Pulleys and tension

What minimum force is required to lift the mass with the pulley system shown?

The word 'force' is right there in the question, so we will certainly start with Newton's 2nd law. But first, let's look at a couple of simpler systems before we try tackling this double pulley.

We need to be sure we know how to deal with tension forces. So, let's look at the simplest possible example: a mass hanging by a chord.

Our intuition tells us that the tension force is equal in magnitude and opposite in direction to the force of gravity. To be explicit, let's write down Newton's 2nd Law for this trivial system anyway.

While this may not have been necessary, it's good to get in the habit of writing out Newton's 2nd Law. As expected, we see the the magnitude of the tension force is equal to the gravitational force. It shouldn't come as a surprise, then, that if T > mg then the mass would be accelerating upwards and if T < mg, it would be accelerating downward. If this is not obvious to you, take a minute to work it out for yourself.

We can now turn to a slightly more interesting system. An Atwood machine consists of two masses suspended from a chord that is wound around a freely-rotating pulley. For now, we will assume that the pulley is massless and frictionless.

Before we try to find the equations of motion for the masses, let's consider how we expect the system to behave. First, it is clear that if the two masses are equal, then there will be no acceleration. And if one of the masses is greater than the other, the greater mass will accelerate downwards while the lesser mass will accelerate upwards. More importantly, because we are assuming the chord is ideal and won't stretch, we can see that the magnitudes of the accelerations will be equal. Moreover, the tension in the chord will be uniform (a consequence of the frictionless pulley).

Now, to verify our intuition, let's start our familiar process of writing down Newton's Laws.

 

Note that I utilized the constraint that the accelerations will be equal and opposite.This puts the equations in a convenient form, because by adding them together (in other words, adding the two left hand sides of the equation and the two right hand sides) the tension term disappears.

We see that our intuition was correct: if the masses are equal, there will be no acceleration, and if the masses are different, the larger mass will accelerate downward.

Now that we have seen some basic examples involving tension and pulleys, we are in good shape to attack the original problem. The first order of business is considering the role that the rope tension plays in this system. We are assuming that the pulleys are massless and frictionless, so we know that the tension everywhere in the rope must be the same --- keep in mind that if that wasn't the case, then different sections of the rope would be accelerating at different rates, and you would wave stretching and/or bunching of the rope, which doesn't happen.

With that in mind, we immediately see that the force pulling on the rope, F, must be equal to the tension in the rope, let's call it T. The lower pulley is being held up by two sections of rope, both with tension T for a total of 2T --- which must equal at least mg to hold up the weight. Therefore, the minimum force required to hold up the weight is mg/2.

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.

A profound and simple guide to solving physics problems

Or, why almost all advice about learning physics is wrong.

Introductory physics is dreaded for one main reason: it makes smart people feel stupid. But it really isn't any more difficult than the other sciences, it's just generally not taught well. Or more specifically, it's taught well enough to attract a handful of future physicists. But the overwhelming majority of students who take intro physics are not going to be physicists, so they need to be introduced to the concepts of physics in a more directed way.

In my experience teaching physics, students think that by the end of the course they have learned a lot, when in fact they've only learned a few things, but have had to apply what they've learned to many specific situations. Physics is difficult because applying general techniques to new and unfamiliar problems is difficult. You will be tested on your problem-solving skills, not your knowledge -- this is the fundamental difference between physics and the other sciences (at least at the introductory level). And this is precisely why many programs use physics to weed out the weaker students: you can't "study" your way through physics in the traditional sense. You absolutely have to roll up your sleeves and practice solving problems, many of which will frustrate you to the point of tears! Or at least, they will if you don't take the approach I outline here...

I want to convince you that in your first semester of introductory physics you will learn exactly three things:

1. Newton's 2nd Law
2. Conservation of linear and angular momentum
3. Conservation of energy

These three laws are all you will need to solve almost every problem you will encounter. Things get tricky when you have to apply these three simple laws to complicated physical systems, but it is imperative that you always keep these laws in mind as you approach each new problem.

In this blog I will provide solutions to standard physics problems, and will show how one should approach each problem with the three laws in mind.