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Free body diagrams and forces tutorial

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Newton's laws

Free body diagrams

QUESTION 1: "A 10 kg box is placed on the floor and pushed with a force of 100N. The friction coefficient is µ=0.5. Draw a Free Body Diagram and determine the acceleration of the box."

What is a "free body diagram" anyways? Basically, it is just a drawing that shows all the force arrows in the problem. Let's try to make one:

To find the acceleration, we need to use Newton's 2nd law:

We can split this up into two equations; one for the x-direction and one for the y-direction:

A forward force of 100N pushes the box in the positive x-direction, while friction opposes this force. The force of gravity just pulls the box straight down. A "normal force" counter acts it and a friction force prevents it from sliding.

But what the heck is the "normal force" exactly? Here's how to think about it:

Let's break out the microscope and look at the individual atoms of the box and the floor. Imagine that the atoms of the floor are connected by tiny springs. If you grab two of them and push them together, they will push back on your hands. If we place a box on top of them, they will get compressed. Therefore, they will push back on the box.

This picture also tells us why the friction force must be calculated the way it is:

A large normal force means that the box is "dug in" to the surface. Obviously this will make it harder to slide along the surface. If we push a box across a horizontal surface, then clearly it is not going to fly up into the air or fall into the ground. Thus the acceleration in the y-direction must be zero:

So the force of gravity and the normal force are equal in magnitude. Let us use this to simplify the x-expression:

So the speed of the box increases by 5m/s for every second we spend pushing it. QUESTION 2: "A box of mass, m, is hanging from the ceiling by two ropes which form a 60 degree angle with the horizontal. Determine the Tension in the ropes." To answer this question we need the following:

  1. A sketch of the setup we are trying to model.

  2. A free body diagram that keeps track of all the forces involved.

  3. Some way of relating the various forces to each other.

  4. A way to check if the result makes sense.

Let's start with 1.

  1. A sketch of the setup we are trying to model.

Just by looking at the picture, we can predict that a heavier box means more tension in the ropes. By the way, "Tension" just means "The amount of force that the ropes pull on the box with". Obviously if the box is heavier, the ropes have to pull harder on it to make sure it stays floating in the air. In reality, if the box gets too heavy, the ropes will snap. Anyways, let's move on to 2.

2. A free body diagram that keeps track of all the forces involved.

We can see that there are 3 forces involved. There are the two tension forces from the two ropes that pull the box upwards and to the sides. There is also the force of gravity that pulls it straight down. Let's move on to 3. We need:

3. Some way of relating the various forces to each other.

Like before, let us apply Newton's 2nd law. We sum up all the forces:

What to do now? Well we know that the box is just hanging from the ceiling. It is not moving left or right, and it is not falling down or floating upwards. In other words, it is NOT accelerating. Therefore, we can write:

^Basically, since the box is not moving in either the x- or y-directions, the forces in the x- and y-directions must sum up to zero. From the two equations, we get:

We need some way of calculating the x- and y-components of the tensions. To do this, we notice that we can split these vectors up using cosine and sine:

Let us use this to rewrite the equations from before:

^Aha, so the two ropes experience the same tension. That of course makes sense. They are set up in exactly the same way. Let us plug this new information in to the y-equation:

NICE! The tension in the ropes is the weight of the box divided by 2 times the sine of the suspension angle. Let's consider the fourth and final step:

4. A way to check if the result makes sense.

Let's think about the original picture:

If we make the suspension angle closer to 90 degrees, the ropes will just hang straight down. In that case,

It makes sense that if we have two ropes holding up a box, then they each take on half of the total weight. If we imagine making the angle closer to 0 degrees, then we have

It also makes sense that if the ropes are pulling at a very flat angle, they need to pull with a lot of force to keep the box in place. QUESTION 3: "A box of mass m is placed on a slope with an angle, A. The friction coefficient between the bo