# Newton's Laws of Motion

## Applications Of The Second Law

(1) Objects, when released, fall to the ground due to the earth's attraction. Newton's universal law of gravity gave the force of attraction between two masses, m and M, as F =GmM/R2 where G is the gravitational constant and R is the distance between mass centers. This force, weight, produces gravitational acceleration g, thus weight = GmM/R2 = mg(2nd Law) giving g = GM/R2. This relationship holds universally. For all objects at the 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)=v2/R where v is the velocity's magnitude and R is the radius of the motion. Hence, the centripetal force F(central) = mv2/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/R2 (acceleration of gravity in example 1 above). Since the moon is about 60 times further from Earth's center than the 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 the earth with the result that at the end of one second the moon is at the same distance from the 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 v2 = 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 the earth at about 200 mi it must be given a tangential velocity of about 18,000 MPH and it would encircle the 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 the earth. Three such satellites, placed 120 degrees apart around the earth, could make instantaneous communication with all points on the 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 the 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 = v2/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.