Newtonian Physics (Light and Matter series, Book 1)
Estimate the total mass of feed required to raise the cows used to make the burgers. How many cm2 is 1 mm2? For instance, a 9. If a certain 7. The ratio of the height to the width is the square root of 2, and this has some useful properties. For instance, if you cut an A4 sheet from left to right, you get two smaller sheets that have the same proportions. There is a whole series of sizes related in this way, all with the same proportions. What would be the dimensions of a B0 box? Is this true? If so, how many molecules? Compare the land areas of the two planets.
Suppose you make a Martini by pouring vermouth into the glass to a depth of 3 cm, and then adding gin to bring the depth to 6 cm. What are the proportions of gin and vermouth?
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What is the volume of each of the small cubes? Problems 67 20 A cute formula from trigonometry lets you find any angle of a triangle if you know the lengths of its sides. The figure on page 71 is a roadmap to the rest of the book.
Even walking, which we consider to be no great feat, requires an intricate series of motions that your cerebrum would be utterly incapable of coordinating. The task of putting one foot in front of the other is controlled by the more primitive parts of your brain, the ones that have not changed much since the mammals and reptiles went their separate evolutionary ways. To make it clear what we are and are not ready to consider, we need to examine and define carefully what types of motion can exist. Walking is thus an example in which there is both a general motion of the whole object and a change in the shape of the object.
Another example is the motion of a jiggling water balloon as it flies through the air. We are not presently attempting a mathematical description of the way in which the shape of an object changes. Motion without a change in shape is called rigid-body motion. It is not immediately obvious, however, what is the most useful way to define the distinction between rotation and motion through space. Imagine that you attempt to balance a chair and it falls over. The center of mass can therefore be defined as the intersection of all the lines made by hanging the pear in this way.
Note that the X in the figure should not be interpreted as implying that the center of mass is on the surface — it is actually inside the pear. Every object has a balance point, referred to in physics as the center of mass.
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For a two-dimensional object such as a cardboard cutout, the center of mass is the point at which you could hang the object from a string and make it balance. In the case of the ballerina who is likely to be three-dimensional unless her diet is particularly severe , it might be a point either inside or outside her body, depending on how she holds her arms. Even if it is not practical to attach a string to the balance point itself, the center of mass can be defined as shown in figure e.
Why is the center of mass concept relevant to the question of classifying rotational motion as opposed to motion through space? We might expect that her motion would be much more complicated than the motion of a small, simply-shaped object, say a marble, thrown up at the same angle as the angle at which she leapt. That is, the motion of the center of mass is the same as the motion the ballerina would have if all her mass was concentrated at a point. By Velocity and Relative Motion restricting our attention to the motion of the center of mass, we can therefore simplify things greatly.
Her center of mass traces a straight line, but a point away from her center of mass, such as her elbow, traces the much more complicated path shown by the dots. By this definition, the tipping chair does have both rotational and center-of-mass motion. Concentrating on the center of mass motion allows us to make a simplified model of the motion, as if a complicated object like a human body was just a marble or a point-like particle. Science really never deals with reality; it deals with models of reality.
When you get a new tire, the mechanic clamps little weights to the rim to balance the wheel. A car wheel that has not been balanced properly has a center of mass that does not coincide with its geometrical center. An object such as the human body does not even have an obvious geometrical center.
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It can be helpful to think of the center of mass as the average location of all the mass in the object. Ballerinas and professional basketball players can create an illu- Section 2. Photo by Dunia Young. The legs are low at the beginning and end of the jump, but come up higher at the middle. Regardless of what the limbs do, the center of mass will follow the same arc, but the low position of the legs at the beginning and end means that the torso is higher compared to the center of mass, while in the middle of the jump it is lower compared to the center of mass.
Our eye follows the motion of the torso and tries to interpret it as the center-of-mass motion of a rigid body. But since the torso follows a path that is flatter than we expect, this attempted interpretation fails, and we experience an illusion that the person is flying horizontally. Here the jumper lowers his legs and upper body at the peak of the jump in order to bring his waist higher compared to the center of mass. Center-of-mass motion in one dimension In addition to restricting our study of motion to center-of-mass motion, we will begin by considering only cases in which the center of mass moves along a straight line.
This will include cases such as objects falling straight down, or a car that speeds up and slows down but does not turn. Note that even though we are not explicitly studying the more complex aspects of motion, we can still analyze the center-of-mass motion while ignoring other types of motion that might be occurring simultaneously. For instance, if a cat is falling out of a tree and is initially upside-down, it goes through a series of contortions that bring its feet under it.
Self-check A Consider a person running, a person pedaling on a bicycle, a person coasting on a bicycle, and a person coasting on ice skates. In which cases is the center-of-mass motion one-dimensional?
Which cases are examples of rigid-body motion? Times can instead be expressed entirely in terms of a single unit, such as hours. Fractions of an hour can be represented by decimals rather than minutes, and similarly if a problem is being worked in terms of minutes, decimals can be used instead of seconds. Self-check B Of the following phrases, which refer to points in time, which refer to Section 2. As with t, x can be negative. Suppose we are describing the motion of a train on tracks linking Tucson and Chicago. As shown in the figure, it is entirely up to you to decide which way is positive.
However important the odometer reading is to car owners and used car dealers, it is not very important in physics, and there is not even a standard name or notation for it. Self-check C A ball hits the floor, bounces to a height of one meter, falls, and hits the floor again.
This is referred to as choosing a coordinate system or choosing a frame of reference. As long as you are consistent, any frame is equally valid. Have you ever been sitting in a train in a station when suddenly you notice that the station is moving backward? Most people would describe the situation by saying that you just failed to notice that the train was moving — it only seemed like the station was moving.
But this shows that there is yet a third arbitrary choice that goes into choosing a coordinate system: valid frames of reference can differ from each other by moving relative to one another. It might seem strange that anyone would bother with a coordinate system that was moving relative to the earth, but for instance the frame of reference moving along with a train might be far more convenient for describing things happening inside the train. Section 2. We can tell this because every two seconds, its position changes by five meters.
In graphical terms, the velocity can be interpreted as the slope of the line. The answer would still have been the same, 2. In example m, the object is moving in the opposite direction: as time progresses, its x coordinate decreases. In graphical terms, a positive slope characterizes a line that goes up as we go to the right, and a negative slope tells us that the line went down as we went to the right.
Note that we are still talking about one-dimensional motion. The graph is not like a map, and the horizontal direction of the graph represents the passing of time, not distance.
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Example n is similar to example l in that the object moves a total of There is no way to characterize the entire graph by a certain velocity or slope, because the velocity is different at every moment. It would be incorrect to say that because the car covered It moved faster than that at the beginning and end, but slower in the middle. There may have been certain instants at which the car was indeed going 2. We assume that our speedometer tells us what is happening to the speed of our car at every instant, but how can we define speed mathematically in a case like this? A mathematical definition that corresponded to the speedometer reading would have to be one that attached a different velocity value to a single point on the curve, i.
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The slope of the tangent line example 1. What is the velocity at the point shown with a dot on the graph? First we draw the tangent line through that point.
To find the slope of the tangent line, we need to pick two points on it. Conventions about graphing The placement of t on the horizontal axis and x on the upright axis may seem like an arbitrary convention, or may even have disturbed you, since your algebra teacher always told you that x goes on the horizontal axis and y goes on the upright axis.