Free Body Diagrams, Newton’s Laws, Friction, Springs, and Circular Motion

Free Body Diagrams and Newton’s Second Law

Introduction to Free Body Diagrams (FBDs)

Free Body Diagrams are essential tools in physics for visualizing the forces acting on an object. They help in applying Newton’s Second Law, which relates the net force on an object to its acceleration.

• Newton’s Second Law: $\vec{F}_{\text{net}} = m\vec{a}$

• Purpose of FBDs: Identify all forces, represent them as vectors, and resolve them into components to analyze motion.

Catalog of Common Forces

Force	Symbol	Formula	Description
Weight	$w$ or $F_g$	$w = mg$	Force due to gravity, always downward
Spring	$F_s$	$F_s = -kx$	Force exerted by a spring, opposite to displacement
Tension	$T$ or $F_T$	Calculated via N2L	Force in a string, direction of string
Normal	$n$ or $F_n$	Calculated via N2L	Perpendicular to surface
Kinetic Friction	$f_k$	$f_k = \mu_k N$	Opposes motion, constant value
Static Friction	$f_s$	$f_s \leq \mu_s N$	Opposes start of motion, variable up to max

Drawing and Interpreting Free Body Diagrams

To construct an FBD, represent the object as a dot or box and draw vectors for each force acting on it. The direction and relative length of each vector indicate the direction and magnitude of the force.

Example: A box sliding right on a rough surface. Which diagram is correct?

Example: A box sliding up a frictionless ramp. Which FBD is correct?

Example: A box compressing a spring. Which FBD is correct?

Component Analysis and Newton’s Second Law

For each FBD, resolve forces into x and y components. Apply Newton’s Second Law in each direction:

• $F_{\text{net},x} = m a_x$

• $F_{\text{net},y} = m a_y$

Friction: Static and Kinetic

Nature of Friction

Friction is a resistive force between two surfaces. It depends on the normal force and the materials in contact (coefficient of friction $\mu$).

• Kinetic Friction: $f_k = \mu_k N$ (object is moving)

• Static Friction: $f_s \leq \mu_s N$ (object is at rest)

Friction in Free Body Diagrams

Friction always acts parallel to the surface and opposite to the direction of motion (or intended motion).

Examples of Friction in FBDs

Box at rest with external force applied:

Box at rest with weaker external force:

Box at rest with stronger external force:

Box moving to the right and speeding up:

Box moving to the left and slowing down:

Box moving to the right at constant velocity:

Springs and Elasticity

Hooke’s Law and Spring Force

The force exerted by a spring is proportional to its displacement from equilibrium:

• Hooke’s Law: $F_{\text{spring}} = -k \Delta x$

• $k$ is the spring constant (N/m)

Spring Compression and Extension

When a spring is compressed or stretched, the force always acts to restore equilibrium.

Elasticity of Materials

Materials deform under stress. The relationship between force, area, and deformation is described by Young’s modulus ($Y$):

• $Y = \frac{k L_0}{A}$

• $F = Y A \frac{\Delta L}{L_0}$

• Tensile stress: $\frac{F}{A}$, Tensile strain: $\frac{\Delta L}{L_0}$

Elastic Properties Table

Material	Young’s Modulus ($10^9$ N/m$^2$)	Bulk Modulus ($10^9$ N/m$^2$)	Shear Modulus ($10^9$ N/m$^2$)
Aluminum	70	76	26
Brass	100	80	40
Concrete	30	13	15
Iron	211	170	82
Nylon	3		4.1
Rubber band	0.005		0.003
Steel	200	140	78
Air		1.41 × 10$^{-4}$
Ethyl alcohol		0.98
Water		2.2
Human ACL	0.1
Human lung	1.5–9.8 × 10$^{-6}$
Pig endothelial cell			2 × 10$^{-5}$

Circular Motion and Centripetal Force

Uniform Circular Motion

When an object moves in a circle at constant speed, it experiences a centripetal acceleration directed toward the center of the circle.

• Centripetal Acceleration: $a_c = \frac{v^2}{r} = 4\pi^2 r f^2$

• Centripetal Force: $F_c = m a_c$

Key Vocabulary

• Centripetal velocity ($v$): Speed along the circle, always tangent to the path.

• Period ($T$): Time for one complete revolution.

• Frequency ($f$): Number of revolutions per second (Hz).

Direction of Forces in Circular Motion

Both the net force and acceleration always point toward the center of the circle. Free body diagrams help identify the forces responsible for centripetal acceleration (e.g., tension, friction, gravity).

Newton’s Second Law for Circular Motion

Apply Newton’s Second Law to circular motion problems by identifying the net inward force and equating it to $m a_c$.

• $F_c = m a_c$

• Common forces: friction (car on road), tension (string), gravity (orbit)

Applications: Orbits and Gravity

Uniform circular motion applies to planetary orbits, where gravity provides the centripetal force.

Example: For a car of mass $m$ making a turn of radius $r$ at speed $v$, the frictional force required is $F_f = m \frac{v^2}{r}$.

Additional info: The study of free body diagrams, friction, springs, and circular motion forms the foundation for analyzing more complex systems in mechanics, including equilibrium, oscillations, and energy conservation.