AP Physics C Mechanics- 2.5 Newton’s Second Law- Study Notes- New Syllabus
AP Physics C Mechanics- 2.5 Newton’s Second Law – Study Notes
AP Physics C Mechanics- 2.5 Newton’s Second Law – Study Notes – per latest Syllabus.
Key Concepts:
- Unbalanced Forces
- Newton’s Second Law of Motion
- Change in Velocity of a System’s Center of Mass
Unbalanced Forces
Unbalanced forces occur when the net external force acting on a system is not equal to zero. In this situation, the forces do not cancel, and the system undergoes an acceleration in the direction of the net force.
- When \( \mathrm{\sum \vec{F} \neq 0} \), the system’s velocity changes — either in magnitude (speed) or direction.
- Unbalanced forces indicate that the system is not in translational equilibrium.
- The motion of the system follows Newton’s Second Law: \( \mathrm{\sum \vec{F} = m\vec{a}} \).
- Unbalanced forces can result from a single unopposed force or several forces whose vector sum is nonzero.
Whenever a system accelerates, at least one unbalanced external force must be acting on it.
Example
A 5 kg box is pulled horizontally by a \( \mathrm{20\,N} \) force while a \( \mathrm{5\,N} \) frictional force opposes the motion. Determine the acceleration of the box and explain why the forces are unbalanced.
▶️ Answer / Explanation
Step 1: Identify the forces on the box:
- Applied force: \( \mathrm{F_{applied} = 20\,N} \) (forward)
- Frictional force: \( \mathrm{F_{friction} = 5\,N} \) (backward)
Step 2: Compute the net force:
\( \mathrm{F_{net} = F_{applied} – F_{friction} = 20 – 5 = 15\,N.} \)
Step 3: Use Newton’s Second Law to find acceleration:
\( \mathrm{a = \dfrac{F_{net}}{m} = \dfrac{15}{5} = 3\,m/s^2.} \)
Step 4: Interpretation:
The net force is nonzero, meaning the forces are unbalanced. Therefore, the box accelerates at \( \mathrm{3\,m/s^2} \) in the direction of the net force.
Newton’s Second Law of Motion
Newton’s Second Law relates the net external force acting on a system to the resulting acceleration of its center of mass. The acceleration of a system’s center of mass is directly proportional to the magnitude of the net force and occurs in the same direction as that force.
\( \mathrm{\sum \vec{F}_{ext} = m \vec{a}_{cm}} \)
- \( \mathrm{\sum \vec{F}_{ext}} \): vector sum of all external forces acting on the system
- \( \mathrm{m} \): total mass of the system
- \( \mathrm{\vec{a}_{cm}} \): acceleration of the system’s center of mass
If no net external force acts on a system, its acceleration is zero; otherwise, it accelerates in the direction of the net force.
Example
A net horizontal force of \( \mathrm{25\,N} \) acts on a cart of mass \( \mathrm{5.0\,kg} \). Determine the acceleration of the cart and its direction of motion.
▶️ Answer / Explanation
Step 1: Apply Newton’s Second Law:
\( \mathrm{\sum F = ma} \)
Step 2: Substitute values:
\( \mathrm{a = \dfrac{F}{m} = \dfrac{25}{5.0} = 5.0\,m/s^2.} \)
Step 3: Direction:
The acceleration is in the same direction as the net applied force.
Result: The cart accelerates at \( \mathrm{5.0\,m/s^2} \) in the direction of the \( \mathrm{25\,N} \) force.
Change in Velocity of a System’s Center of Mass
The velocity of a system’s center of mass changes only when a nonzero net external force acts on the system. If the net external force is zero, the system’s center of mass moves with a constant velocity or remains at rest. This principle directly follows from Newton’s Second Law and is a key condition for translational equilibrium.
\( \mathrm{\sum \vec{F}_{ext} = m \vec{a}_{cm}} \)
Since acceleration is the time rate of change of velocity:
\( \mathrm{\vec{a}_{cm} = \dfrac{d\vec{v}_{cm}}{dt}} \Rightarrow \sum \vec{F}_{ext} = m \dfrac{d\vec{v}_{cm}}{dt}\)
- If \( \mathrm{\sum \vec{F}_{ext} = 0} \), then \( \mathrm{\dfrac{d\vec{v}_{cm}}{dt} = 0} \) — the center of mass moves at constant velocity.
- If \( \mathrm{\sum \vec{F}_{ext} \neq 0} \), the center of mass accelerates in the direction of the net external force.
Key Idea: Internal forces within the system cannot change its overall velocity; only external forces can alter the motion of the center of mass.
Example
A car of mass \( \mathrm{1000\,kg} \) moves at \( \mathrm{20\,m/s} \). When the driver applies the brakes, a net external braking force of \( \mathrm{2000\,N} \) acts opposite to the car’s motion. Find the car’s acceleration and explain the velocity change.
▶️ Answer / Explanation
Step 1: Apply Newton’s Second Law:
\( \mathrm{a = \dfrac{F_{net}}{m} = \dfrac{-2000}{1000} = -2\,m/s^2.} \)
Step 2: Interpretation:
The negative sign indicates acceleration opposite to the direction of motion — a deceleration.
Step 3: Conclusion:
The nonzero external braking force causes the car’s center of mass velocity to decrease at \( \mathrm{2\,m/s^2} \). If no external force acted (i.e., brakes off, frictionless road), the car would continue moving at constant velocity.