Categorize: When the length changes by We will write each of the sentences in the problem as a mathematical equation. This model of growth is reasonable because the lamb gets thicker as it gets longer, growing in threedimensional space.
The The writer of this problem was thinking of the experience of a young girl in the Oneida community of New York State.
Before the dawn of a spring day she helped with the birth of lambs. She was allowed to choose one lamb as her pet, and braided for it a necklace of straw to distinguish it from the others. Then she went into the house, where her mother had made cocoa with breakfast. Many years later she told this story of widening, overlapping circles of trust and faithfulness, to a group of people working to visualize peace.
Over many more years the story is spreading farther. The diameter of our disk-shaped galaxy, the Milky Way, is about 1. The distance to the Andromeda galaxy Fig. If a scale model represents the Milky Way and Andromeda galaxies as dinner plates 25 cm in diameter, determine the distance between the centers of the two plates. Solution Conceptualize: Individual stars are fantastically small compared to the distance between them, but galaxies in a cluster are pretty close compared to their sizes.
A high fountain of water is located at the center of a circular pool as shown in Figure P1. A student walks around the pool and measures its circumference to be How high is the fountain?
Solution Conceptualize: Geometry was invented to make indirect distance measurements, such as this one. It is a fuel savings of ten billion gallons each year. Express this equation in units of cubic feet and seconds. Assume a month is Displacement can be positive, negative or zero. Average speed has no direction and carries no algebraic sign. The magnitude of the average velocity is not the average speed; although in certain cases they may be numerically equal.
This limit is called the derivative of x with respect to t. The instantaneous velocity at any time is the slope of the positiontime graph at that time. Equation 2. This limit is the derivative of the velocity along the x direction with respect to time. A negative acceleration does not necessarily imply a decreasing speed.
Acceleration and velocity are not always in the same direction. The acceleration can also be expressed as the second derivative of the position with respect to time. This is shown in Equation 2. Remember, the relationships stated in Equations 2. Equations 2. A freely falling object experiences an acceleration that is directed downward regardless of the direction or magnitude of its actual motion. Sections 2. From this graph, you should be able to determine the value of the average velocity between two points t1 and t2 and the instantaneous velocity at a given point.
Average velocity is the slope of the chord between the two points and the instantaneous velocity at a given time is the slope of the tangent to the graph at that time.
Section 2. A student at the top of a building of height h throws one ball upward with a speed of vi and then throws a second ball downward with the same initial speed vi. Answer c They are the same, if the balls are in free fall with no air resistance.
This velocity is the same as the initial velocity of the second ball, so after they fall through equal heights their impact speeds will also be the same. If the average velocity of an object is zero in some time interval, what can you say about the displacement of the object for that interval? Answer The displacement is zero, since the displacement is proportional to average velocity.
Two cars are moving in the same direction in parallel lanes along a highway. At some instant, the velocity of car A exceeds the velocity of car B. Does that mean that the acceleration of A is greater than that of B?
Answer No. The position versus time for a certain particle moving along the x axis is shown in Figure P2. Solution Conceptualize: We must think about how x is changing with t in the graph. The slope of the graph line itself is the instantaneous velocity, found, for example, in the following problem 5 part b.
A Solution Conceptualize: velocity. This problem lets you think about the distinction between speed and Categorize: Speed is positive whenever motion occurs, so the average speed must be positive. Velocity we take as positive for motion to the right and negative for motion to the left, so its average value can be positive, negative, or zero.
A position—time graph for a particle moving along the x axis is shown in Figure P 2. Solution Conceptualize: We will have to distinguish between average and instantaneous velocities.
Then for instantaneous velocities we think of slopes of tangent lines, which means the slope of the graph itself at a point. Figure P2. Instantaneous velocity equals the slope of the tangent line, x f — xi 0. This occurs for the point on the graph where x has its minimum value.
Previous problems have displayed position as a function of time with a graph. This problem displays x t with an equation. An object moving with uniform acceleration has a velocity of If its x coordinate 2. Solution Conceptualize: Study the graph. Move your hand to imitate the motion, f irst rapidly to the right, then slowing down, stopping, turning around, and speeding up to move to the left faster than before.
So we can use one of the set of equations describing constantacceleration motion. Make a list of all of the six symbols in the equations: xi , xf , vxi , vxf , ax, and t. Identify ax as the unknown. Choose an equation involving only one unknown and the knowns. That is, choose an equation not involving vx f. In Example 2. Solution Conceptualize: We think of the plane moving with maximum-size backward acceleration throughout the landing… Categorize: …so the acceleration is constant, the stopping time a minimum, and the stopping distance as short as it can be.
Colonel John P. Stapp, USAF, participated in studying whether a jet pilot could survive emergency ejection. He and the sled were safely brought to rest in 1. Determine a the negative acceleration he experienced and b the distance he traveled during this negative acceleration. We have already chosen the straight track as the x axis and the direction of travel as positive.
We expect the stopping distance to be on the order of m. Categorize: We assume the acceleration is constant. Then we have a straightforward problem about a particle under constant acceleration. Our answers for ax and for the distance agree with our order-of-magnitude estimates. For many years Colonel Stapp held the world land speed record. A baseball is hit so that it travels straight upward after being struck by the bat. A fan observes that it takes 3.
We say that the ball is in free fall in its upward motion as well as in its subsequent downward motion and at the moment when its instantaneous velocity is zero at the top. A student throws a set of keys vertically upward to her sorority sister, who is in a window 4. The second student catches the keys 1.
The answer to part b will tell us which it is. Categorize: We model the keys as a particle under the constant free-fall acceleration. Then the hands do not give the keys some unknown acceleration during the motion we consider. A daring ranch hand sitting on a tree limb wishes to drop vertically onto a horse galloping under the tree. The constant speed of the horse is Solution Conceptualize: The man will be a particle in free fall starting from rest. The horse moves with constant velocity.
Solution Conceptualize: Steadily changing force acting on an object can make it move for a while with steadily changing acceleration, which means with constant jerk. Categorize: This is a derivation problem. An inquisitive physics student and mountain climber climbs a He throws two stones vertically downward, 1. Solution Conceptualize: The different nonzero original speeds of the two stones do not affect their accelerations, which have the same value g downward.
This is a pair of free-fall problem s. Categorize: Equations chosen from the standard constant-acceleration set describe each stone separately, but look out for having to solve a quadratic equation.
The equation has two solutions that have to be consid2a ered. In this problem, the other root of the quadratic equation is —3. The stone could have been burped up by a moat monster in the pool, who gave it an upward velocity of In this different story the stone has the same motion from cliff to pool. Kathy tests her new sports car by racing with Stan, an experienced racer. Both start from rest, but Kathy leaves the starting line 1.
Stan moves with a constant acceleration of 3. Find a the time at which Kathy overtakes Stan, b the distance she travels before she catches him, and c the speeds of both cars at the instant Kathy overtakes Stan.
Solution Conceptualize: The two racers travel equal distances between the starting line and the overtake point, but their travel times and their speeds at the overtake point are different. Categorize: We have constant-acceleration equations to apply to the two cars separately. Two objects, A and B, are connected by hinges to a rigid rod that has a length L. The objects slide along perpendicular guide rails, as shown in Figure P2. Assume object A slides to the left with a constant speed v.
Solution Figure P2. B starts moving very rapidly and then slows down, but not with constant acceleration. Differentiation is an operation you can always do to both sides of an equation. It is perhaps a surprise that the value of L does not affect the answer.
Categorize: We must take the motion of each athlete apart into two sections, one with constant nonzero acceleration and one with constant velocity, in order to apply our standard equations.
Analyze: a Laura moves with constant positive acceleration aL for 2. The distance between them is momentarily staying constant at its maximum value when they have equal speeds.
This instant tm is when Healan, still accelerating, has speed At the very end of part c we compared their positions. Do you think that If one set of coordinates is known, values for the other set can be calculated. The associative law of addition states that when three or more vectors are added the sum is independent of the way in which the individual vectors are grouped.
When two or more vectors are to be added, all of them must represent the same physical quantity—that is, have the same units. The negative of vector A is I defined as the vector that when added to A gives zero for the vector sum. The rectangular components of a vector are the projections of the vector onto the respective coordinate axes. Unit vectors are dimensionless and I have a magnitude of exactly 1. A vector A lying in the xy plane, having rectangular components Ax and Ay, can be expressed in unit vector notation.
Unit vectors specify the directions of the vector components. These sums are the components of the resultant vector. Section 3. Determine the magnitude and direction of a vector from its rectangular components. Use unit vectors to express any vector in unit vector notation. Express the resultant vector in unit vector notation. Vector A lies in the xy plane. Both of its components will be negative if it points from the origin into which quadrant? Thus the answer is c.
Is it possible to add a vector quantity to a scalar quantity? Answer Vectors and scalars are distinctly different and cannot be added to each other. It makes no sense to add a football uniform number to a number of apples, and similarly it makes no sense to add a number of apples to a wind velocity.
What are the Cartesian coordinates of this point? Categorize: Trigonometric functions will tell us the coordinates directly. The y coordinate is larger in magnitude because the point is closer to the y axis than to the x axis. The lower left-hand corner of the wall is selected as the origin of a two-dimensional Cartesian coordinate system.
But it is standard to use the tangent. Then a mistake in calculating the radial coordinate cannot also affect the angle coordinate. A surveyor measures the distance across a straight river by the following method Fig.
Then she sights across to the tree. How wide is the river? Solution Conceptualize: Make a sketch of the area as viewed from above. Assume the river banks are straight and parallel, show the location of the tree, and the original location of the surveyor. Figure P3. Now draw the meter baseline, d, and the line showing the line of sight to the tree. For our applications, you will likely never have to use such theorems as the law of cosines or the law of sines.
Why is the following situation impossible? A skater glides along a circular path. Later on, she passes through a point at which the distance she has traveled along the path from the origin is smaller than the magnitude of her displacement vector from the origin. It needs to be the view from a hovering helicopter to see the circular path as circular in shape. To start with a concrete example, we have chosen to draw motion ABC around one half of a circle of radius 5 m.
Categorize: In solving this problem we must contrast displacement with distance traveled. The distance follows the curved path of the semicircle ABC. For any nonzero displacement, less or more than across a semicircle, the distance along the path will be greater than the displacement magnitude.
Finalize: There is nothing special about a circle—it could be any curve. For any motion involving change in direction, the distance traveled is always greater than the magnitude of the displacement. The displacement vectors A and B shown in Figure P3. Report all angles counterclockwise from the positive x axis. Solution Figure P3. You string along one displacement after another, each beginning where the last ended. Visualize the negative of a vector I as an equal-magnitude vector I in the opposite direction.
Visualize —2B as having twice the length of B and being in the opposite direction. Categorize: We must draw with protractor and ruler to construct the additions. Then we must measure with ruler and protractor to read the answers. IWe have of starting point and of scale. A roller-coaster car moves ft horizontally and then rises ft at an angle of It next travels ft at an angle of What is its displacement from its starting point?
Use graphical techniques. Solution Conceptualize: We will draw a side view picture. When adding vectors graphically, the directions of the vectors must be maintained as they start from different points in sequence. In problems about adding other kinds of vectors, a sketch may not seem so real, but it will still be useful as an adjunct to a calculation about vector addition, and the sketch need not be measured to scale to be useful.
Find the magnitude and direction of this vector. Solution Conceptualize: First we should visualize the vector either in our mind or with a sketch. Since the hypotenuse of the right triangle must be greater than either the x or y components that form the legs, we can estimate the magnitude of the vector to be about 50 units. Categorize: We use geometry and trigonometry to obtain a more precise result.
Look out! The vector is not in the fourth quadrant. We should always remember to check that our answers make sense, especially for problems like this where it is easy to mistakenly calculate the wrong angle by confusing coordinates or overlooking a minus sign. If compass directions were stated in this question, we could have reported the vector angle to be Obtain expressions in component form for the position vectors having the polar coordinates a Even without drawing on paper, we are thinking in pictorial terms.
There are in a sense only two vectors to calculate, since parts c , d , and e just ask about the magnitudes and directions of the answers to a and b. Use it yourself whenever you have a choice. Your dog is running around the grass in your back yard. He undergoes successive displacements 3. What is the resultant displacement? He will end up somewhat northwest of his starting point.
Categorize: We use the unit-vector addition method. It is just as easy to add three displacements as to add two. Your observation that the x component is negative and the y component is positive diagnoses the resultant as in the second quadrant. Its location relative to the starting point represents posiI tion vector A. MoveIthree fourths of the way straight back toward O. Its position vector now is C.
Categorize: We use unit-vector notation throughout. There is no adding to do here, but just multiplication of a vector by two different scalars. That gets you to the point with position vector C. Think of this as a very straightforward problem. You will frequently encounter multiplication of a vector by a scalar, as in the relationship of velocity and momentum and the relationship of total force and acceleration. Later in the course you will study two different ways of forming the product of one vector with another vector.
Vector A has a negative x component 3. Solution I the answer Conceptualize: The component description of A is just restated to constitute I to part a. Brace yourself for continued successes.
Find a the resultant in unit-vector notation and b the magnitude and direction of the resultant displacement. Solution Conceptualize: The given diagram shows the vectors individually, but not their addition. The second diagram represents a map view of the I motion of the ball. From it, the magnitude of the resultant R should be about 60 units. Its x component appears to be about 50 units and its y component about 30 units. We model each of the three motions as straight. It works for adding two, three, or any number of vectors.
Perhaps the greatest usefulness of the diagram is checking I positive and negative signs for each component of each vector. The y component of C is negative, for example, because the vector is downward. For each vector it is good to check against the diagram whether the y component is less than, equal to, or greater than the x component. If your calculator is set to radians instead of degrees, the diagram can rescue you. The resultant displacement has a magnitude of cm and is directed at an angle of Find the magnitude and direction of the second displacement.
Categorize: We will use the component method for a precise answer. We already know the total displacement, so the algebra of solving a vector equation will guide us to do a subtraction. A person going for a walk follows the path shown in Figure P3. The total trip consists of four straight-line paths. The average velocity depends on the displacement vector and not on the length of path traveled. The average speed of a particle during any time interval is the ratio of the total distance of travel length of path to the total time.
The magnitude of the instantaneous velocity is called the speed. A particle experiences acceleration when the velocity vector undergoes a change in magnitude, direction, or both. The position vector for a particle in twodimensional motion can be stated using unit vectors.
The position as a function of time is given by Equation 4. Motion in two dimensions with constant acceleration is described by equations for velocity and position which are vector versions of the one-dimensional kinematic equations. The path of a projectile is a parabola. The centripetal acceleration vector is always directed toward the center of the circular path, and, therefore, is constantly changing in direction.
The radial component, ar, is directed toward the center of curvature and arises from the change in direction of the velocity vector. The tangential component, at, is perpendicular to the radius and causes the change in speed of the particle. Galilean transformation equations relate observations made by an observer in one frame of reference A to observations of corresponding quantities made by an observer in a reference frame B moving I with velocity v BA with respect to the f irst.
In Equations 4. A sketch of y versus x for this situation is shown at the right. Sections 4. Section 4. In this case, the particle has a tangential component of acceleration and a radial component of acceleration. Where will the wrench hit the deck? Its vertical motion is affected by gravitation, and is independent of its horizontal motion. The wrench continues moving horizontally with its speed prior to release, and this is the same as the constant forward speed of the boat.
Thus the wrench falls down at a constant small distance from the moving mast. Galileo suggested the idea for this question and gave another argument for answer b : The reference frame of the boat is just as good as the Earth reference frame for describing the motion of the wrench.
In the boat frame the wrench starts from rest and moves in free fall. It travels straight down next to the stationary mast, and lands at the base of the mast. Its average velocity? Answer Its instantaneous velocity cannot be determined at any point from this information.
A spacecraft drifts through space at a constant velocity. Suddenly, a gas leak in the side of the spacecraft gives it a constant acceleration in a direction perpendicular to the initial velocity. The orientation of the spacecraft does not change, so the acceleration remains perpendicular to the original direction of the velocity.
What is the shape of the path followed by the spacecraft in this situation? Answer The spacecraft will follow a parabolic path, just like a projectile thrown off a cliff with a horizontal velocity. For the projectile, gravity provides an acceleration which is always perpendicular to the initial velocity, resulting in a parabolic path. For the spacecraft, the initial velocity plays the role of the horizontal velocity of the projectile. The leaking gas provides an acceleration that plays the role of gravity for the projectile.
If the orientation of the spacecraft were to change in response to the gas leak which is by far the more likely result , then the acceleration would change direction and the motion could become quite complicated. A projectile is launched at some angle to the horizontal with some initial speed vi , and air resistance is negligible. Answer a Yes. The projectile is a freely falling body, because the only force acting on it is the gravitational force exerted by the planet.
Explain whether or not the following particles have an acceleration: a a particle moving in a straight line with constant speed and b a particle moving around a curve with constant speed. I Answer a The acceleration is zero, since the magnitude and direction of v remain conI stant. A motorist drives south at For this 6. Let the positive x axis point east. Analyze: a For each segment of the motion we model the car as a particle under constant velocity.
The total distance and the average speed are scalars, with no direction. Distance must be greater than the magnitude of displacement for any motion that changes direction, and similarly average speed must be greater than average velocity.
After I the f ish swims with constant acceleration for We choose to write separate equations for the x and y components of its motion. We use our old standard equations for constant-acceleration straightline motion, with x and y subscripts to make them apply to parts of the whole motion. Sketch it! The height of the counter is 1. The mug is much more dense than the surrounding air, so its motion is free fall. It is a projectile. Categorize: We are looking for two different velocities, but we are only given two distances.
Our approach will be to separate the vertical and horizontal motions. For convenience, we will set the origin to be the point where the mug leaves the counter.
Since the problem did not ask for the time, we could have eliminated this variable by substitution. This would have been useless. The mug does not travel along this straight line. Its different accelerations in the horizontal and vertical directions require us to treat the horizontal and vertical motions separately. The original speed was an unknown, but the original vertical velocity counts as a known quantity. Its value must be zero because we assume the countertop is horizontal.
With the original velocity horizontal, is the impact velocity vertical? Why not? Because the horizontal motion does not slow down. At the impact point the mug is still moving horizontally with its countertop speed. What is the angle of projection? We guess it is around halfway between them. A placekicker must kick a football from a point Half the crowd hopes the ball will clear the crossbar, which is 3.
When kicked, the ball leaves the ground with a speed of Categorize: Model the football as a projectile, moving with constant horizontal velocity and with constant vertical acceleration. We need a plan to get the necessary information to answer the yes-or-no questions.
In part b we could equally well have evaluated the vertical velocity of the ball at 2. Notice that in part b we found the time interval for the ball to travel 36 m. You can think of the trajectory equation as more complicated, but it just carries out precisely this pair of steps symbolically, instead of requiring a numerical substitution.
The last digit in our answer 0. You might say it is entirely unknown, because the next digit is not quoted for the 3. The athlete shown in Figure P4. The maximum speed of the discus is Determine the magnitude of the maximum radial acceleration of the discus. Solution Conceptualize: The maximum radial acceleration occurs when maximum tangential speed is attained. Visualize the discus as keeping this speed constant for a while, so that its whole acceleration is its radially-inward acceleration.
Categorize: Model the discus as a particle in uniform circular motion. We evaluate its centripetal acceleration from the standard equation proved in the text.
A train slows down as it rounds a sharp horizontal turn, going from The radius of the curve is m. Compute the acceleration at the moment the train speed reaches Assume the train continues to slow down at this time at the same rate. Otherwise it might jump the tracks! Categorize: Since the train is changing both its speed and direction, the acceleration vector will be the vector sum of the tangential and radial acceleration components.
Figure P4. Solution Conceptualize: Visualize one instant in the history of a rubber stopper moving in a circle and speeding up as it does so. The speed of the particle must be changing, and then the radial acceleration must be increasing in magnitude and also swinging around in direction. The whole problem is about one instant of time. A river has a steady speed of 0. A student swims upstream a distance of 1. Since the student can swim 1.
Categorize: The total time interval in the river is the longer time spent swimming upstream against the current plus the shorter time swimming downstream with the current.
But what counts for his average speed is the time he spends with each different speed. In the situation considered here, he spends more than twice as much time with the lower speed, so his average speed is lower than it would be without a current. The student throws a ball into the air along a path that he judges to make an initial angle of How high does she see the ball rise? Solution Conceptualize: The student must throw the ball backward relative to the moving train.
Now we can calculate the maximum height that the ball rises. Here the calculation is fairly simple in each step, but you need skill and practice to string the steps together. The vector diagram of adding train-speed-relative-toground plus ball-speed-relative-to-train to get ball-speed-relative-to-ground deserves careful attention.
The ball is hit at It is possible, perhaps barely possible, that air resistance is negligible, perhaps because a wind is blowing along with the ball in the direction it is hit, or perhaps because a small osmium ball has been substituted for a regulation baseball. Therefore, there must be something wrong with the numbers quoted about the trajectory. Categorize: Given the initial velocity, we can calculate the height change of the ball as it moves m horizontally. So this is what we do, expecting the answer to be inconsistent with grazing the top of the bleachers.
We think of the ball as a particle in free fall moving with constant acceleration between the point just after it leaves the bat until it crosses above the cheap seats. Analyze: The horizontal velocity of the ball is A tennis player might hit a ball 2.
The bombardier releases one bomb. Ignore the effects of air resistance. Where is the plane when the bomb hits the ground? At what angle from the vertical was the bombsight set? Solution Conceptualize: The diagram shows a lot.
At the moment of release, the bomb has the horizontal velocity of the plane. Its motion is part of the downward half of a parabola, just as in the previously 3 solved problem 9, about a mug sliding off the end of a bar. Categorize: We model the bomb as a particle with constant acceleration, equal to the downward free-fall acceleration, from the moment after release until the moment before impact.
Analyze: a We take the origin at the point under the plane at bomb release. We represent the height of the plane as y. The plane and the bomb have the same constant horizontal velocity. An individual motion equation involves only the x component or only the y component of the motion, but the bombsight angle is computed from both of the perpendicular displacement components together. A mouse it has been carrying struggles free from its talons. The hawk continues on its path at the same speed for 2.
To accomplish the retrieval, it dives in a straight line at constant speed and recaptures the mouse 3. We already know the vertical distance y; we just need the horizontal distance during the same time interval minus the 2. We typically make simplifying assumptions to solve complex physics problems, and sometimes these assumptions are not physically possible.
After the idealized problem is understood, we can attempt to analyze the more complex, real-world problem. A car is parked on a steep incline, making an angle of The cliff is Find a the speed of the car when it reaches the edge of the cliff, b the time interval elapsed when it arrives there, c the velocity of the car when it lands in the ocean, d the total time interval the car is in motion, and e the position of the car when it lands in the ocean, relative to the base of the cliff.
Solution Conceptualize: The car has one acceleration while it is on the slope and a different acceleration when it is falling, so. Our standard equations only describe a chunk of motion during which acceleration stays constant.
We imagine the acceleration to change instantaneously at the brink of the cliff, but the velocity and the position must be the same just before point B and just after point B. Adding up the total time interval in motion in part c is not driven by an equation from back in the chapter, but just by thinking about the process.
Solution Conceptualize: The last question asks us to go beyond modeling the skier as moving in free fall, but we use that model in parts a and b.
The takeoff speed is not terribly fast, but the landing slope is so steep that we expect a distance on the order of a hundred meters, and a high landing speed. Mass is an inherent property of an object and is independent of the surroundings and the method of measurement.
The net force or resultant force is the vector sum of all the external forces acting on the object. The orientation of the coordinate system can often be chosen so that the object has a nonzero acceleration along only one direction. Calculations with Equation 5. Weight is not an inherent property of a body; it depends on the local value of g and varies with location.
Remember, the two forces in an action—reaction pair always act on two different objects—they cannot add to give a net force of zero. The equality sign holds when the two surfaces are on the verge of slipping impending motion. The friction force is parallel to the surface on which an object is in contact and is directed opposite the direction of actual or impending motion.
Model the object as a particle and draw a free-body diagram for the object; that is, a diagram showing all external forces acting on the object. For systems containing more than one object, draw separate diagrams for each object.
Do not include forces that the object exerts on its surroundings. Check to be sure that all terms in the resulting equation have units of force.
Remember that you must have as many independent equations as you have unknowns in order to obtain a complete solution. Sections 5. Identify all external forces acting on the system, model each object as a particle, and draw separate free-body diagrams showing all external I I forces acting on each object. Section 5. Recall that you must have as many independent equations as you have unknowns. If an object is in equilibrium, which of the following statements is not true?
Answer Each of the statements can be true and three statements are necessarily true, but both statements d and e are not necessarily true. Statement a must be true because the speed must be constant when the velocity is constant. Statement d need not be true: the object might be at rest or it might be moving at constant velocity.
And e need not be true: the object might be a meteoroid in intergalactic space, with no force exerted on it. Why did this happen? Answer When the bus starts moving, the mass of Claudette is accelerated by the force of the back of the seat on her body. Thus, when the bus starts moving, his feet start accelerating forward, but the rest of his body experiences almost no accelerating force only that due to his being attached to his accelerating feet! Relative to Claudette, however, he is moving toward her and falls into her lap.
Both performers won Academy Awards. What force causes the ball to bounce? A weightlifter stands on a bathroom scale. He pumps a barbell up and down. What happens to the reading on the scale as he does so? What If? What if he is strong enough to actually throw the barbell upward? How does the reading on the scale vary now? Answer If the barbell is not moving, the reading on the bathroom scale is the combined weight of the weightlifter and the barbell.
As a result, he is pushed with more force into the scale, increasing its reading. Near the top of the lift, the weightlifter reduces the upward force, so that the acceleration of the barbell is downward, causing it to come to rest.
If the barbell is held at rest for an interval at the top of the lift, the scale reading is simply the combined weight. As it begins to be brought down, the reading decreases, as the force of the weightlifter on the barbell is reduced.
The reading increases as the barbell is slowed down at the bottom. If we now consider the throwing of the barbell, the variations in scale reading will be larger, since more force must be applied to throw the barbell upward rather than just lift it. Identify action—reaction pairs in the following situations: a a man takes a step b a snowball hits a girl in the back c a baseball player catches a ball d a gust of wind strikes a window.
Answer There is no physical distinction between an action and a reaction. It is clearer to describe the pair of forces as together constituting the interaction between the two objects. Find a the resultant force acting on the object and b the magnitude of the resultant force. A toy rocket engine is securely fastened to a large puck that can glide with negligible friction over a horizontal surface, taken as the xy plane.
The 4. Eight seconds later, its velocity is 8. Solution Conceptualize: The apparatus described is good for demonstrating how a rocket works. Then the Pythagorean theorem gives the magnitude of the force. Analyze: We use the particle under constant acceleration and particle under net force models. An electron of mass 9. It travels in a straight line, and its speed increases to 7.
Assuming its acceleration is constant, a determine the magnitude of the force exerted on the electron and b compare this force with the weight of the electron, which we ignored. Only a very small force is required to accelerate an electron because of its small mass, but this force is much greater than the weight of the electron if the gravitational force can be neglected. In general, it is quite reasonable to ignore the weight of the electron in problems about electric forces. Two forces F1 and F2 act on a 5.
Solution Conceptualize: We are reviewing that forces are vectors. The acceleration will be Figure P5. Categorize: We must add the forces as vectors. Analyze: We use the particle under a net force model. We see that the acceleration is indeed larger in magnitude in part b than in part a and in a direction closer to the x axis.
A bag of cement whose weight is Fg hangs in equilibrium from three wires as shown in Figure P5. For the bag of cement to be in equilibrium, the tension T3 in the vertical wire must be equal to Fg. Figure P5. The magnitudes of T1 and T2 are unknown, but we can take components of these two tensions just as if they were known forces. Then we can eliminate T2 by substitution and solve for T1.
Analyze: We use the particle in equilibrium model. Draw a free-body diagram for the knot where the three ropes are joined. If the right-hand rope is vertical, the tension in the left-hand rope is zero. All this information is contained in the symbolic answer, and not in the numerical answer to problem I The force F2 acting on the object has a magnitude of 5.
DetermineI the magnitude and direction of the one other horizontal force F1 acting on the object. It is merely a coincidence that force F1 has zero for its northward component. I In the system shown in Figure P5. The horizontal surface is frictionless. Consider the acceleration of the sliding object as a function of Fx. When Fx is negative, the acceleration will be negative, and maybe the string will go slack. There can be especially interesting behavior for small positive values of the force, when it is counterbalancing some but not all of the weight of the two kilograms.
Categorize: We use the particle under a net force model for each of the two objects separately. Then we combine the equations. Whenever one block moves a centimeter the other block also moves a centimeter.
The two blocks have the same speed at every instant. The blocks can move together with zero acceleration. When Fx has a larger positive value, the blocks accelerate together in the positive direction.
Their acceleration is a linear function of the force as predicted. In Example 5. Now consider a Starting from rest, the elevator ascends, attaining its maximum speed of 1.
It travels with this constant speed for the next 5. The elevator then undergoes a uniform acceleration in the negative y direction for 1. The acceleration can be found from the change in speed divided by the elapsed time. Analyze: Consider the force diagram of the man shown as two arrows. Therefore, equation [1] gives the force exerted by the scale on the man as N upward, and the man exerts a downward force of N on the scale.
In part b , the force of the scale is no larger than the downward gravitational force, so why does the man keep moving upward? Johannes Kepler invented the term inertia, taken from a Latin word for laziness, to sound like an explanation for this motion in the absence of a net force.
When the total force acting on an object is zero, the object clings to the status quo in its motion. This problem could be extended to a couple of extreme cases. This relationship is described by the theorem written down in Appendix B. I Hint three: The We apply the second law. Only two objects exert forces on the block, namely the incline and the Earth. But the incline exerts both normal and friction forces, and we resolve the gravitational force into its down-incline and into-incline components.
Two blocks connected by a rope of negligible mass are being dragged by a horizontal force Fig. Determine b the acceleration of the system, and c the tension T in the rope.
The tension will be less than one-half of 68 N. Categorize: Because the cord has constant length, both blocks move the same number of centimeters in each second and so move with the same acceleration. The tension in a light string is a constant along its length, and tells how strongly the string pulls on objects at both ends. The solution contains a built-in check. If any of these positive or negative signs were different, it would reveal a mistake. An inventive child named Nick wants to reach an apple in a tree without climbing the tree.
Sitting in a chair connected to a rope that passes over a frictionless pulley Fig. Categorize: Child and chair will be two particles under net forces. Analyze: a Figure P5. His acceleration is quite low. I An object of mass M is held in place by an applied force F and a pulley system as shown in Figure P5. The pulleys are massless and frictionless.
Solution Conceptualize: A pulley system makes it easier to lift a load. We expect T1 to be less than Mg. This will be made clear by. Analyze: Figure P5. The lifting can be done at constant speed, with zero acceleration and total force zero on each object. The tension is constant throughout a light, continuous rope. The ceiling must exert a force greater than the weight of the load. What horizontal force must be applied to a large block of mass M shown in Figure P5. Assume all surfaces and the pulley are frictionless.
Volume 2. Coulomb's law -- Electric fields -- Gauss' law -- Electric potential -- Capacitance -- Current and resistance -- Circuits -- Magnetic fields -- magnetic fields due to currents -- Induction and inductance -- Electromagnetic oscillations and alternating current -- Maxwell's equations Magnetism of matter -- Electromagnetic waves -- Images -- Interference -- Diffraction -- Relativity -- Photons and matter waves -- More about matter waves -- All about atoms -- Conduction of electricity in solids -- Nuclear physics -- Energy from the mucleus -- Quarks, leptons, and the Big Bang.
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