Motionless block on inclined plane

A force is applied to an inclined plane resting on a frictionless surface. What is the force required to keep the small block from sliding down the inclined plane?

In this inaugural problem, I want to stress the importance of keeping in mind the three main Laws, or "Big Ideas" that you will learn in introductory physics.

It should be obvious that the Big Idea for this problem is Newton's 2nd Law. It only involves forces, and there is no time or positional dependence.

Now, before we start rummaging around for formulas, or looking in the book for hints, let's start practicing our new approach to physics problems: let the Big Idea guide us. In fact, let's write it down so we can see it and be inspired.

So, the acceleration of an object depends on the sum of the forces acting on it. In this particular problem, the object we care about is the block. What, specifically, will it's motion be?

Well, it's going to be accelerating horizontally to the right, but since it won't be sliding down, it's vertical acceleration will be zero. We can exploit the vector nature of Newton's 2nd Law, and look at the horizontal and vertical motion separately.

First, let's figure out the type and direction of the forces acting on the block. As is almost always the case, we are assuming that we are doing this experiment on the surface of Earth, so there will be a gravitational force. And since the block is resting on the inclined plane, there will be a normal force associated with the contact between the two objects.

So let's draw a picture. We will use it to write down Newton's 2nd Law for the vertical acceleration, which is zero.

First, be sure you are comfortable with finding components of vectors. In the picture, do you see why the angle between F_N and the vertical component is θ, and why the component has magnitude F_Ncosθ? If not, you should brush up on the definitions of basic trigonometric functions, and basic vector manipulation.

Now we can consider the force on the block in the horizontal direction. It should be apparent from the figure that there will only be one horizontal force: the horizontal component of the normal force. So, let's write down Newton's 2nd Law.

Again, you should see where the sinθ term comes from. The first line is just the statement that whatever acceleration the block has in the horizontal direction, it is caused by the horizontal component of the normal force from the inclined plane. We can combine this equation with the equation from the vertical components, and we find the acceleration necessary to keep the block from sliding down the plane.

Now that we know the acceleration of the block when it's vertical acceleration is zero, we can turn our attention to finding the force, F, applied to the inclined plane. The key is to realize that we can write down Newton's 2nd Law one more time for the system as a whole (since the two objects are touching, we can think of them as a single object).

After writing down Newton's 2nd Law, we substitute the value for a, and we have our answer.

And now that we have an answer, let's make sure it makes sense. First, and most importantly, we should check that the units work out. They do. If you don't see why, then you need to go brush up on basic unit analysis.

We can see that the solution depends on three things: the sum of the masses, the acceleration due to gravity, and tanθ. Since the acceleration due to gravity is a constant, we can ignore it for now. Let's look at the other two.

If the sum of the masses increase, then the required force increases. This makes sense because there is a minimum acceleration of the the blocks that will prevent the smaller block from sliding, and Newton's 2nd Law tells us that if we want to achieve a certain acceleration, and we increase the mass, then the force must increase as well.

The other dependence is on tanθ, which depends on the slope of the incline. First, let's remind ourselves what tanθ looks like: 

When θ=0, tanθ=0 and the required force is zero. And as the the slope of the incline (θ) increases, the required force increases. Try to convince yourself that this is a reasonable prediction, and be sure that you understand why the required force increases with θ.