AcademyForces and Newton's Laws
Academy
Force, Mass, and Acceleration
Level 1 - Physics topic page in Forces and Newton's Laws.
Principle
Newton's second law converts the resultant external force into the system's acceleration.
Notation
\(m\)
mass of the system
\(\mathrm{kg}\)
\(\vec{a}\)
acceleration
\(\mathrm{m\,s^{-2}}\)
\(\sum\vec{F}\)
resultant external force
\(\mathrm{N}\)
\(\sum F_x,\sum F_y\)
resultant force components
\(\mathrm{N}\)
\(a_x,a_y\)
acceleration components
\(\mathrm{m\,s^{-2}}\)
\(\Delta x,v_{0x},t\)
motion quantities after acceleration is known
\(\mathrm{m,\;m\,s^{-1},\;s}\)
Method
The force sum is built first; motion equations are used only after acceleration is known.
Choose system
\[\sum\vec{F}_{\mathrm{external}}=m\vec{a}\]
Internal forces cancel only inside the chosen system.
Resolve components
\[\sum F_x=ma_x,\qquad \sum F_y=ma_y\]
Solve acceleration
\[a_x=\frac{\sum F_x}{m},\qquad a_y=\frac{\sum F_y}{m}\]
Then use motion
\[\Delta x=v_{0x}t+\frac{1}{2}a_xt^2\]
This step assumes the acceleration component is constant.
The free-body diagram shows vertical forces canceling while horizontal forces leave a nonzero resultant.
Reading the diagram by components gives a rightward resultant, so the acceleration is rightward.
Rules
These are the compact results after the external force sum has been built.
Second law
\[\sum\vec{F}=m\vec{a}\]
Component law
\[\sum F_x=ma_x,\qquad \sum F_y=ma_y\]
Acceleration
\[\vec{a}=\frac{\sum\vec{F}}{m}\]
Constant acceleration
\[\Delta x=v_{0x}t+\frac{1}{2}a_xt^2\]
Examples
Question
A
\[4.0\,\mathrm{kg}\]
cart has horizontal forces \[30\,\mathrm{N}\]
right and \[10\,\mathrm{N}\]
left. Find acceleration.Answer
Take right as positive.
\[\sum F_x=30-10=20\,\mathrm{N}\]
\[a_x=\frac{20}{4.0}=5.0\,\mathrm{m\,s^{-2}}\]
right.Checks
- Acceleration has the same direction as the resultant force.
- More mass gives less acceleration for the same resultant force.
- A single applied force is not the resultant unless all other forces cancel.
- Force and acceleration components must use the same sign convention.