PHYSICS -Mechanics- Kinematics: (linear, projectile and simple harmonic motion)

           
MECHANICS: KINEMATICS

Mechanics is the study of motion of a particle or body, the force causing it and the energy possessed by the body. Motion is the change in the position of a body with respect to time and its surroundings.

Motion is basically classified as either a non-periodic motion or a periodic motion. Non-periodic motions are not a repetitive motion. Periodic motion is a repetitive motion that repeats itself after a fixed interval of time known as period.

The study of motion is categorized into four parts for this lecture and they are;

KINEMATICS
DYNAMICS
MECHANICAL ENERGY
FLUID MECHANICS

We will be discussing KINEMATICS in this articket.

 Kinematics is the study of motion without a regard to the force causing the motion. We will be studying kinematics under four headings;

■Linear horizontal motion
■Linear motion under gravity
■Projectile motion
■ Simple Harmonic Motion (Periodic motion)

Linear horizontal motion under
This is the motion of a body from a point to another without being affected by the gravity. A good example is a train moving from a station to another or the motion of a car between two bus terminals.
Terms used in describing this kind of motion include;

the initial speed of the body u,   at time t=0,                                                                                      the final velocity v after time t,                                                                                                          the distance covered S between time t=0 and time t.

The relationships between these four parameters are included in equations generally known as the four equations of motion which you have to know by heart and they are;

The same four equations of motion also apply here with little alteration such as replacing  a with g and also distance  is replaced with height.

Motion-Time Graph

There are basically two types of motion-time graph;

Distance-Time Graph: The slope of distant-time graph represents the average speed.

Velocity-Time Graph:

From the diagram above;

■ AB represents a motion with uniform acceleration.                                                                          
 ■ BC represent a motion whose acceleration has been reduced.                                                            
 ■ CD represents the motion of a body decelerating.                                                                        
 ■ DE represents a motion with constant velocity.                                                                                            
 ■ EO represents the motion of a body decelerating.

 The area under Velocity-Time Graph is the total distance covered and the slope of Velocity-Time Graph represents acceleration.


PROJECTILE MOTION (2-DM MOTION )

This is a non period motion of a body or particle but in two dimensions (combination of linear horizontal motion  component and linear motion under gravity, component) from one fixed point to another moving freely in space in a parabolic manner.

The only parameter introduced here is the angle of projection of the projectile.

We are going to be looking into three cases here.


•Case1: When a body is projected horizontally from a height.

•Case2: When a body is projected at an angle from a horizontal level.

The derivations are shown below

Case3:  When a body is projected at an angle from a height.

The derivations are shown below

SIMPLE HARMONIC MOTION (SHM)

This is the periodic motion of a body about a fixed whose acceleration which is directed toward the fixed point is proportional to the displacement from that point.

A period motion is a repetitive motion of a body that repeat itself after a time interval known as period T. Examples of SHM include the motion of the bob of a simple pendulum, the movement of a mass attached to a spiral spring, the motion of a loaded test tube vertically in a liquid, child's swing in motion, balance wheel of wrist watches, the motion of a sea saw etc.

More parameters are introduced in this kind of motion due to the angular displacement involved in this motion which this gives rise to one of the most important parameters in SHM, and that is the angular frequency  or you call it angular velocity.

This is the constant that surface from the definition of SHM. For different kinds of SHM, if the angular frequency is known, then the value of other parameters can be easily determined.

Therefore, what is peculiar to each motion is the formula for its angular frequency.

The displacement after a period is known as wavelength  while the maximum displacement that the body can attain is called the Amplitude A.

5. Acceleration: The linear acceleration is the change in linear velocity per seconds angular acceleration is the rate of change of angular velocity.

Velocity and Acceleration in SHM from Circular Motion.

Consider a body moving uniformly along a circular path, it can be shown from the diagram that the displacement y at time t and the maximum displacement  are related by the formula,    

Examples of SHM

Energy Stored In SHM

The Energy possessed by a body undergoing a SHM is kinetic energy which is;

Damping, Force Vibration and Resonance

According to Newton’s first law of motion, for a body undergoing a simple harmonic motion, the motion is meant to continue without the body involve coming to rest.

 But the body performing this motion later comes to rest if left to oscillate freely, and this is due to the decrease in energy as the amplitude of oscillation is decreasing due to the friction in the system movement and the viscous drag in air.

This kind of motion is known as damped motion. A free oscillation without damping is only theoretical unless there is something that compensates it.

In other to maintain this oscillation without damping, a restoring force (periodic force to keep the body oscillating) will be needed to restore energy constantly to the system to keep the amplitude constant.

The vibration caused by this periodic force is known as a forced vibration and the system will be compelled to oscillate at a frequency known as forced vibration frequency.

Every objects has their own natural frequency (frequency at which they will oscillate when left freely to vibrate) of vibration, if the frequency of the oscillation caused by the external force coincides with the natural frequency of the body performing the SHM, the amplitude of the system will be amplified and this phenomena is known as Resonance.