Inclined planes Knowledge-Based Study

Study notes

1. Setting Up the Coordinate Axes

   When analyzing an object resting or moving on an inclined plane with an angle of inclination $\theta$, standard horizontal and vertical axes are inconvenient because acceleration occurs parallel to the surface. Instead, the coordinate system is rotated:

   • Parallel axis ($x$-axis): Directed along the slope of the plane.
   • Perpendicular axis ($y$-axis): Directed perpendicular to the surface of the plane.

   Resolving the Weight Vector

   The gravitational force (weight, $W = mg$) always acts vertically downward. By resolving $mg$ along the tilted axes:

   • Perpendicular component: $W_{\perp} = mg \cos\theta$, acting directly into the inclined surface.
   • Parallel component: $W_{\parallel} = mg \sin\theta$, acting down the incline.

   Because there is typically no acceleration perpendicular to the incline ($a_y = 0$), Newton's second law yields $\Sigma F_y = N - mg \cos\theta = 0$, meaning the normal force is:

   $$N = mg \cos\theta$$

   Equation of Motion

   In the idealized case where surface friction is negligible, the only force acting along the inclined axis is the parallel component of gravity.

   Applying Newton's second law along the incline:

   $$\Sigma F_x = mg \sin\theta = ma$$

   Solving for the acceleration gives:

   $$a = g \sin\theta$$

   Key Observations

   • Mass Independence: The acceleration does not depend on the mass of the object. All objects on the same frictionless slope experience the same acceleration.
   • Limiting Cases: If $\theta = 0^\circ$ (horizontal surface), $a = 0$. If $\theta = 90^\circ$ (vertical drop), $a = g$, representing free fall.

   Static Friction and the Angle of Repose

   When friction is present, the static frictional force opposes impending motion down the slope. The maximum static friction is given by:

   $$f_{s,\max} = \mu_s N = \mu_s mg \cos\theta$$

   An object remains stationary as long as $mg \sin\theta \le \mu_s mg \cos\theta$. At the critical angle where motion is about to begin (the angle of repose):

   $$\tan\theta = \mu_s$$

   Kinetic Friction and Acceleration

   Once the object is sliding, kinetic friction opposes the direction of relative velocity:

   • Object sliding down the incline: Kinetic friction $f_k = \mu_k mg \cos\theta$ points up the incline.
     Newton's second law: $mg \sin\theta - \mu_k mg \cos\theta = ma$
     Acceleration: $$a = g(\sin\theta - \mu_k \cos\theta)$$
   • Object pushed or sliding up the incline: Both the component of gravity and kinetic friction point down the incline.
     Newton's second law: $-mg \sin\theta - \mu_k mg \cos\theta = ma$
     Magnitude of deceleration down the slope: $$a = g(\sin\theta + \mu_k \cos\theta)$$