Forces and Inclines

Inclined Planes Objects placed on an incline tend to slide down the surface The rate at which it slides will be dictated by the angle of the incline…greater tilt = greater rate of slide An object that accelerates down an incline must do so because of unbalanced force in that direction

Forces with an inclined surface There are always at least two forces acting on an object on an incline…

Normal Force… Normal force will ALWAYS be perpendicular to the inclined surface…

Weight (force from gravity) ALWAYS straight down ALWAYS at an angle with respect to the inclined surface

Weight (force from gravity) The weight can be resolved into two components. One component is aligned parallel to the inclined surface aligned perpendicular to the inclined surface

Re-define the Axes!! FN Fg y x For purposes of problem solving, redefine the x-axis to line up with the inclined surface and the y-axis to be perpendicular to the inclined surface. This makes the parallel component of the weight in the “x” direction This makes the perpendicular component of the weight and the normal force in the “y” direction y x FN Fgx Fgy Fg

A closer look at the weight vector… The angle between the weight vector and the new y-axis is the same angle as the incline. y x θ θ Fg

A closer look at the weight vector… The perpendicular component is along the new y-axis and its magnitude is given by: y x Fgcosθ θ θ Fg

A closer look at the weight vector… The parallel component is along the new x-axis and its magnitude is given by: y x Fgsinθ θ θ Fg

When the angle gets bigger… The force in the direction down the incline gets larger while the component into the surface gets smaller

Example 1 – Inclined Plane A 25 kg crate is sitting at rest on a ramp inclined at 25 degrees. Find the normal force and the frictional force acting on the crate. Then find µs between the crate and the ramp.

Example 2 – Inclined Plane A 4.5 kg object is accelerating down an inclined plane inclined at 36 degrees and has a coefficient of friction of 0.548. What is the acceleration of the object?

Example 3 – Inclined Plane A 22.7-N block is placed upon an inclined plane which is inclined at a 17.2 degree angle. The coefficient of friction is 0.219. Determine the acceleration of the block if the applied force of 20.6 N is acting up the ramp.

Your Turn #1 – Inclined Planes The infamous Lombard Street in San Francisco has an abnormally steep block which consists of several tight hairpin turns. On average, the roadway on this block of the street is inclined at 16°. Determine the force which would be required to pull a 23-kg wagon and child up the hill at constant speed. Assume the force is exerted parallel to the road and that friction is negligible.

Your Turn #2 – Inclined Planes Ben Laborin exerts a force on an 86-kg crate of books to push it up the ramp of the loading dock at a constant speed of 24 cm/s. The ramp makes an angle of 12° with the horizontal. The coefficient of friction between the crate and the ramp is 0.74. Assuming that Ben is pushing parallel to the inclined plane, determine the force with which he is pushing.