GRAVITY AND FALLING BODIES
Gravity is one of the most familiar forces in nature; its effect on motion has been a subject of discussion for centuries. If an object is dropped from a great height, it can be observed that it falls with ever increasing speed until air resistance balances the effect of gravity, at which time it is said to have reached its terminal velocity. The term free falling bodies is used for objects that are moving freely under the influence of gravity, whether they are moving upward or downward. Any object that has no forces other than gravity acting on it is said to be in free fall, whether it is moving upward, downward, or in any direction.
It is found that if air resistance can be made negligible, then falling bodies will accelerate toward the center of the earth at the same rate, regardless of their mass. The value for the acceleration of gravity, given the symbol g, has been measured on earth as g = 9.8 m/s2. Galileo was the first to demonstrate that all bodies fall at the same rate if air resistance is negligible. ( It is often said that he did this by dropping objects of various masses from the Leaning Tower of Pisa, although there is no historical evidence that he actually used the famed tower.) Galileo’s recorded experiments settled some very old controversies about falling bodies, proving less-popular ideas to be correct.
Even more important than his discoveries about falling objects was his breaking away from old methods of determining truth. Galileo is often credited with being the Father of Modern Science because of his forceful demonstration of the value of observation and the discoveries he made through his ingenious experiments.
The following is a data from one of Galileo’s earliest experiments of a ball rolling down an inclined plane. His data were recorded on his notes. Galileo held a ball at the top of an inclined, grooved board and marked its position. Releasing the ball, he marked its position at the end of equal intervals of time. This is much like dropping a ball from a height, except that the effect of gravity has been “reduced” by allowing the ball to roll slowly down the inclined board rather than falling straight down. The position as measured by Galileo are given in the following table :
Time t (equal intervals) t2 Distance,S(points) S/t2
1 1 33 33.0
2 4 130 32.5
3 9 298 33.1
4 16 526 32.9
5 25 824 33.0
6 36 1192 33.1
7 49 1620 33.1
8 64 2104 32.9
The observations show what was already known quantitatively to Galileo and others of his time – that a rolling (or falling) object picks up speed as it continues to roll (or fall). However, the debt we owe to Galileo is for his careful measurements and his quantitative (mathematical) interpretation of the data. His object was to find a general rule describing how distances increase with increasing time of fall. After some trial and error, and with considerable insight, Galileo realized that the distance traveled was proportional to the square of the elapsed time.
S => h = ½ at
2 ,
for vertical motion, a ==> g ( acceleration due to gravity )
Problems
1. A ball is thrown vertically up with an initial velocity of 15 m/s. How high does the ball rise from its projection point ? How long does it take for this rock to reach the highest point. How high does it go in 2 seconds ? in 3 seconds ? What is the time required to travel a height of 9 m ? 5 m ?
Ans. ( 11.48 m , 1.53 s , 10.4 m , 0.9 m , 0.82 s , 2.24 s , 0.38 s , 2.68 s )
2. A rock is dropped from a bridge 60 m high relative to the water on a river below. How long will it take for the rock to reach the surface of the water ? Calculate the positions of the rock 1s, 2 s, 3s after its release (a) relative to the bridge and (b) relative to the water.
3. A metal sphere is dropped from a 55 m high tower. Determine the height traveled by the sphere in the time interval from 1.5 s to 2.5 s.
FORCE, THE CAUSE OF ACCELERATION; NEWTON’S LAWS OF MOTION
In 1642, several months after Galileo died, Isaac Newton was born. At age 23, Newton developed his famous laws of motion, which completed the overthrow of the Aristotelian idea that had dominated the thinking of the best minds for 2,000 years.
Every acceleration ( change in velocity ) is caused by forces acting on a body. Conversely, if a body does not accelerate, then the total force acting on it is zero even if several forces are present. The apparently simple idea of cause and effect, that forces cause acceleration, didn’t come easily. It was and still is tempting to think of common phenomena as having no cause and simple being “the nature of things”. For example, “Why does water flow downhill?” seems stupid. Yet such question have a serious answers; in this case, the force of gravity causes water to flow downhill. The genius of Newton and others was not only in providing answers to basic questions. But also in simply being curious enough to ask basic questions.
Force is defined intuitively as a push or a pull. If an applied force is the only one thing acting on a body, then the body will accelerate in the same direction as the force. The strength of the force determines the magnitude of the acceleration. If several forces act on a body, then its acceleration is in the same direction as the total force and has magnitude proportional to the total force.
NEWTON’S LAWS OF MOTION
Galileo had a major influence in the study of motion. What Newton did was to write down the relationships between the force and motion in a form that could be used to predict and describe motion. Those relationships were found to apply in every circumstance where an experiment could be performed to test them and came to be known as Newton’s laws of motion.
The First Law : Inertia (mass). A body rest remain at rest or in motion in a straight line with a constant velocity unless acted upon by an outside force. The property of a body that causes it to remain at rest or to maintain a constant velocity is called inertia. The law was a refinement of Galileo’s idea --- in the absence of force, a moving object will continue moving. Galileo considered the tendency of things to resist change in motion as inertia. Inertia is a measure of how difficult is it to set a body into motion, or if it is already moving, how difficult is it to stop.
The Second Law: The acceleration produced by forces acting on a body is directly proportional to and in the same direction as the net external force and inversely proportional to the mass of the body.
a = Fnet/m ==> Fnet= ma , m ==> mass and a ==> acceleration
Newton’s Second law gives a precise definition of force that is consistent with our intuitive notions of a force as a push or a pull. A large force produces a large acceleration, a large mass requires a large force to make it accelerate at the same rate as a small mass, and a body will accelerate in the same direction as the net force on it.
The Third Law : Action – Reaction. Whenever one body exerts a force on a second body, the second body exerts a force back on the first that is equal in magnitude and opposite in direction. This is paraphrase as, “For every action there is equal and opposite reaction”.
One force is called the action force and the other is the reaction force. In every interaction, the forces always occur in pairs. The action and the reaction pair of forces make up the interaction between two things. We know that forces can cancel when they are equal and act in the opposite direction on the same object. Even though action and reaction are equal and oppositely directed, they do not cancel each other for they are acting on different bodies.
An example is a swimmer that exerts a force on the side of the pool. By Newton’s third law, the side of the pool exerts a force back on the swimmer – an external force. If friction is negligible between the swimmer and the water, she will then move in a direction opposite to the force she exerted on the side of the pool with an acceleration proportional to the force she exerted.
Cars accelerate forward by exerting backward forces on the ground. The reaction force of the ground acts as an external force on the car in the forward direction.
UNITS OF FORCE
1. Newton – is the force required to give a mass of 1 kilogram an acceleration of 1 m/ s
2.
1 newton ( N ) = 1 kg-m/s
2
2. Dyne – is the force required to give a mass of 1 gram an acceleration of 1 cm/ s
2.
1 dyne = 1 g-cm/ s
2
3. Pound – is the force required to give a mass of 1 slug an acceleration of 1 ft/ s
2.
1 lb = 1 slug-ft/ s
2 = 4.448 N
WEIGHT, FRICTION, TENSION, AND OTHER CLASSES OF FORCES
The weight of an object is the gravitational force exerted on it by the earth. When an object is dropped near the earth’s surface, it is accelerated by the gravitational force with an acceleration g, thus by Newton’s second law, the weight w becomes
w = mg. ==> m = w/ g
We see in this equation the relation between mass and weight : Weight is a force proportional to the mass of a body and g is the constant of proportionality. Here, g is taken to be positive, since the direction of forces are indicated with plus or minus sign. Weight depends on the location of the object, since the acceleration of gravity varies with location. As you go higher, g decreases so that weight also is decreasing. On the moon g ==> 1/6 of the earth’s g.
Center of gravity. The force of gravity on solids can be considered to act on a single point, called center of gravity (c.g.).For symmetrical objects, c.g. is at its geometric center. For asymmetrical objects, the c.g. is closer to the more massive part of the body. A closer related concept is the center of mass (c.m.), is the point at which all of the mass in a body can be considered to be located.
Newton’s Universal Law of Gravitation. The law states that there a force of attraction between any two masses that is proportional to the product of the masses and inversely proportional to the square of the distance between their centers of mass.
F = G m
1 m
2/r
2
where G ==> Newton’s Universal constant of gravitation
G = 6.67 x 10
–11 N.m
2/kg
2 ,
m
1, m
2 ==> masses in kg and
r ==> distance between the centers of mass in meter.
FRICTION
Friction is any force that opposes every effort to start to slide or roll one body over another body. Frictional forces are specially important to us in our daily lives, for without them we could not walk or hold things with our hands; cars wouldn’t be able to start or stop; nails and screws would be useless. Frictional forces are not fundamental forces like gravity or electromagnetism, but arise as reaction to other applied forces. Friction is proportional to the force exerted by one substance on another perpendicular to the surface between them---that is, the normal force (perpendicular force). The mathematical expressions are :
1. f = u
kF
N , u
k==> coefficient of kinetic friction , F
N ==>Normal force
2. f = u
s F
N , u
s ==>coefficient of static friction
Equation 1 is used for the friction between moving substance and equation 2 for stationary substances.
Coefficient of friction is the ratio of the force of friction f to the normal force, F
N.
PRINCIPLES OF FRICTION
1. The force of friction always act in a direction opposite to the direction of motion, for objects in
relative notion --- that is, sliding or rolling.
2. The frictional force is proportional to the normal (perpendicular ) force between the two surfaces in
contact.
3. Frictional force is approximately independent of the area of contact between the surfaces.
4. The frictional force depends on the particular material that make up the surfaces.
* Synovial fluid – a fluid which looks like blood plasma which lubricates the joints and limbs of the body.
ADVANTAGES OF FRICTION
1. Walking would be impossible without friction.
2. Pulley driven machines depend on friction for their operation.
3. Friction prevents belts from slipping off their pulley.
4. Friction between the tires and the road prevents skidding of vehicles.
5. Clutch, bolts and nuts, nails, screws, matches, brakes, etc. depends on friction.
DISADVANTAGES OF FRICTION
1. Wearing out of parts of machines, thus causing extra expenses for maintenance.
2. It causes expansion on machine parts and heat loses thus reducing the efficiency of machines.
TENSION
A tension is any force carried by a flexible string, rope, cable, chain, etc. Because the medium carrying the force is flexible, it can only pull and can exert no force except along its length. Tension comes from a Latin word meaning “to stretch thin”. In muscle systems the fibrous cords that carry forces exerted by muscles to other parts of the body are called tendons. Tension is due to the cohesive atomic and molecular electromagnetic forces acting in a string.
For a body suspended on a string with zero or constant speed upward or downward, the tension is given by T = w = mg. If the body accelerates downward on a string, the tension is given by T + ma = mg and if the body has an upward acceleration on a string, the tension is given by T = mg + ma.
Problems :
1. Determine the weight of a 50 kg person on earth. On the moon if g is 1/6 of the earth’s g.
2. Determine the mass of a box if a force of 80 N is able to accelerate it at 1.25 m/ s
2.
3. Calculate the mass of a flea in grams if its weight is 5 x 10
–6 N.
4. Find the acceleration of a rocket with mass of 1.2 x 10
6 kg if its engine exerts a net force of 2 x 10
6 N.
5. A 70 kg gymnast climbs on a rope. Determine the tension in the rope if
(a) he climbs at constant speed,
(b) he has an upward acceleration of 0.5 m/s
2 ; and
(c) he goes downward with a downward acceleration of 0.5 m/s
2.
6. Determine the force of gravitation between the earth and the sun and between the earth and the
moon.
m
E = 5.99 x 10
24 kg , m
S = 1.99 x 10
30 kg , m
M = 7.36 x 10
22 kg
S
earth–sun = 149.6 x 10
9 m , S
earth–moon = 3.84 x 10
8 m , Radius of earth = 6.367 x 10
6 m
7. A man weighs a fish of mass m on a spring scale attached to the ceiling of a elevator. Show that if the elevator accelerates in either direction , the spring scale gives a reading different from the weight of the fish. What is the reading on the scale if the elevator moves up or down at constant speed?
8. How much torque do you exert if you push perpendicularly on a door with a force of 30 N at a distance of 0.85 m from its hinges?