AP Physics 1 Questions: Inclined Planes and Component Decomposition

📚 Quick Study Guide

• 📐 Inclined Plane Basics: An inclined plane is a flat surface tilted at an angle, $\theta$, to the horizontal. Objects on inclined planes experience gravitational force, normal force, and sometimes friction.
• 💪 Component Decomposition: The key to solving inclined plane problems is to decompose the gravitational force ($F_g$) into components parallel ($F_{g\parallel}$) and perpendicular ($F_{g\perp}$) to the inclined plane.
• ➗ Formulas:
  • $F_{g\parallel} = F_g \sin(\theta) = mg \sin(\theta)$
  • $F_{g\perp} = F_g \cos(\theta) = mg \cos(\theta)$
  • Normal Force: $F_N = F_{g\perp} = mg \cos(\theta)$ (if no other vertical forces are present)
  • Friction: $F_f = \mu F_N = \mu mg \cos(\theta)$, where $\mu$ is the coefficient of friction.
• 📝 Net Force: Apply Newton's Second Law, $\Sigma F = ma$, separately for forces parallel and perpendicular to the plane.
• 💡 Tips:
  • Always draw a free-body diagram.
  • Choose your coordinate system wisely (parallel and perpendicular to the plane).
  • Pay attention to the direction of friction (opposes motion or intended motion).

🧪 Practice Quiz

1. A block of mass $m$ is placed on an inclined plane with an angle of $\theta$ to the horizontal. What is the magnitude of the component of the gravitational force acting parallel to the inclined plane?
   1. $mg \cos(\theta)$
   2. $mg \sin(\theta)$
   3. $mg \tan(\theta)$
   4. $mg$
2. A box slides down a frictionless inclined plane of angle $\theta$. What is the acceleration of the box?
   1. $g \cos(\theta)$
   2. $g \sin(\theta)$
   3. $g \tan(\theta)$
   4. $g$
3. A block is at rest on an inclined plane. The angle of inclination is increased until the block begins to slide. What force is overcome when the block starts to move?
   1. Kinetic friction
   2. Static friction
   3. Normal force
   4. Gravitational force
4. A block of weight $W$ rests on an inclined plane that makes an angle $\theta$ with the horizontal. The normal force exerted on the block by the plane is:
   1. $W$
   2. $W \sin(\theta)$
   3. $W \cos(\theta)$
   4. $W \tan(\theta)$
5. A skier is sliding down a slope with a constant velocity. Which of the following is true about the net force acting on the skier?
   1. The net force is constant and non-zero.
   2. The net force is zero.
   3. The net force is increasing.
   4. The net force is decreasing.
6. A block is pushed up an inclined plane with a force $F$. The work done by gravity on the block as it moves a distance $d$ up the plane is:
   1. $mgd$
   2. $mgd \cos(\theta)$
   3. $-mgd \sin(\theta)$
   4. $-mgd \cos(\theta)$
7. Two blocks, one with mass $m$ and the other with mass $2m$, are placed on identical inclined planes. Both blocks are released from rest at the same height. Assuming there is no friction, which block reaches the bottom of the incline first?
   1. The block with mass $m$
   2. The block with mass $2m$
   3. They reach the bottom at the same time
   4. It depends on the angle of inclination