Inclined Planes and Atwood Machines — AP Physics 1
AP Physics 1 · Dynamics · 14 min read
1. Frictionless Inclined Planes★★☆☆☆⏱ 3 min
An inclined plane constrains motion to only be parallel to its sloped surface. To simplify calculations, we rotate the coordinate system so that the positive $x$-axis is parallel to the incline (default pointing down) and the positive $y$-axis is perpendicular to the surface, pointing outward.
Weight $mg$ acts straight down, so we decompose it into parallel and perpendicular components. Using similar triangles, the angle between the weight vector and the negative $y$-axis equals the incline angle $\theta$, giving the components below:
Parallel (x-direction, down incline): $mg \sin\theta$
Perpendicular (y-direction, into incline): $mg \cos\theta$
Since there is no acceleration perpendicular to the incline, net force in $y$ is zero, so normal force $N = mg \cos\theta$. For frictionless inclines, net parallel force gives $a = g \sin\theta$.
2. Inclined Planes with Friction★★★☆☆⏱ 4 min
Friction always opposes motion (or the tendency for motion). Static friction acts on stationary objects with maximum magnitude $f_s \leq \mu_s N$, while kinetic friction acts on sliding objects with fixed magnitude $f_k = \mu_k N$, where $\mu_k < \mu_s$.
Normal force does not change with friction: $N = mg \cos\theta$ still holds because there is no acceleration perpendicular to the incline. For an object sliding down, friction acts up the incline, giving acceleration $a = g(\sin\theta - \mu_k \cos\theta)$. An object will start sliding if $\tan\theta > \mu_s$, the critical condition for sliding.
3. Ideal Atwood Machines★★★☆☆⏱ 3 min
To solve, write Newton's second law for each mass separately, then eliminate tension. If $m_1 > m_2$, $m_1$ accelerates down, $m_2$ accelerates up. Align positive direction with acceleration, add the equations to solve for acceleration:
a = g \frac{m_1 - m_2}{m_1 + m_2}, \quad T = 2g \frac{m_1 m_2}{m_1 + m_2}
4. Modified Atwood Machines (Incline + Atwood Combinations)★★★★☆⏱ 4 min
The most common AP exam problem combines both topics: one mass on an incline connected to a hanging mass over a pulley, called a modified Atwood machine. The same ideal assumptions apply: uniform tension, equal acceleration magnitude for both masses.
Common Pitfalls
Why: Students forget the similar triangles relationship between the incline angle and force component angle.
Why: Students default to friction opposing the weight component, not the direction of motion.
Why: Students confuse zero-acceleration equilibrium with accelerating motion.
Why: Students default to the flat ground rule, forgetting only the perpendicular weight component presses against the incline.
Why: Students forget an inextensible string requires equal displacement for both masses, so equal acceleration magnitude.
Why: Students memorize $f_s = \mu_s N$ and forget this is only the maximum value.