Mechanics and Motion: Kinematics and Dynamics of Particles, Lecture notes of Physics

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CHAPTER TWO

KINEMATICS AND

DYNAMICS OF PARTICLES

 (^) Mechanics ;-is the study of the physics of motions and how it relates to the physical factors that affect them, like force, mass, momentum and energy.  (^) Dynamics ;- which deals with the motion of objects with its cause – force;  (^) kinematics ;- describes the possible motions of a body or system of bodies without considering the cause.  (^) Alternatively, mechanics may be divided according to the kind of system studied.  (^) The simplest mechanical system ;-is the particle, defined as a body so small that its shape and internal structure are of no consequence in the given problem.  (^) More complicated ;- is the motion of a system of two or more particles that exert forces on one another.

Definition: Kinematical Quantities  (^) Position: - The location of an object with respect to a chosen reference point.  (^) Displacement : - The change in position of an object with respect to a given reference frame.

Distance (S) :- The length of the path followed by the object.

Average Velocity ( 𝒗 av) :- Is the total displacement divided by the total time.

Type equation here.

Average Speed :- is the total distance traveled by the object divided by the total

elapsed time.

𝑉 𝑎𝑣

=

𝑡𝑜𝑡𝑎𝑙𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑠 𝑡𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙(△𝑡)

…………………………………..2.1. 3

Average and Instantaneous Accelerations If the velocity of a particle changes with time, then the particle is said to be accelerating. Average acceleration: is the change in velocity (∆𝑣) of an object divided by the time interval during which that change occurs. 𝑎 av = ∆𝑣 ∆𝑡 𝑡𝑓−𝑡𝑖

……………………………….2.1.

Instantaneous acceleration : -The limit of average acceleration as △t approaches zero.

2.1.2. Motion with Constant Acceleration For motion with constant acceleration,  (^) The velocity changes at the same rate throughout the motion.  (^) Average acceleration over any time interval is equal to the instantaneous instant of time. acceleration at any 𝑎 = ∆𝑣 = ∆𝑡 𝑡 𝑣𝑓 −𝑣𝑖 , assuming t i = 0 By definition, 𝑣 (^) 𝑎𝑣 = ∆𝑟 ∆𝑡 for t=0 then, ∆𝑟 = 𝑣 (^) 𝑎𝑣t Rearranging this equation gives, 𝑣 = 𝑣𝑖+ 𝑎𝑡  (^) For motion with constant acceleration, average velocity can be written as:

𝑣 𝑎𝑣

=

𝑣𝑓+ 𝑣𝑖 2

𝑟_f − 𝑟𝑖= 𝑣𝑡𝑖 + 𝑎𝑡^2

𝑎𝑣

 Again, ∆𝑟 = 𝑣

t

but, 𝑣𝑎𝑣

=

𝑣𝑓+ 𝑣𝑖 2

and

𝑡=

𝑣𝑓−𝑣𝑖 𝑎

after substituting,

 For 2D motion, 𝑎 = 𝑎𝑥𝑖 + 𝑎𝑦𝑗, 𝑣𝑓 = 𝑣𝑥𝑓𝑖 + 𝑣𝑦𝑓𝑗, 𝑣=𝑖 𝑣𝑥𝑖𝑖+ 𝑣𝑦𝑖𝑗

= ∆𝑟

𝑣𝑓+𝑣𝑖 𝑣𝑓−𝑣𝑖 2 𝑎

𝑣𝑓^2 = 𝑣𝑖^2 + 2𝑎∆𝑟

Example 1 A track covers 40m in 8.5s while smoothly slowing down to a final speed of 2.8m/s. Find ; a) Its original speed b) its acceleration

Solution

We are given

that,

S = 40m , t = and vf = 2.8m/s

 a) v

i =?

2

From ∆𝑟 =

𝑣𝑓+𝑣𝑖

𝑡,

we get , vi=

2 ∆𝑟 𝑡

− 𝑣𝑓

vi =

2𝑥 40 𝑚 8. 5 𝑚

- 2.8m/s

vi = 6.6m/s

 b) 𝑎 =?

𝑎 =

∆𝑣 ∆𝑡 = 𝑣𝑓−𝑣𝑖 𝑡 = 6.6𝑚/𝑠 −2.8 𝑚 / 𝑠 8.5𝑠

𝒂 = 𝟎.

9

8.5s

2.1.3. Free Fall Motion  (^) The motion of an object near the surface of the Earth under the only control of the force of gravity is called free fall.  (^) In the absence of air resistance, all objects fall with constant acceleration, g, toward the surface of the Earth.  (^) The acceleration due to gravity varies with latitude, longitude and altitude on Earth„s surface. 2.1.4. Projectile Motion

• (^) Projectile is any object thrown obliquely into the space.
• (^) The object which is given an initial velocity and afterwards follows a path determined by the gravitational force acting on it is called projectile and the motion is called projectile motion. - (^) A stone projected at an angle, - (^) a bomb released from an aero plane, - (^) a shot fired from a gun, - (^) a shot put or javelin thrown by the athlete are examples for the projectile.

 (^) Consider a body projected from a point ' O' with velocity 'u'.  (^) The point ' o ' is called point of projection and ' u ' is called velocity of projection.  (^) Velocity of Projection (u): the velocity with which the body projected.  (^) Angle of Projection ( 𝜽 ): The angle between the direction of projection and the horizontal plane passing through the point of projection is called angle of projection.  (^) Trajectory (OAB): The path described by the projectile from the point of projection to the point where the projectile reaches the horizontal plane passing through the point of projection is called trajectory.

 The trajectory of the projectile is a parabola.

 (^) For projectile motion ay = -g ax= 0 (Because there is no force acting horizontally)

Home Activities

1. A ball is thrown with an initial velocity of 𝑢 = (10 i +15 j ̂) m/s. When it

reaches

the top of its trajectory, neglecting air resistance, what is its

a) velocity? b) Acceleration?

2. An astronaut on a strange planet can jump a maximum horizontal distance of

15m if his initial speed is 3m/s. What is the free fall acceleration on the planet?

2.2. Particle Dynamics and Planetary Motion 2.2.1. The Concept of Force as A Measure of Interaction  (^) In physics, any of the four basic forces; gravitational , electromagnetic , strong nuclear and weak forces govern how particles interact.  (^) All other forces of nature can be traced to these fundamental interactions. The fundamental interactions are characterized on the basis of the following four criteria: o (^) The types of particles that experience the force, o (^) The relative strength of the force, o (^) The range over which the force is effective, and o (^) The nature of the particles that mediate the force.

2.2.2. Types of Forces

 (^) Forces are usually categorized as contact and non-contact. i) Contact Force  (^) It is a type of force that requires bodily contact with another object. And it is further divided into the following.

1. Muscular force ; exists only when it is in contact with an object. 2. Frictional Forces ; is the resisting force that exists when an object is moved or move on a surface. 3. Normal Force ; 4. Applied Force ; is a force that is applied to a person or object. 5. Tension Force ; Tension is the force applied by a fully stretched cable or wire on to an object. 6. Spring Force ; is Force exerted by a compressed or stretched spring. tries to anchored 7. Air Resisting Force ; is wherein objects experience a frictional force when moving through the air.

ii) Non-Contact Force

 (^) It is a type of force that does not require a physical contact with the other object.  (^) It is further divided into the following types of forces:

1. Gravitational Force ; Gravitational force is an attractive force that can be defined by Newton„s law of gravity. 𝐀 𝐀

𝑮 𝒎 𝟏

𝒎 𝟐

𝑭 = 𝒓 𝟐

. It is a force exerted by large bodies such as planets and stars. 2. Magnetic Force ; The types of forces exerted by a magnet on magnetic objects. 3. Electrostatic Force ; The types of forces exerted by all electrically charged bodies on another charged bodies in the universe.

2.2.3. Newton’s Laws of Motion and Application  (^) Laws of motions are formulated for the first time by English physicist Sir Isaac Newton in

 (^) Newton„s laws continue to give an accurate account of nature, except for very small bodies such as electrons or for bodies moving close to the speed of light.

Newton’s First law of Motion :

 (^) “Everybody continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it.”  (^) This is sometimes called the Law of Inertia.  (^) Essentially, it makes the following two points:

• (^) An object that is not in motion will not move until a force acting upon it.
• (^) An object in constant motion will not change its velocity until a force acts upon it.

Newton's Second law of Motion :

 (^) The acceleration acquired by a point particle is directly proportional to the net force acting on the particle and inversely proportional to its mass and the acceleration is always in the direction of the net force.  (^) Mathematically, Σ F = m a

Newton's Third law of Motion:

 (^) States that ; “For every action there is always an equal and opposite reaction.” FBA = - FAB or FAB + FBA = 0

Note that:

 Action and reaction forces are always exist in pair.

 A single isolated force cannot exist.

 Action and reaction forces act on different objects.

 (^) Frictional force refers to the force generated by two surfaces that are in contact and either at rest or slide against each other.  (^) These forces are mainly affected by the surface texture and amount of force impelling them together.  (^) The angle and position of the object affect the amount of frictional force.  (^) If an object is placed on a horizontal surface against another object, then the frictional force will be equal to the weight of the object.  (^) If an object is pushed against the surface, then the frictional force will be increased and becomes more than the weight of the object.  (^) Generally friction force is always proportional to the normal force between the two interacting surfaces.  (^) Mathematically ; Ffrict ∝ Fnorm Ff = μFN ……………………………………………….(2.2.2) Where ; μ- is the coefficient of friction. Forces of Friction

 (^) frictional forces have two types; a. Static friction : exists between two stationary objects in contact to each other. Mathematically ; 𝑓𝑠= 𝜇 𝑠𝑁 b. Kinetic friction : arises when the object is in motion on the surface. 𝑓𝑘 = 𝜇𝑘𝑁 Where 𝜇𝑘 - is called the coefficient of kinetic friction. The values of 𝜇𝑘 and 𝜇𝑠depend on the nature of the surfaces, but 𝝁 (^) 𝒌 is generally less than 𝝁𝒔****.  (^) The coefficients of friction are nearly independent of the area of contact between the surfaces. Example A 25.0-kg block is initially at rest on a horizontal surface. A horizontal force of 75.0 N is required to set the block in motion. After it is in motion, a horizontal force of 60.0 N is required to keep the block moving with constant speed. Find the coefficients of static and kinetic friction from this information.

Solution

Application of Newton’s Laws of Motion  (^) In this section we apply Newton„s laws to objects that are either in equilibrium or accelerating along a straight line under the action of constant external forces.  (^) The following procedure is recommended when dealing with problems involving Newton„s laws:

1. Draw a sketch of the situation.
2. 2. Consider only one object (at a time), and draw a free-body diagram for that body, showing all the forces acting on that body.
3. Newton's second law involves vectors, and it is usually important to resolve vectors into components.
4. For each body, Newton's second law can be applied to the x and y components separately.
5. Solve the equation or equations for the unknown(s).

Example

A bag of cement of weight 300 N hangs from three ropes as shown in the figure below.

Two of the ropes make angles of 𝜃 1 = 53.0° and 𝜇 2 = 37.0° with the horizontal. If the

system is in equilibrium, find the tensions T 1 , T 2 , and T 3 in the ropes.