knowledge, science, and the universe chapter 1 - Physical Science ...

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SECTION 3–3 Weight and Acceleration 31 15 m/sec 0 m/sec 15 m/sec 9.8 m/sec 9.8 m/sec 15 m/sec 19.6 m/sec 19.6 m/sec 15 m/sec 29.4 m/sec 29.4 m/sec 15 m/sec 39.2 m/sec 39.2 m/sec figure 3.3 The green ball is thrown to the right at 15 m/sec at the same time the blue ball is dropped. However, both will hit the ground at the same time because the acceleration of all falling bodies is the same regardless of their initial velocities. instead of vertically, with an initial speed of 15 m/sec, as indicated in Figure 3.3. This time it follows a curved path as it falls. Curved motion appears more complicated, but can be simplified by separating the path of the ball into horizontal and vertical parts. At the end of the first second, the ball is still moving horizontally with a speed of 15 m/sec, but in addition it is falling with a vertical speed of 9.8 m/sec downward. After another second, the downward speed has increased to 19.6 m/sec, while the horizontal motion remains unchanged. As things progress the horizontal part of the motion never changes, while the downward velocity continues to increase at the rate of 9.8 m/sec every second because the force of gravity pulls only in that direction. This change in the downward speed of the ball is the same as if the ball had no initial motion at all. It may not be obvious, but two balls positioned at the same height, with one dropped at the same instant as the second ball is thrown horizontally, will hit the ground at the same time because the downward acceleration on both is the same. In all three examples—dropping a rock, throwing a ball into the air, and throwing a ball horizontally—the initial velocities are all different but the accelerations, and therefore the way the velocities change, are all the same. 3–3 wEIGhT ANd ACCELERATION Now suppose you were to drop both a basketball- sized boulder and a baseball- sized rock from the same height. You might expect the boulder to drop more rapidly than the rock, because it is much heavier. But upon dropping the two rocks you find that, despite their great difference in size and weight, their accelerations are exactly the same. Their weights do not affect how they accelerate! Look again at Newton’s Second Law of Motion. The large boulder has greater mass than the smaller rock so it is harder to accelerate. Yet they both accelerate at the same rate so the force on the boulder has to be greater than the force on the rock. You are familiar with this, of course. More massive things weigh more and weight just refers to the force of gravity on an object. Weight and mass are exactly proportional, with the proportionality factor being the acceleration g. If “w” is weight, Newton’s second law for this case is simply w = mg which clearly expresses the proportionality. So weight The force of gravity on an object.
32 C h A p T E R 3 The Gravitational Interaction figure 3.4 In the absence of friction all objects fall with the same acceleration regardless of their mass. if the mass of one object is two times the mass of another, its weight is exactly twice as great. When you step on a scale to measure your mass, you are really measuring the force of gravity on you—your weight. This is an accurate mass estimate only because of this propor - tionality.* We must point out that objects all fall at the same rate of acceleration, regardless of their weight, only if the resistance from air can be ignored. The effect of air friction, considered in more detail in Chapter 5, will cause lighter objects, like a leaf, to accelerate more slowly than a heavier rock. But even light objects such as feathers and leaves have exactly the same acceleration when they fall in the absence of air resistance. Any two objects, lead weights or popcorn, when dropped in a vacuum simultaneously from the same height, will hit the ground at the same time, traveling at the same speed (Figure 3.4). Hopefully you made certain no one was at the base of the cliff before tossing off the boulder! You know from experience, of course, that a falling boulder lands with more force than a small rock does and could cause serious injury. The above discussion just quantifies this common sense. 3–4 CIRCuLAR MOTION ANd ThE MOON’S ORBIT The Italian scientist Galileo Galilei, so the story goes, proved that all objects fell at the same rate by dropping two unequal masses off the leaning Tower of Pisa. He probably did not do that actual experiment, but he did know gravity’s acceleration is constant for all things and so did Newton, experimenting a century later, in a much more science- friendly society. How did Newton go from this beginning to a law describing the force of gravity A popular legend claims that Newton saw an apple fall in his orchard and thus “discovered” gravity. What Newton did was look with a questioning mind at falling objects like apples and then ask if the interaction that made these earthly things fall was the same interaction that * Actually, you are measuring the force of the scale pushing back on you, which by Newton’s Third Law is equal to the force of gravity pulling down on you. Read more on this in Chapter 5. held the Moon in its orbit. Up to that time the motion of the Moon was considered to be unrelated to the gravity on Earth. The Moon was thought to be embedded in some mysterious cosmic material that kept it firmly in place, much like a marble affixed inside a sphere of glass centered on Earth. Newton used mathematics to show that if the following two hypotheses were true, then the interaction of gravity alone was responsible for pulling apples to Earth and holding the Moon in its orbit: 1. The force of Earth’s gravity on the Moon is essentially perpendicular to the Moon’s velocity. 2. The force of gravity diminishes with distance as 1/d 2 where “d” is the distance between the centers of any two gravitating bodies: Earth and an apple on the one hand, or Earth and the Moon on the other. Our Moon circles Earth every 27.3 days. Its speed is roughly uniform, but the direction of travel is constantly curving in accordance with a centripetal acceleration pointed at Earth. In Chapter 2 we learned that when an object changes its direction of travel there must be a net sideways force acting on it. It doesn’t matter if it is a ball twirled on the end of a string, a car turning a circular corner, or the Moon in its orbit. All of these experience a centripetal acceleration caused by a force directed toward the center of the circular path. Newton pointed out that if a cannon on the edge of a cliff were fired horizontally, the cannon ball would accelerate downward in an arc, like the green ball in Figure 3.3. If the projectile were fired faster with a greater initial horizontal velocity, it would travel farther horizontally before hitting the ground. In a wonderful intuitive leap, Newton pointed out that if the cannonball were fired fast enough from a tall- enough mountain, it would never hit the ground, because Earth’s round surface would curve away at the same rate the cannonball fell. The cannonball would be forever falling but its forward velocity would never let it land; it would be in orbit! The Moon is doing just that. Its forward speed is such that it always remains the same distance from Earth (Figure 3.5). Newton realized that to calculate the
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