2 Forces and Newton’s Laws

Learn how to represent forces as vectors, draw free-body diagrams, and apply Newton’s laws to equilibrium and motion.

Forces as vectors

A is an interaction that can change an object’s velocity. Because it has both magnitude and direction, is a . Forces on the same object combine to form the :

\vec F_{\text{net}}=\sum_i\vec F_i.

Choose coordinate axes that make the problem easier to solve. A with magnitude $F$ at angle $\theta$ above the positive horizontal direction has components

F_x=F\cos\theta,\qquad F_y=F\sin\theta.

The signs of the components depend on the directions chosen as positive. For example, a of $50\,\mathrm{N}$ at $30^\circ$ above horizontal has components of approximately $43\,\mathrm{N}$ horizontally and $25\,\mathrm{N}$ vertically. Add forces component by component to find the :

F_{\text{net},x}=\sum_i F_{i,x},\qquad F_{\text{net},y}=\sum_i F_{i,y}.

Takeaway: Resolve angled forces into components, then add components along each axis.

Free-body diagrams

A (FBD) isolates one object and represents every external on it with an arrow. To make one:

Choose the object and draw it as a dot or simple shape.
Add and label each external , with the arrow pointing in its direction.
Choose coordinate axes; when useful, align them with the object’s motion or a surface.
Resolve angled forces into components if that simplifies the equations.

Include forces acting on the chosen object. Do not include forces the object exerts on other objects, and do not add the as if it were another physical . If analyzing several objects, draw a separate diagram for each one.

Takeaway: A clear diagram helps identify which forces belong in the sum for one object.

Newton’s laws

Newton’s laws connect forces to motion.

Newton’s first law (inertia): In an inertial reference frame, an object remains at rest or continues with constant velocity when its net external is zero. Zero means zero acceleration, not necessarily zero velocity.
: The net external determines acceleration:
$\vec F_{\text{net}}=m\vec a.$
Apply the law independently along each coordinate direction:
$\sum_i F_{i,x}=ma_x,\qquad \sum_i F_{i,y}=ma_y.$
Acceleration points in the direction of the . For the same , an object with greater mass has smaller acceleration.
Newton’s third law: When object $A$ exerts a on object $B$ , object $B$ simultaneously exerts a of equal magnitude in the opposite direction on object $A$ . These forces act on different objects, so they do not cancel each other on a single object’s .

Takeaway: Use the sum for the object you selected; Newton’s third-law partners belong on separate object diagrams.

and constant velocity

Translational occurs when the is zero, so acceleration is zero:

\vec F_{\text{net}}=\vec 0,\qquad \vec a=\vec 0.

An object in static is at rest. In dynamic , it moves with constant velocity. In either case, the components balance along each coordinate direction:

\sum_i F_{i,x}=0,\qquad \sum_i F_{i,y}=0.

These conditions describe acceleration, not necessarily the object’s current velocity. An object moving at constant velocity can be in just as an object at rest can.

Common models

Choose a model according to the interaction:

Weight is Earth’s gravitational on an object. Near Earth’s surface it points vertically downward and has magnitude $mg$ . Mass is measured in kilograms; weight is a measured in newtons.
is a contact perpendicular to a surface. It is not necessarily equal to weight; its magnitude depends on the other forces and the acceleration perpendicular to the surface.
Tension is a pulling transmitted by a taut cord or rope. It points along the cord and away from the object.
Applied is a direct push or pull exerted by a person or another object.
Spring for an ideal spring follows Hooke’s law:
$\vec F_s=-k\vec x.$
Here, $\vec x$ is displacement from the relaxed position and $k$ is spring stiffness. The negative sign indicates that the spring is restoring.
acts parallel to the surfaces in contact and resists relative slipping, or the tendency to slip. Static can adjust up to a maximum:
$f_s\leq\mu_sN.$
Once the surfaces slide, a common model for kinetic is
$f_k=\mu_kN.$
Static is not always at its maximum, and ’s direction is determined by relative slipping at the contact—not necessarily by the object’s overall motion.

Takeaway: Identify the interaction before assigning a , and check its direction and physical role.

Worked example: a block on an incline

Consider a $5.0\,\mathrm{kg}$ block on a frictionless ramp tilted at $30^\circ$ above horizontal. Choose one axis along the ramp and another perpendicular to it. The block’s contains its weight, $mg$ , vertically downward and the , $N$ , perpendicular to the ramp.

Resolve the weight into a component down the ramp and a component into the ramp:

mg\sin 30^\circ\quad\text{and}\quad mg\cos 30^\circ.

There is no acceleration perpendicular to the ramp, so the balances the perpendicular component of weight:

N=mg\cos 30^\circ\approx 42\,\mathrm{N}.

Along the ramp, the component of weight produces the acceleration:

ma=mg\sin 30^\circ,

a=g\sin 30^\circ\approx 4.9\,\mathrm{m/s^2}.

The balances only the perpendicular component of weight; it does not cancel the component parallel to the ramp.

Takeaway: Choose axes suited to the surface, then apply separately in each direction.