB} = m_A\ a_A\).
Using Second law for \(B\) gives: \(F_{BA} = m_B\ a_B\).
According to the third law: \(F_{AB} = F_{BA}\); the magnitudes are equal.
From the physical constraint: \(a_A = a_B = a\); the two objects move together.
Therefore,

\[F = (m_A + m_B) a \implies a = \frac{F}{m_A + m_B}.\]

Try to draw the vertical forces on blocks \(A\) and \(B\) as a practice exercise.