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LINEAR MOMENTUM AND ITS CONSERVATIONLINEAR MOMENTUM AND ITS CONSERVATIONThe linear momentum The linear momentum PP of a of a particle of mass particle of mass mm moving moving with a velocity with a velocity vv is defined is defined to to be the product of the mass be the product of the mass and velocity:and velocity:Linear momentum is a Linear momentum is a vector vector quantityquantity because it equals the because it equals the product of a scalar quantity product of a scalar quantity m m and a and a vector quantity vvector quantity v. Its . Its direction is along v, it has direction is along v, it has dimensions dimensions ML/T, and its SI ML/T, and its SI unit is kg m/s.unit is kg m/s.

* Using Newton’s * Using Newton’s second law of motion, second law of motion, we can conclude that we can conclude that “ “ The time rate of The time rate of change of the linear change of the linear momentum of a momentum of a particle is equal to the particle is equal to the net force acting on the net force acting on the particleparticle””

Conservation of Momentum forConservation of Momentum for a Two-Particle Systema Two-Particle SystemApplying Newton’s second law Applying Newton’s second law to each particle, we can writeto each particle, we can write

Newton’s third law tells us that Newton’s third law tells us that FF1212 and F and F2121 are equal in are equal in magnitude and opposite in magnitude and opposite in direction. That is, they form an direction. That is, they form an action–reaction pair Faction–reaction pair F1212 , F , F2121. . We can express this condition We can express this condition asas

where pwhere pllii and p and p2i2i are the initial values and p are the initial values and p1f1f and p and p2f2f the the final values of the momentum final values of the momentum during the time interval during the time interval dt dt over which the reaction pair interacts.over which the reaction pair interacts.Conservation of momentum Conservation of momentum Whenever two or more particles in an isolated Whenever two or more particles in an isolated system interact, the total momentum of the system system interact, the total momentum of the system remains constant.remains constant.

Because the time derivative of the total Because the time derivative of the total momentum pmomentum ptot tot p p11 p p22 is is zero, we conclude zero, we conclude that the that the total momentum of the system must remain total momentum of the system must remain constantconstant:

An object moves with An object moves with simple harmonic motion simple harmonic motion whenever its acceleration is whenever its acceleration is proportional to its displacement proportional to its displacement from some equilibrium position from some equilibrium position and is oppositely directed.and is oppositely directed.

SIMPLE HARMONIC MOTIONSIMPLE HARMONIC MOTION

Applying Newton’s second law Applying Newton’s second law to the motion of the block, to the motion of the block, together with equation relates together with equation relates a force that is proportional to a force that is proportional to the displacement which is the displacement which is given by Hooke’s law , so :given by Hooke’s law , so :

The frequency:The frequency: represents the number of represents the number of oscillations that the particle makes per unit timeoscillations that the particle makes per unit time

In general, a particle moving In general, a particle moving along the along the x axis exhibits simple x axis exhibits simple harmonic motion harmonic motion when when x, the x, the particle’s displacement from particle’s displacement from equilibrium, varies in time equilibrium, varies in time according according to the relationshipto the relationship

The period The period T of the motion T of the motion : is the : is the time it takes for time it takes for the particle to go through one full cycle.the particle to go through one full cycle.

The units of The units of f are cycles per second sf are cycles per second s-1-1, , or hertz (Hz).or hertz (Hz).

THE PENDULUMTHE PENDULUMWhen is small, a simple When is small, a simple pendulum oscillates in pendulum oscillates in simple harmonic motion simple harmonic motion about the equilibrium about the equilibrium position position θθ. The restoring . The restoring force is force is mg sin mg sin θθ , the , the component component of the of the gravitational force tangent gravitational force tangent to the arc.to the arc.

if we assume that if we assume that θθ is small, is small, we can use the approximation we can use the approximation sin sin θθ = = θθ ; thus the equation of ; thus the equation of motion for the simple motion for the simple pendulum becomespendulum becomes

Where Where θθ max max is the is the maximum angular maximum angular displacement displacement

and the angular frequency isand the angular frequency is

Therefore, can be written asTherefore, can be written as

The period of the motion isThe period of the motion is

The period and frequency of a simple pendulum The period and frequency of a simple pendulum depend only on the length of the string and the depend only on the length of the string and the acceleration due to gravity.acceleration due to gravity.

The simple pendulum can be used as a The simple pendulum can be used as a timekeeper because its period depends timekeeper because its period depends only on its length and the local value of only on its length and the local value of g. g. It is also a convenient device for making It is also a convenient device for making precise measurements of the free-fall precise measurements of the free-fall acceleration.acceleration.

The Foucault pendulum at the The Foucault pendulum at the Franklin Institute in Philadelphia. Franklin Institute in Philadelphia. This type of pendulum was first This type of pendulum was first used by the French physicist used by the French physicist Jean Jean Foucault Foucault to verify the Earth’s to verify the Earth’s rotation experimentally. As the rotation experimentally. As the pendulum swings, the vertical pendulum swings, the vertical plane in which it oscillates plane in which it oscillates appears to rotate as the bob appears to rotate as the bob successively knocks over the successively knocks over the indicators arranged in a circle on indicators arranged in a circle on the floor. the floor. In reality, the plane of oscillation In reality, the plane of oscillation is fixed in space, and the Earth is fixed in space, and the Earth rotating beneath the swinging rotating beneath the swinging pendulum moves the indicators pendulum moves the indicators into position to be knocked down, into position to be knocked down, one after the other.one after the other.