Applied Physics 101Lecture # 02

MechanicsPhysical sciences that deals with the study of the motion of object is called MechanicsMECHANICSDynamicsKinematicsDescription of the motion of objects without reference to force, e.g. motion of particle in straight line such as falling stone, an accelerating train etc..Description of the motion of objects with reference to some forces, that produce and/or change the motion of objectsConsidering a complex object as a single particle will allow us to ignore the possible internal motions such as rotary motion of object or vibrations of parts.

Description of motionMotion of objectsGraphical representation Mathematical equations There are two methods for the study of kinematics of particles, both will be discussed Better for solving problems More precise than mathematical equations Provides more physical insight

Concentrated mass at some position in space is called particle and can be described by its position (r) with some reference point and its mass (m). Complete description of the motion of a particle can be obtained from the mathematical dependence of its position (x) [with respect to the origin (O) of some frame of reference] on time (t).

This means that motion and/or position of the particle is the function of time, i.e. X (t).

*Particle at rest (No motion at all): When a particle occupy a position and doesn't change it with respect to time is said to be in rest. All the time particle remains at the same positionMathematical representation of rest position X (t) = A .(1)Note: Here x, is dependent variable while t, is independent variableDescription of motion

Motion at constant speed (Velocity): The rate of change of position of a particle is called velocity, it will be positive if the motion is in the direction of increasing, x or will be negative when moving in the opposite direction. In case of constant motion, plating position against time will give a straight line with a constant slope.Graphical representation of motion The slope of any function tell us about the rate of change, Here the slope of position means rate of change of position, and is called velocity. Thus the constant motion of particle can be mathematically represented by equation of st- line such as, x (t) = A + Bt .(2)Remind you equ. of straight line y = mx + b

AccelerationThe rate of change of velocity (in direction and magnitude) is called acceleration. Thus here in the graph the slope is also changing.(So acceleration can be define as the rate of change of slope). These graph are curve rather than a st- lineThus mathematically we can illustrate the graph i.e. the acceleration of particle as x (t) = A + Bt +Ct2 ..(3)Note: Single derivatives of position with respect to time gives velocity and double derivatives of position with respect to time gives acceleration of the particle

Average velocityWhen the motion of particle is changing either with respect to direction or magnitude, it is convenient to describe it with average velocity (v).Consider a particle at time t1 is at x1 and at time t2 is at x2, The average velocity over the interval of time can be define as is the change in the position and is called the displacement that occurs during the time interval,Where:The average velocity (v) is simply the slope the straight line that connects the end points of the intervals, where the actual behavior between X1 and X2 are of no concerns

Problem 1You drive a car down a straight road for 5.2 mi @ 43 mi/hr, at which point you run out of gas. You walk 1.2 mi farther to the nearest gas station in 27 min. What is your average velocity from the time you start yours car to the time you arrived to the gas station?Solution:The net distance covered 5.2 mi +1.2 mi = 6.4 mi x =The net time ist = t1 + t2 = 34.3min = 0.75hrWhere;Thus from Equation 4, the average velocity is vx

Instantaneous velocityThe average velocity of a particle describes the overall behavior during some interval of time but it will not helpful in determining the details of its motion. In order to know the details of motion of particle at every instant of time we use the instantaneous velocity, which is the motion of particle during a small instant of time.When the interval of time gets smaller and smaller such that in the limiting case t tends to zero, then the line connecting the end points of the interval approaches to the tangents to the x (t) curve at a point and the average velocity comes close to the slope of x (t), which defines the instantaneous velocity v.Equation 5 can be written in the derivative form asThus instantaneous velocity is just the rate of change of position with time at a small interval

Making the time smaller and smaller by moving the ends of time t2 closer to t1, such that the interval tends to zero and the chard become a tangent at the smaller instant of time.

Accelerated motionThe velocity of a particle can be change with respect to time as the particle continue its motion, this change in the velocity may be due to change in speed or due to change in the direction of motion, in either case the changing velocity is called acceleration. The average acceleration can be given asSimilar to the case of average velocity , the average acceleration (a) tells us nothing about the variation of v (t) with time t during the small interval of time t.Constant acceleration: When the change in the velocity is the same in the equal interval of time, then it is called constant acceleration, e.g. acceleration due to earth gravity (g = 9.8m/s) is constant near the earth surface

Variable acceleration: When the change in velocity (v) is not the same for equal interval of time then we have the case of variable acceleration.Instantaneous acceleration: The change in the velocity of a particle at a small instant of time such that t tends to zero is called instantaneous acceleration and is given as It must be noted that acceleration may be positive or negative independent of whether velocity is positive or negative. We can have a positive acceleration with a negative velocity, because acceleration is the change in the velocity. Further when the acceleration and velocity have opposite signs, so that the speed of particle is decreasing we called this deceleration.

The acceleration defined by equation 8 is just the slope of v(t) graph. If v(t) is a constant then a = 0 (figure 1); if v(t) is a straight line, then acceleration is constant (figure 2) and is equal to the slope of st- line; if v(t) is a curve (figure 3); then acceleration will be some function of t, and can be obtained by the derivative of v(t).vNo acceleration

Motion with constant accelerationObjects falling near the earth surface and braking cars are some typical examples of constant acceleration, while swinging pendulum ball, and rain drops falling against air resistance are examples of motion with variable acceleration. We will drive some equations concerning the motion of particles with constant acceleration.Let an object starts its motion with initial velocity vo at t = 0 and later some interval of time t it has velocity v, acceleration of this particle can be give as v = vo + at (9)This equation gives velocity as a function of time, this equation is in the form of st- line equation y = mx + b, which describe the graph of a st- line, here a is the slope and vo is the intercept (value of v at t = 0)

If the plot of v against t is a straight line then the average velocity occurs midway through the interval and is equal to the average of the two endpoints at time t = 0 and time t, i.e.Using equation 9 here we will getNow the average velocity can be given as follows; assuming that particle moves from position xo at t = 0Thus from equation 11 and 12 we can obtainx = xo + vot + at2 (13)

We can find the position of particle at all subsequent time if we know the values of acceleration a and the initial conditions i.e. the position xo and the velocity vo at time t = 0The net distance x xo is called displacement and we chose xo = 0 for convenience.There are four variables (x, v, a, and t) and two initial conditions xo and vo, In order to find time independent equation we can use equations 9 and 13 and deduce the following equation which is independent of timev2 = vo2 + 2a (x xo) (14)

Set of Equations that can be used for motion with constant accelerationv = vo +at (9) v2 = vo2 + 2a (x xo) (14) x = xo + vot + 1/2at2 (13) x = xo +1/2 (vo + v )t (15) x = xo + vt -1/2at2 (16) The above set of equations can be simplified further by choosing xo = 0 to be the origin of co ordinate system. You can choose either direction of the co ordinate axis to be positive. All the quantities i.e. velocity, acceleration and displacements must be positive in that direction and will be negative in the opposite direction.o-y+y-x+x

Summery of the Kinamaticsgiven a constant acceleration we can integrate to get explicit v and axavttt

Motion of free falling bodiesThe most common example of motion with constant acceleration is the motion of body falling toward the earth. For this we consider motion of body in vacuum and with no air resistance that may affect the motion.All bodies despite of their shape, size and composition fall with the same acceleration at the same point near the earth surface. This acceleration is called free fall acceleration or acceleration due to gravity and is given by g = 9.8 m/s.If the distance of fall is small as compare to the earth radius 6400km then the acceleration due to gravity i.e. g is constant near the earth surface.The direction of acceleration due to gravity is always down whether the velocity of the particle is up or downward. The set of equations (from 9 to 16) can be used for free falling bodies as well with the following considerations.

Replace a with g consider the free fall as the y-axis Take the upward direction to be the positive Y- axisReplace a with g in the above set of equations because acceleration due to gravity is always downward and that direction we have chosen as y axis.v = vo - gt .(17) y = yo + vot - 1/2gt2 .(18)

Chose the starting point as origin, andy = - 1/2gt2, y = -1/2(9.8m/s2)(1.0s)using equation 17, 18 v = vo - gtPut vo = 0, t = 1s to find the positiony = -4.9m/sTo find the velocity put vo = 0 v = 0 9.8m/s2(1.0s)Vo = -9.8m/sy = yo + vot - 1/2gt2 Sample problem 5A body is dropped from rest and falls freely. Determine the position and velocity of the body after 1.0, 2.0, 3.0 and 4.0 s have elapsed ?

tYvasmm/sm/s20009.81.o-4.9-9.8-9.82.0-19.6-19.6-9.83.0-44.1-29.4-9.84.0-78.4-39.2-9.8