people approaching this subject may think of math as something tedious and overly abstract. However, in the context of physics, math comes to life. A quadratic equation may seem a little dry, but when you’re using it to work out the correct angle to fire a rocket at for the perfect trajectory, you may find it more palatable! Chapter 2 explains all the math you need to know to perform basic physics calculations.
So what are you going to get out of physics? If you want to pursue a career in physics or in an allied field such as engineering, the answer is clear: You’ll need this knowledge on an everyday basis. But even if you’re not planning to embark on a physics-related career, you can get a lot out of studying the subject. You can apply much of what you discover in an introductory physics course to real life:
✔ In a sense, all other sciences are based upon physics. For example, the structure and electrical properties of atoms determine chemical reactions; therefore, all of chemistry is governed by the laws of physics. In fact, you could argue that everything ultimately boils down to the laws of physics!
✔ Physics does deal with some pretty cool phenomena. Many videos of physical phenomena have gone viral on YouTube; take a look for yourself. Do a search for “non-Newtonian fluid,” and you can watch the creeping, oozing dance of a cornstarch-water mixture on a speaker cone.
✔ More important than the applications of physics are the problem-solving skills it arms you with for approaching any kind of problem. Physics problems train you to stand back, consider your options for attacking the issue, select your method, and then solve the problem in the easiest way possible.
Observing Objects in Motion
Some of the most fundamental questions you may have about the world deal with objects in motion. Will that boulder rolling toward you slow down? How fast do you have to move to get out of its way? (Hang on just a moment while we get out our calculator… .) Motion was one of the earliest explorations of physics.
When you take a look around, you see that the motion of objects changes all the time. You see a motorcycle coming to a halt at a stop sign. You see a leaf falling and then stopping when it hits the ground, only to be picked up again by the wind. You see a pool ball hitting other balls in just the wrong way so that they all move without going where they should. Part I of this book handles objects in motion – from balls to railroad cars and most objects in between. In this section, we introduce motion in a straight line and rotational motion.
Speeds are big with physicists – how fast is an object going? Thirty-five miles per hour not enough? How about 3,500? No problem when you’re dealing with physics. Besides speed, the direction an object is going is important if you want to describe its motion. If the home team is carrying a football down the field, you want to make sure they’re going in the right direction.
When you put speed and direction together, you get a vector – the velocity vector. Vectors are a very useful kind of quantity. Anything that has both size and direction is best described with a vector. Vectors are often represented as arrows, where the length of the arrow tells you the magnitude (size), and the direction of the arrow tells you the direction. For a velocity vector, the length corresponds to the speed of the object, and the arrow points in the direction the object is moving. (To find out how to use vectors, head to Chapter 4.)
Everything has a velocity, so velocity is great for describing the world around you. Even if an object is at rest with respect to the ground, it’s still on the Earth, which itself has a velocity. (And if everything has a velocity, it’s no wonder physicists keep getting grant money – somebody has to measure all that motion.)
If you’ve ever ridden in a car, you know that velocity isn’t the end of the story. Cars don’t start off at 60 miles per hour; they have to accelerate until they get to that speed. Like velocity, acceleration has not only a magnitude but also a direction, so acceleration is a vector in physics as well. We cover speed, velocity, and acceleration in Chapter 3.
Plenty of things go round and round in the everyday world – CDs, DVDs, tires, pitchers’ arms, clothes in a dryer, roller coasters doing the loop, or just little kids spinning from joy in their first snowstorm. That being the case, physicists want to get in on the action with measurements. Just as you can have a car moving and accelerating in a straight line, its tires can rotate and accelerate in a circle.
Going from the linear world to the rotational world turns out to be easy, because there’s a handy physics analog (which is a fancy word for “equivalent”) for everything linear in the rotational world. For example, distance traveled becomes angle turned. Speed in meters per second becomes angular speed in angle turned per second. Even linear acceleration becomes rotational acceleration.
So when you know linear motion, rotational motion just falls in your lap. You use the same equations for both linear and angular motion – just different symbols with slightly different meanings (angle replaces distance, for example). You’ll be looping the loop in no time. Chapter 7 has the details.
When Push Comes to Shove: Forces
Forces are a particular favorite in physics. You need forces to get motionless things moving – literally. Consider a stone on the ground. Many physicists (except, perhaps, geophysicists) would regard it suspiciously. It’s just sitting there. What fun is that? What can you measure about that? After physicists had measured its size and mass, they’d lose interest.
But kick the stone – that is, apply a force – and watch the physicists come running over. Now something is happening – the stone started at rest, but now it’s moving. You can find all kinds of numbers associated with this motion. For instance, you can connect the force you apply to something to its mass and get its acceleration. And physicists love numbers, because numbers help describe what’s happening in the physical world.
Physicists are experts in applying forces to objects and predicting the results. Got a refrigerator to push up a ramp and want to know if it’ll go? Ask a physicist. Have a rocket to launch? Same thing.
Have you ever watched something bouncing up and down on a spring? That kind of motion puzzled physicists for a long time, but then they got down to work. They discovered that when you stretch a spring, the force isn’t constant. The spring pulls back, and the more you pull the spring, the stronger it pulls back.
So how does the force compare to the distance you pull a spring? The force is directly proportional to the amount you stretch the spring: Double the amount you stretch the spring, and you double the amount of force with which the spring pulls back.
Physicists were overjoyed – this was the kind of math they understood. Force proportional to distance? Great – you can put that relationship into an equation, and you can use that equation to describe the motion of the object tied to the spring. Physicists got results telling them just how objects tied to springs would move – another triumph of physics.
This particular triumph is called simple harmonic motion. It’s simple because force is directly proportional to distance, and so the result is simple. It’s harmonic because it repeats over and over again as the object on the spring bounces up and down. Physicists were able to derive simple equations that could tell you exactly where the object would be at any given time.
But that’s not all. Simple harmonic motion applies to many objects in the real world, not just things on springs. For example, pendulums also move in simple harmonic motion. Say you have a stone that’s swinging back and forth on a string. As long as the arc it swings through isn’t too high, the stone on a string is a pendulum; therefore, it follows simple harmonic motion. If you know how long the string is and how big of an angle the swing covers, you can predict where the stone will be at any time. We discuss simple harmonic motion in