由于摩擦的力永远是平行的to the surface upon which the object is traveling. It will come directly from the center of the object, and be pointed in the opposite direction to the motion of the object.

In this diagram,Vis pointed parallel to the surface and opposite to the direction of motion.

Remember, in a force diagram, forces come from the center of the object. The force of gravity willalwaysbe straight down from the center of the object.

In this diagram,Wis pointed directly downward.

When working in an inclined plane, when we break our force due to gravity into components, the angle at the top that triangle will be equal to the angle of the inclined plane.

In the diagram, the total force due to gravity is given byW.Xrepresents the horizontal component, and is parallel to the surface of the plane.Yrepresents the vertical component, and is perpendicular to the surface of the plane.XandYthus create a right angle, withWas the hypotenuse. By turning this triangle such that the right angle aligns with the right angle of the inclined plane (betweenQandP), we can see thatWaligns with the incline surface andYaligns with the base,P. Based on these alignments, the angle betweenWandYmust be equal to the angle between the surface andP;因此,.

The normal force is always perpendicular to the surface on which the object is placed, and is pointed away from said surface.

In this diagram,Zis the only force that is perpendicular to the surface and in the upward direction. This force must counteract the vertical force of gravity, which will be perpendicular to the surface in the downward direction (Y).

In this diagram,Vrepresents the force due to friction. The equation for the force due to friction is, whereis the coefficient of friction.

In this case,Zrepresents the normal force. We can re-write the equation for friction:

Zcan be re-written in terms of the angle, but will always need to be multiplied by the coefficient of friction in order to give an equation forV.

The other equations are true.

-XandYform a right angle, so the Pythagorean theorem applies.

-Zis the normal force, which is, by definition, equal and opposite the vertical force of gravity.

-Wis the total force of gravity, which will be equal to the mass times the acceleration of gravity.

- the triangle formed byW,X, andYis similar to the triangle formed by the surface,Q, andP, meaning that these angles must be equal.

In this problem,YandZare equal, but opposite forces, andZis our normal force.

If we can solve forY, then we can findZ.

We can use our understanding of trigonometry to find an equation forY.

If we plug infor the angle, we see:

Since we are solving forY, we can multiply both sides byW.

Now that we know an equation forY, we can return to our original equation to solve forZ.

There is no way for us to know the distance the object travels. Even though the force diagram places our object roughly halfway up the plane, we do not know the position where the object starts or where the object ends. We cannot calculate a change in distance.

We can use our understanding of trigonometry to find this relationship, sinceXandYform a right angle.

If we plug infor the angle, we see thatXis opposite ofandWis the hypotenuse.

Since we're looking forX, we can multiply both sides byW.

For this problem, we need to break the forces into horizontal and vertical components. The only difference with an inclined plane is that you have to translate the horizontal plane to be parallel to the surface upon which the object is traveling, and the vertical plane to be perpendicular to the surface upon which the object is traveling.

Luckily, this is already done in the diagram, with the force of gravity (W) broken into the vertical component (Y) and horizontal component (X).

Now, we need to sum the forces in each plane. The forces in the vertical plane will be perpendicular to the surface:ZandY.

SinceZandYare opposite and equal forces:

There is no net force in the vertical plane. This makes sense, as we would not expect the object to have any movement perpendicular to the surface on which it is sliding.

Now we can sum the horizontal forces:VandX.

Note thatXis positive andVis negative. This is becauseVrepresents the force of friction, and is opposite to the force ofX.

Since there is no force in the vertical plane, this gives our final net force:.