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NEWTON’S LAW OF UNIVERSAL GRAVITATIONNEWTON’S LAW OF UNIVERSAL GRAVITATION

Before 1687, clear under-Before 1687, clear under-standing of the forces causing standing of the forces causing plants and moon motions plants and moon motions was not available. In that was not available. In that year, Isaac Newton knew, year, Isaac Newton knew, from his first law, that a net from his first law, that a net force had to be acting on the force had to be acting on the Moon because without such a Moon because without such a force the Moon would move force the Moon would move in a straight-line path rather in a straight-line path rather than in its almost circular than in its almost circular orbit. orbit.

Newton reasoned that Newton reasoned that this force was the this force was the gravitational attraction gravitational attraction exerted by the Earth exerted by the Earth on the Moon.on the Moon.

every particle in the Universe every particle in the Universe attracts every other particle attracts every other particle with a force that is directly with a force that is directly proportional to the product of proportional to the product of their masses and inversely their masses and inversely proportional to the square of proportional to the square of the distance between them.the distance between them.

Newton’s law of universal gravitation Newton’s law of universal gravitation states thatstates that

If the particles have masses If the particles have masses mm11 and and mm22 and are separated and are separated by a distance by a distance r r , the , the magnitude magnitude of this of this gravitational force isgravitational force is

where where G G is a constant, called the universal is a constant, called the universal gravitational constant, that has been measured gravitational constant, that has been measured experimentally. Its value in experimentally. Its value in SISI units is units is

MOTION WITH CONSTANT ACCELERATIONMOTION WITH CONSTANT ACCELERATION

Velocity vector as a function of timeVelocity vector as a function of time

Position vector as a function of Position vector as a function of timetime

The instantaneous acceleration a is defined as The instantaneous acceleration a is defined as the limiting value of the ratio ∆the limiting value of the ratio ∆v/v/∆∆t as t as ∆∆t t approches zero :approches zero :

)(222ifif rravv Velocity vector as a Velocity vector as a

function of Position function of Position vectorvector

* As long as the book * As long as the book is not moving, is not moving, f = F. f = F. Because the book is Because the book is stationary, we stationary, we call this call this frictional force the frictional force the force of static friction force of static friction ffss

* When the book is in * When the book is in motion, we call the motion, we call the retarding force the retarding force the force of kinetic friction force of kinetic friction ffkk

FORCES OF FRICTIONFORCES OF FRICTION

• • The direction of the force of The direction of the force of static friction between any two static friction between any two surfaces in contact with each surfaces in contact with each other is opposite the direction of other is opposite the direction of relative motion and can have relative motion and can have valuesvalues where the dimensionless where the dimensionless constant constant µµss is called the is called the coefficient of static friction coefficient of static friction and and n n is the magnitude of the normal is the magnitude of the normal force. force. When When ffs s = f= fs,maxs,max = µ = µssn. n. The The inequality holds when the inequality holds when the applied force is less than µapplied force is less than µssn.n.

• • The direction of the force The direction of the force of kinetic friction acting on of kinetic friction acting on an object is opposite the an object is opposite the direction of the object’s direction of the object’s sliding motion relative to sliding motion relative to the surface applying the the surface applying the frictional force and is given frictional force and is given by where by where µµkk is the is the coefficient of kinetic coefficient of kinetic friction.friction.

Work done by a constant forceWork done by a constant force

The work The work W W done on done on an object by an agent an object by an agent exerting a constant exerting a constant force on force on the object is the object is the product of the the product of the component of the force component of the force in the direction of the in the direction of the displacement and the displacement and the magnitude of the magnitude of the displacement:displacement:

WorkWork is a scalar quantity, and its units are force is a scalar quantity, and its units are force multiplied by length. Therefore, the SI unit of multiplied by length. Therefore, the SI unit of work is the work is the Newton meter (Nm)Newton meter (Nm). This . This combination of units is used so frequently that it combination of units is used so frequently that it has been given a name of its own: the has been given a name of its own: the joule (J).joule (J).

Work expressed as a dot productWork expressed as a dot product

scalar product scalar product allows us to indicate how F and d allows us to indicate how F and d interact in a way that depends on how close to interact in a way that depends on how close to parallel they happen to beparallel they happen to be

KINETIC ENERGYKINETIC ENERGYIf the particle is displaced a If the particle is displaced a distance distance d, the net work d, the net work done by the total force done by the total force ΣΣF isF is

when a particle undergoes when a particle undergoes constant acceleration we have ,constant acceleration we have ,

where where vvii is the speed at t= 0 is the speed at t= 0 and vand vff is the speed at time t. is the speed at time t. After substituting we get: After substituting we get:

In generalIn general, the kinetic energy , the kinetic energy K of a particle of K of a particle of mass m moving with a speed v mass m moving with a speed v is defined asis defined as

Kinetic energy is a scalar quantity Kinetic energy is a scalar quantity and has the same units as work.and has the same units as work.

It is often convenient to It is often convenient to write this equation in the write this equation in the form:form:

Potential Energy and Conservation of EnergyPotential Energy and Conservation of EnergyWe introduced the concept of We introduced the concept of kinetic energykinetic energy, , which is the energy associated with the motion of which is the energy associated with the motion of an object.an object.Potential energy Potential energy U U • Is the energy associated with the arrangement Is the energy associated with the arrangement of a of a system system of objects that exert forces on each of objects that exert forces on each other.other.• It can be thought of as stored energy that can It can be thought of as stored energy that can either do work or be converted to kinetic energy.either do work or be converted to kinetic energy.

SystemSystem Is consists of two or more objects Is consists of two or more objects that exert forces on one anotherthat exert forces on one another

Gravitational potential energy is the Gravitational potential energy is the potential energy of the object–Earth potential energy of the object–Earth system.system.

The product of the magnitude of The product of the magnitude of the gravitational force the gravitational force mgmg acting on an object acting on an object and the and the height height yy of the object of the object

As an object falls toward the Earth, As an object falls toward the Earth, the Earth exerts a gravitational the Earth exerts a gravitational force force mgmg on the on the object, with theobject, with thedirection of the force being the same as the direction direction of the force being the same as the direction of the object’s motion.of the object’s motion.

Gravitational Potential Energy Gravitational Potential Energy UUgg

Let us now directly relate the work done on an Let us now directly relate the work done on an object by the gravitational force to the object by the gravitational force to the gravitational potential energy of the object–Earth gravitational potential energy of the object–Earth system.system.

where we have used the fact thatwhere we have used the fact that

From this result we conclude that : 1- The work done on any object by the gravitational force is equal to the negative of the change in the system’s gravitational potential energy.

2- This result demonstrates that it is only the 2- This result demonstrates that it is only the differencedifference in the gravitational potential in the gravitational potential energy at energy at the initial and final locations that matters. This the initial and final locations that matters. This means that we are free to place the origin of means that we are free to place the origin of coordinates in any convenient location.coordinates in any convenient location.

3- the object falls to the 3- the object falls to the Earth is the same as the work Earth is the same as the work done were the object to start done were the object to start at the same point and slide at the same point and slide down an incline to the Earth. down an incline to the Earth. Horizontal motion does not Horizontal motion does not affect the value of affect the value of WWgg

CONSERVATIVE AND NONCONSERVATIVE FORCES CONSERVATIVE AND NONCONSERVATIVE FORCES Conservative forces Conservative forces have two important properties:have two important properties:

1. A force is conservative if the work it does on a particle 1. A force is conservative if the work it does on a particle moving between any two points is independent of the moving between any two points is independent of the path taken by the particle.path taken by the particle.2. The work done by a conservative force on a particle 2. The work done by a conservative force on a particle moving through any closed path is zero. moving through any closed path is zero. (A closed path is (A closed path is one in which the beginning and end points are identical.)one in which the beginning and end points are identical.)

The The gravitational force gravitational force is one example of a is one example of a conservative force, and the conservative force, and the force that a spring force that a spring exertsexerts on any object attached to the spring is on any object attached to the spring is another.another.

Non-conservative ForcesNon-conservative ForcesA force is non-conservative if it causes a change in A force is non-conservative if it causes a change in mechanical energy mechanical energy E, E, which we define as the sum sum of kinetic and potential energiesof kinetic and potential energies. For exampleFor example, if a book is sent sliding on a horizontal surface that is not frictionless, the force of kinetic friction reduces the book’s kinetic energy. The type of energy associated with temperature is internal energy.

CONSERVATION OF MECHANICAL ENERGYCONSERVATION OF MECHANICAL ENERGYAn object held at some height h above the floor has no kinetic energy. The gravitational potential energy of the object–Earth system is equal to mgh. If the object is dropped, as it falls, its speed and thus its kinetic energy increase, while the potential energy of the system decreases. The sum of the kinetic and potential energies remains constant.

This is an example of the This is an example of the principle of conservation principle of conservation of mechanical energy.of mechanical energy.

Because the total mechanical energy E of a system is defined as the sum of the kinetic and potential energies, we can write

It is important to note that this equation is valid only It is important to note that this equation is valid only when no energy is added to or removed from the when no energy is added to or removed from the system. Furthermore, there must be no non-system. Furthermore, there must be no non-conservative forces doing work within the system.conservative forces doing work within the system.

We can state the principle of conservation of energy as and so we have