AcademyApplying Force Models
Academy
Particle Dynamics with Newton's Second Law
Level 1 - Physics topic page in Applying Force Models.
Principle
Newton's second law turns the external force model for a system into its acceleration.
Notation
\(m\)
mass of the chosen system
\(\mathrm{kg}\)
\(\vec a\)
acceleration of the chosen system
\(\mathrm{m\,s^{-2}}\)
\(\sum\vec F\)
resultant external force
\(\mathrm{N}\)
\(\vec W\)
weight force
\(\mathrm{N}\)
\(g\)
gravitational field strength
\(\mathrm{m\,s^{-2}}\)
\(\theta\)
slope angle
rad or deg
Method
Derivation 1: Choose the system
Newton's second law applies to one chosen system. Only external forces on that system belong in the force sum.
Vector law
\[\sum\vec F_{\mathrm{external}}=m\vec a\]
Internal forces cancel only inside the chosen system.
Resolve by axes
\[\sum F_x=ma_x,\qquad \sum F_y=ma_y\]
Use kinematics after
\[v=v_0+at\]
This step is valid only after the force model gives a constant acceleration.
Derivation 2: Derive the frictionless-slope acceleration
For a block on a frictionless slope, choose one axis parallel to the slope and one perpendicular to it. Gravity is the only force with a component down the slope.
Parallel force balance
\[\sum F_{\parallel}=mg\sin\theta=ma_{\parallel}\]
Slope acceleration
\[a_{\parallel}=g\sin\theta\]
Perpendicular balance
\[\sum F_{\perp}=N-mg\cos\theta=0\]
There is no acceleration through the surface.
The free-body diagram shows the actual forces only: weight downward and the normal force perpendicular to the surface. The component equations above are built from these forces.
Rules
These are the compact results from the method above.
Second law
\[\sum \vec F=m\vec a\]
Component law
\[\sum F_x=ma_x,\qquad \sum F_y=ma_y\]
Weight vector
\[\vec W=m\vec g\]
Frictionless slope
\[a=g\sin\theta\]
Examples
Question
A
\[4.0\,\mathrm{kg}\]
cart is pulled forward by \[18\,\mathrm{N}\]
while a \[6\,\mathrm{N}\]
resistance acts backward. Find \(a\).Answer
Choose forward as positive. The net force is
\[\sum F=18-6=12\,\mathrm{N}\]
so \[a=\frac{12}{4.0}=3.0\,\mathrm{m\,s^{-2}}\]
Checks
- Only external forces belong in the net-force sum for the chosen system.
- Acceleration points in the direction of the resultant force.
- Zero acceleration means zero net force, not zero individual forces.
- The normal force is found from the perpendicular force equation.