# Newton’S laws of Motion

# Newton’S laws of Motion

Applications of the second law

Third law of motion or law of action-reaction

Earthly and heavenly motions were of great interest to English scientist Isaac Newton (1643–1727). He formulated three mathematical laws, which allowed a complete analysis of dynamics, the study of moving objects. His laws relate all aspects of motion to force and mass. Newton’s three laws of motion, although shown by Einstein in the early twentieth century to be an inadequate description of the natural world under some conditions, are so accurate under most conditions that they are still used today throughout science and engineering.

## First law of motion

Galileo Galilei’s (1564–1642) observation that without externally applied force a moving body would tend to move forever in a straight line—or, if at rest, never move at all—challenged Aristotle’s notion that the natural state of motion on Earth was one of rest. Galileo deduced that it was a property of matter to maintain its state of linear motion (or rest), a property first called “inertia” by Johannes Kepler in the early 1600s. Newton, grasping the meaning of inertia and recognizing that Aristotle should have concentrated not on what keeps a body in motion but on what changes a body’s state of motion, set forth a first law of motion codifying the concept of inertia: A body at rest remains at rest or a body in constant motion remains in constant motion along a straight line unless acted on by an external force.

## Examples of the first law

Why use seat belts? When you are riding in a car, you and the car have the same motion. When the brakes are applied, the brakes stop the car. What stops you? During a slow stop, a force exerted by the friction of your body against the seat: in a fast stop, such as during a crash, a force exerted either by a seat belt, the steering wheel, the dashboard, or the wind-shield. From a human point of view, the seat belt is obviously preferable. When the accelerator is depressed with the car in gear the motor turns the wheels and the car moves forward. What moves you forward? As the car moves forward the seat presses against you and pushes you forward.

Force is required not only to change your velocity, but your direction. Say that while you are riding in the front passenger seat of a car, the driver suddenly turns left: what about you? A force exerted on your body by the seat changes your direction of motion, which would otherwise continue in a straight line. Or, in a fast turn, the door by your elbow may also contribute force to changing your state of motion. But there is never a change in motion without application of a force. That is the first law.

## Second law of motion

Force produces a change in the state of motion; that is, an acceleration. Newton found that the greater a body’s mass, the greater the force required to produce a certain amount of acceleration. He also found that when applying equal force to two different masses, the ratio of their accelerations was inversely proportional to the ratio of their masses. Newton’s second law of motion is therefore as follows: A net force acting on a body produces an acceleration; the acceleration is inversely proportional to its mass and directly proportional to the net force and in the same direction.

This law can be put mathematically F = ma where F is the net force, m is the mass, and a the acceleration. The second law is a cause-effect relationship. The net force acting on a body is determined from all forces acting and the resultant acceleration calculated (assuming a known mass). From the acceleration, velocity and distance traveled can be determined for any time.

## Applications of the second law

(1) Objects, when released, fall to the ground due to Earth’s attraction. Newton’s universal law of gravity gave the force of attraction between two masses, m and M, as F =GmM/R^{2} where G is the gravitational constant and R is the distance between mass centers. This force, weight, produces gravitational acceleration g, thus weight = GmM/R^{2} = mg(2nd Law) giving g = GM/R^{2}. This relationship holds universally. For all objects at Earth’s surface, g =32 ft/sec/sec or 9.8 m/sec/sec downward and on Jupiter 84 ft/sec/sec. Since the dropped object’s mass does not appear, g is the same for all objects. Falling objects have their velocity changed downward at the rate of 32 ft/sec each second on earth. Falling from rest, at the end of one second the velocity is 32 ft/sec, after 2 seconds 64 ft/sec, after 3 seconds 96 ft/sec, etc.

For objects thrown upward, gravitational acceleration is still 32 ft/sec/sec downward. A ball thrown upward with an initial velocity of 80 ft/sec has a velocity after one second of 80-32= 48 ft/sec, after two seconds 48-32= 16 ft/sec, and after three seconds 16-32= -16 ft/sec (now downward), etc. At 2.5 seconds the ball had a zero velocity and after another 2.5 seconds it hits the ground with a velocity of 80 ft/sec downward. The up and down motion is symmetrical.

(2) Friction, a force acting between two bodies in contact, is parallel to the surface and opposite the motion (or tendency to move). By the second law, giving a mass of one kilogram (kg) an acceleration of 1 m/sec/sec requires a force of one Newton (N). However, if friction were 3 N, a force of 4 N must be applied to give the same acceleration. The net force is 4N (applied by someone) minus 3N (friction) or 1N.

Free fall, example (1), assumed no friction. If there were atmospheric friction it would be directed upward since friction always opposes the motion. Air friction is proportional to the velocity; as the velocity increases the friction force (upward) becomes larger. The net force (weight minus friction) and the acceleration are less than due to gravity alone. Therefore, the velocity increases less rapidly, becoming constant when the friction force equals the weight of the falling object (net force=0). This velocity is called the terminal velocity. A greater weight requires a longer time for air friction to equal the weight, resulting in a larger terminal velocity.

(3) A contemporary and friend of Newton, Halley, observed a comet in 1682 and suspected others had observed it many times before. Using Newton’s new mechanics (laws of motion and universal law of gravity) Halley calculated that the comet would reappear at Christmas, 1758. Although Halley was dead, the comet reappeared at that time and became known as Halley’s comet. This was a great triumph for Newtonian mechanics.

Using Newton’s universal law of gravity (see example 1) in the second law results in a general solution (requiring calculus) in which details of the paths of motion (velocity, acceleration, period) are given in terms of G, M, and distance of separation.

While these results agreed with planetary motion known at the time there was now an explanation for differences in motions. These solutions were equally valid for applying to any systems body: Earth’s moon, Jupiter’s moons, galaxies, truly universal.

(4) Much of Newton’s work involved rotational motion, particularly circular motion. The velocity’s direction constantly changes, requiring a centripetal acceleration. This centripetal acceleration requires a net force, the centripetal force, acting toward the center of motion. Centripetal acceleration is given by a(central)=v^{2}/R where v is the velocity’s magnitude and R is the radius of the motion. Hence, the centripetal force F(central) = mv^{2}/R, where m is the mass.

These relationships hold for any case of circular motion and furnish the basis for ‘thrills’ experienced on many amusement park rides such as ferris wheels, loop-the-loops, merry-go-rounds, and any other means for changing your direction rather suddenly. Some particular examples follow.

(a.) Newton asked himself why the moon did not fall to Earth like other objects. Falling with the same acceleration of gravity as bodies at Earth’s surface, it would have hit Earth. With essentially uniform circular motion about Earth, the moon’s centripetal acceleration and force must be due to earth’s gravity. With gravitational force providing centripetal force, the centripetal acceleration is a(central) = GM/R^{2} (acceleration of gravity in example 1 above). Since the moon is about 60 times further from Earth’s center than Earth’s surface, the acceleration of gravity of the moon is about. 009 ft/sec/sec. In one second the moon would fall about.06 inch but while doing this it is also moving away from Earth with the result that at the end of one second the moon is at the same distance from Earth, R.

(b.) With the gravitational force responsible for centripetal acceleration, equating the two acceleration expressions given above gives the magnitude of the velocity as v^{2} = GM/R with the same symbol meanings. For the moon in (a) its velocity would be about 2,250 MPH. This relationship can be universally applied.

Long ago it was recognized that this analysis could be applied to artificial “moons” or satellites. If a satellite could be made to encircle Earth at about 200 mi it must be given a tangential velocity of about 18,000 mph and it would encircle Earth every 90 seconds; astronauts have done this many times since 1956, 300 years after Newton gave the means for predicting the necessary velocities.

It was asked: what velocity and height must a satellite have so that it remains stationary above the same point on Earth’s surface; that is, have the same rotation period, one day, as Earth. Three such satellites, placed 120 degrees apart around Earth, could

make instantaneous communication with all points on Earth’s surface possible. From the fact that period squared is proportional to the cube of the radius and the above periods of the moon and satellite and the moon’s distance, it is found that the communication satellite would have to be located 26,000 mi from Earth’s center or 22,000 mi above the surface. Its velocity must be about 6,800 mph. Many such satellites are now in space around Earth.

(c.) When a car rounds a curve what keeps it on the road? Going around a curve requires a centripetal force to furnish the centripetal acceleration, changing its direction. If there is not, the car continues in a straight line (first law) moving outward relative to the road. Friction, opposing outward motion, would be inward and the inward acceleration a(cent) =friction/mass = v^{2}/R. Each radius has a predictable velocity for which the car can make the curve. A caution must be added: when it is raining, friction is reduced and a lower velocity is needed to make the curve safely.

## Third law of motion or law of action-reaction

Newton questioned the interacting force an outside agent exerted on another to change its state of motion. He concluded that this interaction was mutual so that when you exert a force on something you get the feeling the other is exerting a force on you. Newton’s third law of motion states: When one body exerts a force on a second body, the second body exerts an equal and opposite force on the first body.

In the second law, only forces exerted on a body are important in determining its acceleration. The third law speaks about a pair of forces equal in magnitude and opposite in direction, which are exerted on and by two different bodies. This law is useful in determining forces acting on an object by knowing forces it exerts. For example, a book sitting on a table has a net force of zero. Therefore, an upward force equal to its weight must be exerted by the table on the book. According to the third law the book exerts an equal force downward on the table. When two objects are in contact they exert equal and opposite forces on each other and these forces are perpendicular to the contacting surface.

## Examples of the third law

(1) What enables us to walk? To move forward parallel to the floor we must push backward on the floor with one foot. By the third law, the floor pushes forward, moving us forward. Then the process is repeated with the other foot, etc. This cannot occur unless there is friction between the foot and floor and on a frictionless surface we would not be able to walk.

### KEY TERMS

**Centripetal acceleration—** Produces a change in the direction of velocity and always perpendicular to the velocity vector. This, in turn is caused by a centripetal force.

**Force—** Influence exerted on an object by an outside agent which produces an acceleration changing the object’s state of motion.

**Inertia—** Property of matter whereby any change in state of motion is opposed. Quantitatively measured by mass.

(2) How can airplanes fly at high altitudes and space crafts be propelled? High altitude airplanes utilize jet engines; that is, engines burn fuel at high temperatures and expel it backward. In expelling the burnt fuel a force is exerted backward on it and it exerts an equal forward force on the plane. The same analysis applies to space crafts.

(3) A father takes his eight-year-old daughter to skate. The father and the girl stand at rest facing each other. The daughter pushes the father backwards. What happens? Whatever force the daughter exerts on her father he exerts in the opposite direction equally on her. Since the father has a larger mass his acceleration will be less than the daughter’s. With the larger acceleration the daughter will move faster and travel farther in a given time.

## Resources

### BOOKS

Croce, Nicholas. *Newton and the Three Laws of Motion.* New York: Rosen Publishing Group, 2005.

Hewitt, Paul. *Conceptual Physics.* Englewood Cliffs, NJ: Prentice Hall, 2001.

Munson, Bruce, et al. *Fundamentals of Mechanics.* 4th ed. New York: John Wiley & Sons, 2002.

Scheck, Florian. *Mechanics: From Newton’s Laws to Deterministic Chaos.* New York: Springer, 2006.

### OTHER

*National Aeronautics and Space Administration.* “Newton’s Laws of Motion.” March 16, 2006. <http://www.lerc.nasa.gov/WWW/K-12/airplane/newton.html> (accessed Oct. 1, 2006).

Billy W. Sloope

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