Let s Look Again at the Sled in Fig 6 7

Learning Objectives

By the end of this section, you volition exist able to:

• Define net strength, external forcefulness, and organisation.
• Sympathize Newton's second law of move.
• Apply Newton's second police force to make up one's mind the weight of an object.

Newton'southward second law of motion is closely related to Newton's beginning law of move. It mathematically states the cause and effect human relationship between strength and changes in motion. Newton's second law of movement is more quantitative and is used extensively to summate what happens in situations involving a strength. Earlier we can write down Newton's second law as a simple equation giving the exact relationship of force, mass, and dispatch, we demand to acuminate some ideas that have already been mentioned.

Get-go, what practice we mean by a change in move? The answer is that a change in motion is equivalent to a modify in velocity. A change in velocity means, by definition, that in that location is an acceleration. Newton'southward first police force says that a internet external force causes a change in motion; thus, nosotros see that a net external force causes dispatch.

Another question immediately arises. What do we mean past an external force? An intuitive notion of external is right—an external strength acts from outside the system of interest. For instance, in Figure i(a) the organisation of interest is the carriage plus the kid in information technology. The 2 forces exerted by the other children are external forces. An internal strength acts between elements of the system. Once more looking at Figure 1(a), the strength the child in the wagon exerts to hang onto the wagon is an internal strength betwixt elements of the system of interest. Only external forces affect the motion of a system, according to Newton'southward first law. (The internal forces actually cancel, as nosotros shall see in the next section.) Y'all must ascertain the boundaries of the system earlier you can determine which forces are external. Sometimes the system is obvious, whereas other times identifying the boundaries of a organisation is more than subtle. The concept of a arrangement is fundamental to many areas of physics, as is the correct application of Newton's laws. This concept will exist revisited many times on our journey through physics.

Figure 1. Different forces exerted on the same mass produce different accelerations. (a) Two children push a wagon with a child in it. Arrows representing all external forces are shown. The system of interest is the wagon and its rider. The weight westward of the arrangement and the back up of the footing North are besides shown for completeness and are assumed to cancel. The vector f represents the friction acting on the carriage, and it acts to the left, opposing the move of the railroad vehicle. (b) All of the external forces interim on the arrangement add together to produce a net force, F_cyberspace . The free-body diagram shows all of the forces interim on the system of interest. The dot represents the center of mass of the system. Each strength vector extends from this dot. Considering there are two forces acting to the right, we draw the vectors collinearly. (c) A larger net external strength produces a larger acceleration (a′ > a) when an adult pushes the child.

Now, information technology seems reasonable that dispatch should exist direct proportional to and in the aforementioned direction as the cyberspace (full) external force acting on a system. This assumption has been verified experimentally and is illustrated in Figure ane. In part (a), a smaller force causes a smaller acceleration than the larger force illustrated in part (c). For completeness, the vertical forces are besides shown; they are assumed to abolish since there is no acceleration in the vertical direction. The vertical forces are the weight due west and the back up of the ground N, and the horizontal force f represents the strength of friction. These volition be discussed in more detail in later sections. For now, nosotros volition define friction equally a force that opposes the motion past each other of objects that are touching. Figure 1(b) shows how vectors representing the external forces add together to produce a cyberspace strength,

F_net .

To obtain an equation for Newton's second law, we first write the relationship of acceleration and cyberspace external forcefulness as the proportionality

[latex]\text{a}\propto{\text{F}_{cyberspace}}\\[/latex]

where the symbol ∝ means "proportional to," and F_net is the internet external force. (The net external force is the vector sum of all external forces and can be determined graphically, using the head-to-tail method, or analytically, using components. The techniques are the same every bit for the addition of other vectors, and are covered in Two-Dimensional Kinematics.) This proportionality states what we take said in words—acceleration is direct proportional to the internet external force. Once the system of interest is called, it is important to identify the external forces and ignore the internal ones. It is a tremendous simplification not to have to consider the numerous internal forces interim between objects within the system, such equally muscular forces within the kid's body, let alone the myriad of forces between atoms in the objects, but by doing so, we can easily solve some very complex bug with only minimal mistake due to our simplification.

Now, information technology too seems reasonable that dispatch should be inversely proportional to the mass of the system. In other words, the larger the mass (the inertia), the smaller the dispatch produced past a given strength. And indeed, as illustrated in Figure ii, the same net external forcefulness applied to a car produces a much smaller acceleration than when applied to a basketball. The proportionality is written as

[latex]\text{a}\propto{\frac{1}{yard}}\\[/latex]

where m is the mass of the system. Experiments take shown that acceleration is exactly inversely proportional to mass, simply as information technology is exactly linearly proportional to the net external force.

Figure ii. The same force exerted on systems of different masses produces dissimilar accelerations. (a) A basketball player pushes on a basketball game to brand a pass. (The consequence of gravity on the ball is ignored.) (b) The same player exerts an identical forcefulness on a stalled SUV and produces a far smaller acceleration (fifty-fifty if friction is negligible). (c) The free-body diagrams are identical, permitting direct comparison of the 2 situations. A series of patterns for the free-body diagram will emerge every bit yous do more problems.

It has been constitute that the acceleration of an object depends simply on the net external forcefulness and the mass of the object. Combining the two proportionalities just given yields Newton's second law of motion.

Newton's Second Law of Motility

The acceleration of a organisation is direct proportional to and in the same direction every bit the net external force acting on the system, and inversely proportional to its mass. In equation grade, Newton'due south second law of movement is

[latex]{\text{a}}=\frac{{{\text{F}}}_{\text{internet}}}{1000}\\[/latex].

This is often written in the more than familiar grade

F _cyberspace=thousand a.

When only the magnitude of forcefulness and acceleration are considered, this equation is but

F_net = ma.

Although these last two equations are really the same, the first gives more insight into what Newton's second law means. The constabulary is a cause and outcome human relationship amid three quantities that is not simply based on their definitions. The validity of the 2nd constabulary is completely based on experimental verification.

Units of Force

F _cyberspace=k a is used to define the units of force in terms of the three basic units for mass, length, and fourth dimension. The SI unit of measurement of force is called the newton (abbreviated Northward) and is the force needed to advance a 1-kg organization at the charge per unit of 1 m/south². That is, sinceF _net=m a,

i N = 1 kg ⋅ m/southward^two.

While most the entire world uses the newton for the unit of force, in the United States the nearly familiar unit of strength is the pound (lb), where one N = 0.225 lb.

Weight and the Gravitational Strength

When an object is dropped, information technology accelerates toward the eye of Earth. Newton's 2d constabulary states that a net strength on an object is responsible for its acceleration. If air resistance is negligible, the cyberspace force on a falling object is the gravitational force, normally called its weight w. Weight can be denoted as a vector w because it has a management; down is, by definition, the direction of gravity, and hence weight is a downwardly force. The magnitude of weight is denoted as due west . Galileo was instrumental in showing that, in the absence of air resistance, all objects autumn with the same acceleration g. Using Galileo's effect and Newton'southward second police force, nosotros can derive an equation for weight.

Consider an object with mass m falling downwards toward Earth. It experiences only the downward force of gravity, which has magnitude west. Newton'south second law states that the magnitude of the internet external force on an object isF_internet = ma. Since the object experiences only the downward force of gravity, F _net=west. We know that the acceleration of an object due to gravity is m, or a=thou. Substituting these into Newton'south 2nd police force gives

Weight

This is the equation for weight—the gravitational force on a mass m:

w = mg

Since g = ix.80 m/s² on Globe, the weight of a 1.0 kg object on Earth is 9.8 North, as we meet:

w = mg = (i.0 kg)(nine.80m/s²)=ix.eight North.

Call back that thou tin can take a positive or negative value, depending on the positive direction in the coordinate organization. Be certain to take this into consideration when solving problems with weight.

When the net external force on an object is its weight, we say that information technology is in complimentary-autumn. That is, the simply force acting on the object is the strength of gravity. In the real world, when objects fall downwards toward Earth, they are never truly in gratis-autumn considering there is e'er some upwardly force from the air acting on the object.

The dispatch due to gravity chiliad varies slightly over the surface of Earth, so that the weight of an object depends on location and is not an intrinsic property of the object. Weight varies dramatically if one leaves Earth's surface. On the Moon, for example, the acceleration due to gravity is but i.67 m/s². A 1.0-kg mass thus has a weight of ix.8 Due north on World and only about 1.7 N on the Moon.

The broadest definition of weight in this sense is that the weight of an object is the gravitational force on it from the nearest large trunk, such as World, the Moon, the Sun, and so on. This is the almost common and useful definition of weight in physics. Information technology differs dramatically, however, from the definition of weight used by NASA and the popular media in relation to space travel and exploration. When they speak of "weightlessness" and "microgravity," they are actually referring to the phenomenon nosotros call "complimentary-autumn" in physics. We shall use the above definition of weight, and we volition make careful distinctions between costless-fall and actual weightlessness.

Information technology is important to exist aware that weight and mass are very different physical quantities, although they are closely related. Mass is the quantity of matter (how much "stuff") and does not vary in classical physics, whereas weight is the gravitational force and does vary depending on gravity. It is tempting to equate the two, since most of our examples take place on World, where the weight of an object just varies a lilliputian with the location of the object. Furthermore, the terms mass and weight are used interchangeably in everyday language; for instance, our medical records oft prove our "weight" in kilograms, but never in the right units of newtons.

Mutual Misconceptions: Mass vs. Weight

Mass and weight are frequently used interchangeably in everyday language. Withal, in scientific discipline, these terms are distinctly different from one another. Mass is a measure of how much matter is in an object. The typical measure of mass is the kilogram (or the "slug" in English units). Weight, on the other hand, is a measure of the force of gravity acting on an object. Weight is equal to the mass of an object (m) multiplied by the acceleration due to gravity (g). Like whatsoever other strength, weight is measured in terms of newtons (or pounds in English units).

Assuming the mass of an object is kept intact, it volition remain the aforementioned, regardless of its location. However, because weight depends on the acceleration due to gravity, the weight of an object can change when the object enters into a region with stronger or weaker gravity. For example, the acceleration due to gravity on the Moon is 1.67m/s² (which is much less than the acceleration due to gravity on Globe, 9.80m/south²). If you measured your weight on Earth and then measured your weight on the Moon, you would discover that y'all "weigh" much less, fifty-fifty though yous do not await any skinnier. This is because the force of gravity is weaker on the Moon. In fact, when people say that they are "losing weight," they really mean that they are losing "mass" (which in plough causes them to weigh less).

Take-Dwelling house Experiment: Mass and Weight

What do bath scales measure? When you lot stand on a bathroom scale, what happens to the scale? It depresses slightly. The scale contains springs that shrink in proportion to your weight—similar to condom bands expanding when pulled. The springs provide a measure of your weight (for an object which is not accelerating). This is a force in newtons (or pounds). In nigh countries, the measurement is divided by 9.eighty to give a reading in mass units of kilograms. The scale measures weight merely is calibrated to provide data virtually mass. While standing on a bathroom scale, push downwardly on a table side by side to you lot. What happens to the reading? Why? Would your scale measure the same "mass" on Earth as on the Moon?

Instance ane. What Acceleration Can a Person Produce when Pushing a Lawn Mower?

Suppose that the net external force (push minus friction) exerted on a lawn mower is 51 N (nearly 11 lb) parallel to the footing. The mass of the mower is 24 kg. What is its acceleration?

Figure 3. The internet strength on a lawn mower is 51 N to the right. At what rate does the lawn mower advance to the right?

Strategy

Since F_net and m are given, the dispatch tin be calculated directly from Newton'southward 2d law as stated inF _net=1000 a.

Solution

The magnitude of the acceleration a is [latex]{\text{a}}=\frac{{{\text{F}}}_{\text{net}}}{m}\\[/latex]. Entering known values gives

[latex]a = \frac{51\text{ N}}{24\text{ kg}}\\[/latex]

Substituting the units kg ⋅ k/s² for N yields

[latex]a=\frac{\text{51 kg}\cdot {\text{m/s}}^{two}}{\text{24 kg}}=2.1 \text{ yard/southward}^{2}\\[/latex].

Give-and-take

The direction of the acceleration is the aforementioned direction as that of the net force, which is parallel to the ground. There is no information given in this example about the private external forces interim on the system, but we can say something about their relative magnitudes. For case, the force exerted by the person pushing the mower must be greater than the friction opposing the motion (since we know the mower moves forwards), and the vertical forces must cancel if there is to exist no acceleration in the vertical direction (the mower is moving just horizontally). The acceleration found is small-scale plenty to be reasonable for a person pushing a mower. Such an try would not last too long because the person'due south acme speed would soon exist reached.

Case 2. What Rocket Thrust Accelerates This Sled?

Prior to manned space flights, rocket sleds were used to test shipping, missile equipment, and physiological effects on human subjects at high speeds. They consisted of a platform that was mounted on i or two rails and propelled past several rockets. Calculate the magnitude of force exerted by each rocket, called its thrust T, for the iv-rocket propulsion organization shown in Effigy four. The sled's initial dispatch is 49m/s2, the mass of the system is 2100 kg, and the force of friction opposing the motion is known to be 650 Due north.

Figure iv. A sled experiences a rocket thrust that accelerates it to the right. Each rocket creates an identical thrust T. As in other situations where there is only horizontal dispatch, the vertical forces cancel. The ground exerts an upward force North on the system that is equal in magnitude and contrary in management to its weight, w. The system here is the sled, its rockets, and rider, so none of the forces betwixt these objects are considered. The arrow representing friction (f) is drawn larger than calibration.

Strategy

Although there are forces acting vertically and horizontally, we assume the vertical forces cancel since in that location is no vertical dispatch. This leaves us with just horizontal forces and a simpler 1-dimensional problem. Directions are indicated with plus or minus signs, with correct taken every bit the positive direction. See the free-body diagram in the figure.

Solution

Since acceleration, mass, and the force of friction are given, nosotros start with Newton'southward second police force and look for ways to detect the thrust of the engines. Since we accept defined the management of the strength and acceleration as interim "to the right," we need to consider but the magnitudes of these quantities in the calculations. Hence we begin with

F_internet = ma,

where F_net is the net forcefulness along the horizontal direction. We can see from Effigy 4 that the engine thrusts add, while friction opposes the thrust. In equation form, the cyberspace external forcefulness is

Substituting this into Newton's second police gives

Using a little algebra, we solve for the full thrust ivT:

Substituting known values yields

[latex]4T=\text{ma}+f=\left(\text{2100 kg}\right)\left({\text{49 1000/s}}^{2}\right)+\text{650 North}\\[/latex]

[latex]4T=1.0\times {10}^{5}\text{N}\\[/latex],

and the private thrusts are

[latex]T=\frac{1.0\times {\text{x}}^{5}\text{N}}{4}=2.half-dozen\times {10}^{iv}\text{N}\\[/latex].

Discussion

The numbers are quite large, then the issue might surprise you. Experiments such every bit this were performed in the early 1960s to test the limits of human being endurance and the setup designed to protect human subjects in jet fighter emergency ejections. Speeds of g km/h were obtained, with accelerations of 45 g'southward. (Recall that g, the dispatch due to gravity, is 9.eighty m/southward². When we say that an acceleration is 45 k's, it is 45×9.eighty m/sⁱⁱ, which is approximately 440 1000/south².) While living subjects are not used any more, land speeds of 10,000 km/h take been obtained with rocket sleds. In this example, as in the preceding 1, the arrangement of involvement is obvious. Nosotros will see in later examples that choosing the system of interest is crucial—and the pick is not e'er obvious.

Newton's second law of motion is more than than a definition; it is a relationship among dispatch, force, and mass. It can help us make predictions. Each of those physical quantities tin can be defined independently, so the second constabulary tells the states something basic and universal nigh nature. The next department introduces the third and concluding law of move.

Section Summary

• Dispatch, a, is defined as a change in velocity, meaning a modify in its magnitude or direction, or both.
• An external force is one interim on a system from outside the system, as opposed to internal forces, which act between components within the system.
• Newton's second law of movement states that the acceleration of a system is directly proportional to and in the same management as the net external force acting on the system, and inversely proportional to its mass.
• In equation form, Newton's 2nd law of motion is [latex]{\text{a}}=\frac{{{\text{F}}}_{\text{net}}}{m}\\[/latex].
• This is frequently written in the more familiar form:F _net=m a.
• The weight west of an object is defined as the force of gravity acting on an object of mass k. The object experiences an acceleration due to gravity g:

  w = grand g.

• If the only strength acting on an object is due to gravity, the object is in free fall.
• Friction is a force that opposes the move past each other of objects that are touching.

Conceptual Questions

1. Which statement is right? (a) Net force causes motion. (b) Net forcefulness causes change in motion. Explain your answer and give an case.

2. Why can we fail forces such as those holding a body together when we apply Newton's second law of motion?

3. Explain how the choice of the "system of involvement" affects which forces must be considered when applying Newton's second police of motility.

4. Depict a situation in which the net external force on a organization is not zip, yet its speed remains constant.

v. A system can have a nonzero velocity while the net external force on it is nada. Draw such a state of affairs.

6. A rock is thrown straight up. What is the net external forcefulness interim on the rock when information technology is at the tiptop of its trajectory?

7. (a) Give an example of different cyberspace external forces acting on the aforementioned system to produce different accelerations. (b) Give an example of the same internet external force acting on systems of different masses, producing different accelerations. (c) What police accurately describes both effects? State it in words and as an equation.

8. If the acceleration of a organization is zero, are no external forces interim on it? What nigh internal forces? Explicate your answers.

9. If a constant, nonzero force is applied to an object, what can y'all say about the velocity and acceleration of the object?

10. The gravitational force on the basketball in Figure 2 is ignored. When gravity is taken into account, what is the management of the internet external forcefulness on the basketball—higher up horizontal, beneath horizontal, or notwithstanding horizontal?

Problems & Exercises

You may assume data taken from illustrations is accurate to three digits.

ane. A 63.0-kg sprinter starts a race with an acceleration of iv.xx m/southward². What is the net external force on him?

2. If the sprinter from the previous problem accelerates at that rate for 20 m, and then maintains that velocity for the residuum of the 100-one thousand dash, what will be his fourth dimension for the race?

3. A cleaner pushes a 4.50-kg laundry cart in such a way that the net external force on it is 60.0 North. Calculate the magnitude of its dispatch.

4. Since astronauts in orbit are apparently weightless, a clever method of measuring their masses is needed to monitor their mass gains or losses to adjust diets. 1 way to exercise this is to exert a known force on an astronaut and mensurate the acceleration produced. Suppose a net external force of fifty.0 N is exerted and the astronaut's acceleration is measured to be 0.893 thousand/s². (a) Summate her mass. (b) By exerting a force on the astronaut, the vehicle in which they orbit experiences an equal and contrary force. Discuss how this would affect the measurement of the astronaut'south acceleration. Propose a method in which recoil of the vehicle is avoided.

five. In Figure 3, the internet external strength on the 24-kg mower is stated to be 51 N. If the force of friction opposing the motion is 24 Northward, what force F (in newtons) is the person exerting on the mower? Suppose the mower is moving at 1.5 yard/s when the force F is removed. How far will the mower go before stopping?

6. The same rocket sled drawn in Figure 5 is decelerated at a rate of 196 m/s². What forcefulness is necessary to produce this deceleration? Presume that the rockets are off. The mass of the system is 2100 kg.

Figure 5.

seven. (a) If the rocket sled shown in Effigy half-dozen starts with only 1 rocket called-for, what is the magnitude of its acceleration? Presume that the mass of the system is 2100 kg, the thrust T is 2.four × 10⁴ N, and the forcefulness of friction opposing the move is known to be 650 N. (b) Why is the acceleration not ane-fourth of what it is with all rockets burning?

Figure six.

8. What is the deceleration of the rocket sled if it comes to residuum in i.1 s from a speed of m km/h? (Such deceleration acquired i test bailiwick to black out and have temporary blindness.)

9. Suppose 2 children push horizontally, just in exactly opposite directions, on a third child in a railroad vehicle. The offset child exerts a force of 75.0 Due north, the 2nd a force of ninety.0 Due north, friction is 12.0 North, and the mass of the third kid plus wagon is 23.0 kg. (a) What is the system of involvement if the acceleration of the kid in the wagon is to exist calculated? (b) Draw a gratuitous-body diagram, including all forces acting on the system. (c) Calculate the dispatch. (d) What would the dispatch be if friction were 15.0 North?

10. A powerful motorbike tin produce an dispatch of [latex]3.50{\text{m/s}}^{2}\\[/latex] while traveling at xc.0 km/h. At that speed the forces resisting motion, including friction and air resistance, total 400 North. (Air resistance is coordinating to air friction. Information technology always opposes the motion of an object.) What is the magnitude of the force the motorcycle exerts backward on the ground to produce its acceleration if the mass of the motorbike with rider is 245 kg?

eleven. The rocket sled shown in Figure 8 accelerates at a rate of 49.0 chiliad/s². Its passenger has a mass of 75.0 kg. (a) Summate the horizontal component of the force the seat exerts against his body. Compare this with his weight by using a ratio. (b) Calculate the management and magnitude of the total force the seat exerts against his body.

Figure eight.

12. Echo the previous problem for the situation in which the rocket sled decelerates at a rate of 201 m/sⁱⁱ. In this trouble, the forces are exerted past the seat and restraining belts.

13. The weight of an astronaut plus his space suit on the Moon is just 250 N. How much exercise they weigh on Earth? What is the mass on the Moon? On Earth?

14. Suppose the mass of a fully loaded module in which astronauts take off from the Moon is 10,000 kg. The thrust of its engines is 30,000 Northward. (a) Calculate its the magnitude of acceleration in a vertical takeoff from the Moon. (b) Could it lift off from World? If not, why not? If it could, calculate the magnitude of its acceleration.

Glossary

acceleration:

the rate at which an object's velocity changes over a period of time

gratis-fall:

a state of affairs in which the just strength acting on an object is the force due to gravity

friction:

a forcefulness by each other of objects that are touching; examples include rough surfaces and air resistance

cyberspace external force:

the vector sum of all external forces acting on an object or arrangement; causes a mass to accelerate

Newton's second police force of motion:

the internet external force F _net on an object with mass 1000 is proportional to and in the aforementioned direction as the dispatch of the object, a, and inversely proportional to the mass; defined mathematically as [latex]\mathbf{\text{a}}=\frac{{\mathbf{\text{F}}}_{\text{net}}}{m}\\[/latex]

organization

defined by the boundaries of an object or collection of objects being observed; all forces originating from exterior of the organisation are considered external forces

weight

the force w  due to gravity acting on an object of mass [latex]m[/latex] ; defined mathematically as: west = mg , where g  is the magnitude and direction of the dispatch due to gravity

Selected Solutions to Problems & Exercises

1. 265 N

3. thirteen.3 grand/s²

vii. (a) 12 1000/s²(b) The dispatch is not i-quaternary of what information technology was with all rockets burning considering the frictional force is nonetheless as large as it was with all rockets burning.

9. (a) The system is the child in the carriage plus the wagon.

Figure 9

(b)

(c) a = 0 . 130 1000/s^two in the direction of the second child's push.

(d) a = 0.00 m/s²

11. (a) 3.68 × 10³ N. This force is 5.00 times greater than his weight. (b) 3750 Due north; 11.3º above horizontal

13. 1.5 × ten³ Due north , 150 kg , 150 kg

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