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Scalar and vector quantities. Force, velocity and acceleration. Newton’s laws of motion. Law of Universal Gravitation. Coefficient of friction.

Scalar and vector quantities. In physics some things such as temperature, mass, length, volume, density, time, distance and speed possess only “magnitude”. Other things such as force, velocity and acceleration possess both magnitude and direction. Those things which possess only magnitude are referred to as scalar quantities; those which have both magnitude and direction are referred to a vector quantities. Vector quantities must be added, subtracted and multiplied in accordance with the rules for operating on vectors. It is common practice to write vector quantities in bold type, which we will do.

Def. Force. A push or pull exerted on a body. It is a vector quantity possessing both magnitude and direction.

Velocity and speed. In common usage speed and velocity have the same meaning. In physics, however, they have different meanings. In physics, velocity is a vector quantity possessing both magnitude and direction. Velocity can be defined as the rate of motion in a particular direction. Speed is simply rate of motion. It tells how far an object moves per unit of time. An object’s speed corresponds to the magnitude of its velocity.

Def. Acceleration. Rate of change of velocity with respect to time. It is a vector quantity possessing both magnitude and direction.

Examples.

1. Consider a bee flying through the air. At a particular instant it may be flying in a particular
direction with a velocity V_{1}. A second later it may be flying in a different direction with a
velocity V_{2}. The change in velocity over that one second interval is given by the vector
difference V_{2} - V_{1} and this is its acceleration over that one second time interval.

2. If we are driving on an expressway and our direction of movement stays constant and we step
on the accelerator so that the vehicle starts increasing in speed at a rate of 3 miles/sec per second
then the vehicle is accelerating at a rate of 3 miles/sec^{2}.

Mathematically, acceleration is given as a = dv/dt where a is acceleration and v is velocity.

Def. Uniformly accelerated motion. Motion in which the acceleration is constant over time.

In the general case, acceleration varies with time but in some problems, such as those involving falling bodies and the acceleration of gravity, the acceleration remains constant over time.

To explain various observed natural phenomena including the orbiting of the earth and other planets about the sun and the phenomenon of material things on earth having ‘weight’ and tending to fall toward the earth, Sir Isaac Newton (1643 - 1727) postulated the existence of a general attractive force existing between any two material objects in the universe in his Law of Universal Gravitation which states:

Law of Universal Gravitation. Every body in the universe attracts every other body with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them i.e.

where m_{1 }and m_{2} are the masses of the two bodies, r is the distance between them, F is the force,
and G is a proportionality constant called the gravitational constant.

[Side note. Who would ever suspect that two bodies of mass (not touching each other) would exert a force on each other? How could they do it? By what mechanism could two bodies that were not even touching each other exert forces on each other? It sounds absurd. Who would ever suspect such a thing? What line of thought would have led Newton to postulate a force existing between any two bodies of mass in the universe? Answer: Copernicus had already proposed the theory that the earth and the other planets revolved around the sun in circular orbits. If one assumes that this theory is correct, a natural question to ask is, “What causes the planets to move in a circle? What keeps them from flying in a straight line off into space?” If the assertion is true that they move in orbits about the sun, then there must be some centripetal force pulling them toward the sun. What is this force and where does it come from? The assumption that they move in orbits about the sun implies the existence of a force, a centripetal force, drawing them towards the sun. Thus if Copernicus’ theory is true, bodies of mass must attract each other. And if any two bodies of mass in the universe attract each other, could this be the explanation for the observed fact that objects in daily life have weight and tend to fall to the earth? Could this phenomenon be simply a consequence of the attraction of any two bodies in the universe for each other?

From the above line of reflective thought, we see how a little reflective thought can lead to big conclusions. Newton never answered the question of, “What is this force and where does it come from?” However, Kepler had published his laws, and assuming Kepler’s laws as valid, Newton was able to derive a mathematical expression for the strength of the force.

Kepler’s Laws. Kepler arrived at the following laws from study of astronomical observations:

Law 1. Each planet moves around the sun in an elliptic path with the sun at one focus of the ellipse.

Law 2. As a planet moves in its orbit, a line drawn from the sun to the planet sweeps out equal areas in equal intervals of time.

Law 3. The squares of the periods of the planets are proportional to the cubes of their mean distances from the sun.

To derive the expression for the strength of the attractive force, Newton first showed that a
moving particle subjected to a succession of discrete blows all directed toward a fixed point will
move in such a way as to sweep out equal areas in equal times. Then, since the time interval Δt
can be chosen arbitrarily small, as Δt approaches zero, the succession of blows become a
continuously acting centripetal force and the broken curve for the path becomes a smooth one.
Newton then showed that the converse was also true i.e. if equal areas are swept out in equal
time, the force acting on the particle must be a centrally directed one. He thus showed that the
assumption of a centripetal force was consistent with Kepler’s second law. Newton then proved
that for bodies moving along conic sections (such as hyperbolas, parabolas, ellipses, and circles)
the centripetal force at any instant must be proportional to the inverse square of the distance from
body to the focus i.e. he showed that they were acted upon by a law of the general form F = c/r^{2}
where c is a constant and r is the distance from the body to the center of force.]

Newton also postulated three laws of motion which can be stated as follows:

Newton’s First Law of Motion (Law of Inertia). A body remains in a state of rest or of uniform motion in a straight line unless acted upon by some unbalanced force.

A body’s natural state, if there are no unbalanced forces acting on it, is either a state of rest or of uniform motion in a straight line.

Note. We specify unbalanced force in our statements above because a body may have several forces acting on it but the body may not be moving because they cancel each other out i.e. each force is opposed by a balancing force.

Newton’s Second Law of Motion. An unbalanced force acting on a body produces an acceleration of the body. This acceleration is in the direction of the force and in proportion to the force and is inversely proportional to the mass.

Newton’s second law can be stated as

2) F = kma

where

F = applied force

m = mass

a = acceleration

k = a proportionality constant

To repeat, Newton’s second law states

(1) If a force is applied to a body, the body will be accelerated in the direction of the force and the magnitude of the acceleration will be proportional to the size of the force. In other words, when we apply a force to a body we cause a change in the velocity of the body in the direction of the force. This change in velocity can take the form of speeding the body up, slowing it down or changing its direction and this change of velocity is the acceleration referred to. The amount of acceleration ( i.e. the amount of change in velocity), is proportional to the magnitude of the force. In other words,

(2) For a given force F, the acceleration produced by the force is inversely proportional to the mass of the body i.e.

where m is the mass of the body. Thus if the mass of the body is doubled, the acceleration produced by a given force is reduced to half.

As a consequence, we have

It would be nice if we could simplify F = kma by making k = 1, since the law would then be of the simpler form F = ma. We can do this if we are willing to define one of the quantities in the equation in terms of the other two. This is what physicists have done. Consequently, in terms of the units we are about to define, Newton’s second law can be written

3) F = ma

Units of force, mass and acceleration used in F = ma.

MKS (meter-kilogram-second) system. The fundamental unit of mass is the kilogram, the
fundamental unit of acceleration is m/sec^{2}, and the fundamental unit of force is a derived unit
called the newton. One newton of force is defined to be that unbalanced force that will produce
an acceleration of 1 m/sec^{2} in a mass of 1 Kg.

CGS (centimeter-gram-second) system. The fundamental unit of mass is the gram, the
fundamental unit of acceleration is cm/sec^{2}, and the fundamental unit of force is a derived unit
called the dyne. One dyne of force is defined to be that unbalanced force that will produce an
acceleration of 1 cm/sec^{2} in a mass of 1 gram.

English system. The fundamental unit of force is the pound, the fundamental unit of
acceleration is ft/sec^{2}, and the fundamental unit of mass is a derived unit called the slug. One
slug of mass is defined to be that mass which when acted on by a 1 lb force will acquire an
acceleration of 1 ft/sec^{2}.

Thus the following are the three sets of units that can be used with F = ma:

MKS System: F (newtons) = m (kilograms) × a (m/sec^{2})

CGS System: F (dynes) = m (grams) × a (cm/sec^{2})

English System: F (pounds) = m (slugs) × a (ft/sec^{2})

Newton’s second law, F= ma, and weights of objects. The mass m of a body is a measure of the inertia of the body and the weight w of a body is the pull due to the force of gravity acting on the body and varies with location.

MKS System. In accordance with Newton’s second law, F = ma, one newton of force
accelerates one kilogram of mass 1 m/sec^{2}. (By definition, one newton is that amount of force
that accelerates a kilogram of mass 1 m/sec^{2}. ). The acceleration of gravity is 9.8 m/sec^{2}.

The weight of a body is the force exerted by the earth on the body and this force accelerates the
body at a rate of 9.8 m/sec^{2}. Thus the gravitational force on a 1 Kg object must be a force of 9.8
newtons. Hence the weight that we associate with a 1 Kg mass is a force of 9.8 newtons.

Just how much force is one newton? One newton of force corresponds to a weight of 1/9.8 = 0.102 kilograms.

If w is weight, m is mass, and g is the acceleration of gravity, we have

w (newtons) = m (kilograms) × g (m/sec^{2})

CGS System. One dyne of force accelerates one gram of mass one cm/sec^{2} . (By definition,
one dyne is that amount of force that accelerates a gram of mass 1 cm/sec^{2}. ). The acceleration
of gravity is 980 cm/sec^{2}.

The weight of a body is the force exerted by the earth on the body and this force accelerates the
body at a rate of 980 cm/sec^{2}. Thus the gravitational force on a 1 gram object is a force of 980
dynes. Hence the weight that we associate with a 1 gram mass is a force of 980 dynes.

Just how much force is one dyne? One dyne of force corresponds to a weight of 1/980 = 0.00102 grams.

If w is weight, m is mass, and g is the acceleration of gravity, we have

w (dynes) = m (grams) × g (cm/sec^{2})

English system. One pound of force accelerates one slug of mass one foot/sec^{2} (By
definition, one slug is the amount of mass that is accelerated by a pound of force one foot/sec^{2}).
The acceleration of gravity is 32 ft/sec^{2}.

The weight of a body is the force exerted by the earth on the body and this force accelerates the
body at a rate of 32 ft/sec^{2}. Since a one pound object is accelerated by the force of gravity at a
rate of 32 ft/sec^{2} and the gravitational force acting on it is 1 lb, its mass in slugs must be 1/32
slugs. The weight that we associate with a 1 lb object is a force of 1 lb.

How much mass is one slug? The mass of a one pound object is 1/32 slugs so one slug corresponds to the mass of an object weighing 32 pounds.

If w is weight, m is mass, and g is the acceleration of gravity, we have

w (pounds) = m (slugs) × g (ft/sec^{2})

Example 1. What force, in newtons, is required to accelerate a small cart with a mass of 15 kg at
a rate of 3 m/sec^{2} if we assume no frictional forces?

Solution. Substituting into the formula F = ma,

F = 15 × 3 = 45 newtons

Example 2. A car weighs 3200 lb. What force is needed to give it an acceleration of 10 ft/sec^{2},
assuming no frictional forces?

Solution. The mass of the car is 3200/32 = 100 slugs. From F = ma we obtain

F = 100 ×10 = 1000 lb.

Example 3. A body of mass 3 kg is acted upon by a force of 15 newtons. What is its acceleration?

Solution. Using F = ma we get a = F/m = 15/3 = 5 m/sec^{2}

Newton’s Third Law of Motion. Whenever one body exerts a force on another, the second body exerts a force equal in magnitude and opposite in direction on the first. This law is often called the law of action and reaction. One force is called the action and the other force the reaction.

Examples. A 2 lb book sits on the table. The book exerts a force of 2 lb on the table and the
table exerts an equal and opposite force of 2 lb on the book. A man presses his hand against a
wall with a force of 20 lb. The wall presses back against his hand with a force of 20 lb. A man
pushes against a 64 lb block sitting on a frictionless surface with a force of 20 lb. The block
pushes back with a force of 20 lb and also accelerates in accordance with the formula F = ma. It
has a mass of 64/2 = 2 slugs and is accelerated at a rate of a = F/m = 20/2 = 10 ft/sec^{2}.

Friction. When one attempts to slide or roll one body over another he finds that there is a force, called friction, that opposes the motion. The cause of frictional forces is not fully understood and is an important field of research. One explanation for friction is that it is caused by irregularities in the surfaces of bodies, the idea being that as the uneven surfaces slide against each other they tend to interlock and resist movement. Although friction may be reduced by polishing surfaces, experiments have shown that if surfaces are made too smooth the friction between them actually increases.

If we hook a spring balance to an object and pull horizontally we find that usually more force is needed to start the object sliding than to keep it sliding. Thus we must distinguish between the friction when a body is not sliding and when it is sliding. We call one friction static friction and the other kinetic or sliding friction.

Experiments on friction reveal the following facts about it:

1. Frictional forces act parallel to the surfaces that are sliding over one another and in the direction opposite to that of the motion.

2. The amount of friction depends on both the kind of material and its surface.

3. In the case of objects moving at high speeds, as in the case of a bullet moving through a gun barrel, friction decreases with increasing speed. However, at medium speeds, the sliding friction is nearly independent of speed.

4. Friction is practically independent of the area of contact between the surfaces. For example, the force needed to slide a brick across a table is almost the same whether the brick lies on its side, on its edge, or on end.

5. Friction is directly proportional to the force pressing the two surfaces together.

Def. Kinetic or sliding friction. The tangential force between two surfaces when one surface is sliding over another.

Def. Static friction. The tangential force between two surfaces when the two surfaces are not sliding relative to each other. The tangential force between two surfaces just before one surface starts to slide over the other is called the maximum force of static friction.

Def. Coefficient of kinetic or sliding friction. The coefficient of kinetic or sliding friction μ_{k} is
the ratio of the force necessary to move one surface over the other with uniform velocity to the
normal force pressing the two surfaces together i.e.

where

f_{k} = force required to just balance the kinetic friction force

N = normal force pressing the two surfaces together

The force f_{k} is the force of kinetic friction and is given by f_{k} = μ_{k}N.

Def. Coefficient of static friction. The coefficient of static friction μ_{s} is the ratio of the
maximum force of static friction to the normal force pressing the two surfaces together i.e.

where

f_{s} = maximum force of static friction

N = normal force pressing the two surfaces together

Example. A block weighs 100 lb. A force of 40 lb is required to keep it in uniform motion on a horizontal surface. What is the coefficient of friction?

Solution. Substituting into the formula μ_{k} = f_{k}/N we get

References

Dull, Metcalfe, Brooks. Modern Physics.

Schaum. College Physics.

Sears, Zemansky. University Physics.

Semat, Katz. Physics.

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