If an object is not horizontal to the ground, it may be on an inclined plane. We need to adjust both the calculations and the free-body diagram when determining the net force on an object on an inclined plane. Learn how in this lesson.
When you sit at the top of a slide, you are perpendicular to the ground. But if you scoot far enough forward, you will end up sliding down the slide because it is a tilted surface. So down you go until you reach the bottom, where you (hopefully) end up perpendicular to the ground once again.
The reason you travel down the slide is because of an unbalanced force. This happens for most objects - unbalanced forces cause them to travel down tilted surfaces, which in physics we call inclined planes. Things like balls, toy cars, and dinner plates will all slide down an inclined plane as gravity pulls them back to Earth from their elevated position at the top of the plane. This is because the force of gravity is an object's weight, and it is what keeps us on the ground instead of floating around in the air.
As with objects that are flat on the ground, the normal force, which is a force that acts in the direction perpendicular to a surface, also acts in these situations. Usually, when we're standing on flat ground, the normal force works in the direction opposite to gravity because the ground is horizontal.
But as an object travels down an inclined plane, the normal force is tilted along with the tilt of the plane. So instead of being perpendicular to the ground, it's now perpendicular to the surface of the inclined plane. In fact, the normal force will always be perpendicular to the surface the object is on, no matter the angle of the surface.
Determining Net Force
This makes things a bit tricky for determining the net force acting on an object because gravity is still directed downward, even when the normal force is not directed upward. We can see how this presents a problem by using a free-body diagram to illustrate the forces.
As you can see, the normal force acts perpendicular to the surface of the plane but gravity still acts downward and perpendicular to the horizontal ground.
What we need to do is take that one gravity vector and break it into two components. One component is parallel to the inclined surface and the other is perpendicular to the surface but in the opposite direction of the normal force.
Now the perpendicular gravity vector balances the normal force and the parallel vector accelerates the object down the inclined plane. Because the normal force and the perpendicular force are balanced, the net force on the object is the parallel component of the gravity vector (in the absence of friction, of course).
Since we split the gravity vector into two components, we calculate the magnitude of each force a bit differently than we would for an object on a horizontal surface. On a horizontal surface, the gravitational force is equal to the object's weight, as is the normal force.
But now the parallel force is the mass of the object * g * sin (theta), where g is the acceleration due to gravity (9.8 m/s^2), and theta is the angle of the inclined plane. For the perpendicular vector, the magnitude is the mass of the object * g * cos (theta).
Let's give this a try. Say you have a 500 kg object sliding down an inclined plane tilted upward from the ground at a 45° angle. You can ignore the downward gravitational force because we split it into its components and instead solve for each of these.
So the parallel component is:
500 kg * 9.8 m/s^2 * sin (45)
which gives us 3465 N of force.
The perpendicular component is:
500 kg * 9.8 m/s^2 * cos (45)
which also gives us 3465 N of force.
We know the normal force is also 3465 N because it balances the perpendicular force. Since there is no force acting in the direction opposing the parallel component, the net force is equal to the parallel force, so 3465 N down the incline.
You can also determine the acceleration of the object by dividing the net force by the mass of the object. So in this case, 3465 N / 500 kg = 6.93 m/s^2.
Friction Affects Net Force
Okay, so that was a nice 'ideal' example where friction was not a factor. But since friction is the force that opposes motion and is experienced by objects quite often, we can't ignore it any longer. The good news is that it's not difficult to add it into our vector mixture.
In order to determine the net force on the object when friction is acting, simply find the vector sum of the two parallel forces (remember, the normal force and the perpendicular force balance each other). All you need to do is to add the parallel vectors, which have opposite directions. For example, if the parallel force due to gravity is 50 N down the incline and friction is 20 N up the incline, the sum is 30 N, which is the net force. Since the force acting down the incline is greater, the net force acts in that direction.
For our previous example, if friction was acting against the object sliding down the 45° incline with a force of 465 N, then our net force would be: F net = 3465 N down - 465 N up, which comes to 3000 N down the incline.
One thing that might be helpful when solving these types of problems is to rotate the free-body diagram so that it looks like a normal, horizontal situation. As you can see below, by rotating the angle of the diagram, this looks much more like an object sitting on a horizontal surface. Rotating the diagram is also much easier than trying to solve this problem with a tilted head!
Rotated free-body diagram
In physics, we refer to tilted surfaces as inclined planes. As an object travels down an inclined plane, the normal force acting on it also tilts because the normal force is a force perpendicular to the surface.
When we stand on flat ground, the normal force acts upward, but when we tilt, the normal force tilts with us.
But because gravity, which is the weight of an object, doesn't tilt, we need to separate it into two components: one that is parallel to the surface of the inclined plane and one that is perpendicular to the surface but opposite in direction to the normal force.
Once we determine the magnitude of each of these forces, we can solve for the net force that is causing the object to travel down the inclined plane as well as the acceleration of the object as it travels towards the ground.
Review this lesson's facts in order to achieve these goals:
- Define inclined plane and normal force in terms of physics
- Summarize the relationship between the normal force and gravity
- Find the net force and acceleration of an object traveling on an inclined plane