1206EL - Concepts in Physics

Monday - September 14th, 2009

Notes• Please review math rules on webpage

• Components and unit vectors as well as scalar (dot) product (you will need them for the assignment)

• Please find all notes and the first assignment on the following webpage:

• http://www.sno.phy.queensu.ca/~tine

• Next 4 lectures will be given by Dr. Chris Jillings

more notes

• First assignment posted today

• It is due on Wednesday, September 23rd

• Please bring to box before my office (511)

• There will be 8 assignments in total, each one is worth 25 points

• You will need 100 points to be be accepted for final exam

• Please mark your name clearly on your solutions as well as the exercise number.

Kinematics• Kinematics is the study of movements

• We will start with simple examples today and then make is more complicated over the next few lectures

Basics 1. Time..The absolute measure of the orderly succession of events. Unit of time

is the second s. 2. Particle..a body is assumed with negligible dimensions 3. Rigid body..A body with all internal points fixed relative to each other

Important parameters and their units:

• d = distance travelled in time -(m) • v = average velocity - (m/s) • a = acceleration - (m/s2) • t = time - (s) • θ = Rotation angle - (radians)

Particle Kinematics.

This discipline of mechanics deals with the displacement of particles over time without reference to the forces that cause the motion, velocity and acceleration of the particle.

Linear motion - definition

Linear motion is the most basic of all motions. Uninterrupted objects will continue to move in a straight line indefinitely. Under every day circumstances gravity and friction conspire to bring objects to rest.

Linear motion is measured in two parts. Speed, and direction. Together these make up the velocity.

Remember these things?

Remember these things?

y = mx + b

• This equation is called the slope-intercept form for a line. • The graph of this equation is a straight line. • The slope of the line is m. • The line crosses the y-axis at b. • The point where the line crosses the y-axis is called the y-intercept. • The x, y coordinates for the y-intercept are (0, b).

linear motion

Considering a particle moving along a path in space from position A to position B over a time interval. The position vector r locates the particle relative to the reference frame (say cartesian xyz) . The distance the particle moves along the path we call d. The change in position of the particle over the time is its linear displacement and is identified by the vector ( r B - r A )

The speed of the particle over a time period = (d 2 - d 1 ) /(t 2 - t 1 ) = the speed of the particle

As the time interval reduces to zero the change in position Δ r /Δ t = ( rB - rA ) / (t2 - t1 ) => v = the instantaneous velocity The instantaneous velocity is tangential to the particle path

The acceleration is the change of velocity over time and we call it a.

velocity v can be expressed as: v = u + at , where u is the initial velocity

Your turn1. A charged particle in an accelerator starts from rest, accelerates at 1.5 ms-2 for 3 s and then continues at a steady speed for a further 6 s.

Draw the v-t graph and find the total distance travelled.

Your turn1. A charged particle in an accelerator starts from rest, accelerates at 1.5 ms-2 for 3 s and then continues at a steady speed for a further 6 s.

Draw the v-t graph and find the total distance travelled.

a

b

h

A speedboat starts from rest, accelerating at 2 ms-2 for 20 s. It then continues at a steady speed for a further 30 s and decelerates to rest in 30 s. Find:

(a) the distance travelled in m,

(b) the average speed in ms-1 and,

(c) the time taken to cover half the distance.

and another one

A speedboat starts from rest, accelerating at 2 ms-2 for 20 s. It then continues at a steady speed for a further 30 s and decelerates to rest in 30 s. Find:

(a) the distance travelled in m,

and another one

A speedboat starts from rest, accelerating at 2 ms-2 for 20 s. It then continues at a steady speed for a further 30 s and decelerates to rest in 30 s. Find:

(b) the average speed in ms-1 and,

and another one

A speedboat starts from rest, accelerating at 2 ms-2 for 20 s. It then continues at a steady speed for a further 30 s and decelerates to rest in 30 s. Find:

(c) the time taken to cover half the distance.

and another one

Vertical free fallFree Fall is one of the most important accelerated movements and we can observe it in our daily life. All of us have dropped things before and they always make their way down. Especially in the winter we can slip and fall.

Many years ago (> 2000 y) people thought about description of motions. Most of those idea’s turned out

to be wrong by now. But having a wrong concept is very typical and can help to find the correct answers.

Aristoteles describes the cosmos and the motions therein in his books. With small changes these ideas were still valid until the 16th century - so overall about 2000 year.

Only with new methods in science, where experiments play an important and deciding role was it possible to find new answers.

Aristoteles - 384 to 322 b.c.

free fall

Galileo 1564-1642

In Galileo's attack on the Aristotelian cosmology, few details were actually new. However, his approach and his findings together provided the first coherent presentation of the science of motion. Galileo realized that, out of all the observable motions in nature, free-fall motion is the key to the understanding of all motions of all bodies. But Galileo is also in many ways typical of scientists in general. His approach to the problem of motion makes a good "case" to be used in the following sections as an opportunity to discuss strategies of inquiry that are still used in science.

Galileo says: My purpose is to set forth a very new science dealing with a very ancient subject. There is, in nature, perhaps nothing older than motion, concerning which the books written by philosophers are neither few nor small; nevertheless, I have discovered some properties of it that are worth knowing that have not hitherto been either observed or demonstrated. Some superficial observations have been made, as for instance, that the natural motion of a heavy falling body is continuously accelerated; but to just what extent this acceleration occurs has not yet been announced .... Other facts, not few in number or less worth knowing I have succeeded in proving; and, what I consider more important, there have been opened up to this vast and most excellent science, of which my work is merely the beginning, ways and means by which other minds more acute than mine will explore its remote corners.

free fall Galileo chooses a definition of uniform acceleration

Two New Sciences deals directly with the motion of freely falling bodies. In studying the following paragraphs from it, we must be alert to Galileo's overall plan. First, he discusses the mathematics of a possible, simple type of motion (which we now call uniform acceleration or constant acceleration). Then he proposes that heavy bodies actually fall in just that way. Next, on the basis of this proposal, he derives a prediction about balls rolling down an incline. Finally, he shows that experiments bear out these predictions. By Aristotelian cosmology is meant the whole interlocking set of ideas about the structure of the physical universe and the behavior of all the objects in it.

Here is what scientist all over the world still do: • -presenting a definition • -stating an assumption (or hypothesis) • -deducing predictions from the hypothesis • -experimentally testing the predictions

Galileo’s inclined experiment

This picture painted in 1841 by G. Bezzuoli, attempts to reconstruct an experiment Galileo is alleged to have made during his time as lecturer at Pisa. Off to the left and right are men of ill will: the blasé Prince Giovanni de Medici (Galileo had shown a dredging-machine invented by the prince to be unusable) and Galileo's scientific opponents. These were leading men of the universities; they are shown here bending over a book of Aristotle, where it is written in black and white that bodies of unequal weight fall with different speeds. Galileo, the tallest figure left of center in the picture, is surrounded by a group of students and followers.

Information Implied:When an object is propelled into the air, it is assumed that all other forces acting on the object except gravity are negligible. This means that:

• We neglect any effects due to air resistance on the object. • We neglect any effects due to the Earth's rotation. • We assume that the object does not rise high enough for the acceleration of gravity to

change .With these assumptions the body's acceleration is both constant and downward regardless of its direction of motion or its height above the ground. This means that object's acceleration is downwards regardless of weather the object is moving upwards or downwards,

a = -g = -9.80 m/s2 .

Frame of Reference:The freefall equation above is derived by assuming the upward y-direction is the positive direction and the clock starts at time t = 0. The sign of the acceleration is negative and it equal to a = -g .

The Law of Falling Bodies:In the absence of air resistance, any two bodies that are dropped from rest at the same moment will reach the ground at the same time regardless of their mass.

A stronger statement is: The acceleration of all objects is the same in the absence of air resistance. As long as an object is in freefall - regardless if it is going up, down or sideways - its acceleration is equal to 9.80 m/s2downwards.

Derivation of the Free-Fall Equations:Since we usually associate the vertical direction with the y-axis, we will transform the generic equations for constant acceleration by first replacing " x " with " y ", " a " with "-g ", and " v " with " vy ".

Then,

Choosing a Suitable Free Fall Frame of Reference (Axis).- A stone is dropped from a balloon that is descending at a uniform rate of 12 m/s when it is 1000 m from ground. Calculate the velocity and position of the stone after 10 s and the time it takes the stone to hit the ground.

Your turn

Choosing a Suitable Free Fall Frame of Reference (Axis).- A stone is dropped from a balloon that is descending at a uniform rate of 12 m/s when it is 1000 m from ground. Calculate the velocity and position of the stone after 10 s and the time it takes the stone to hit the ground.

Your turn

We must remember some of the kinematics formulas:

v = vi + aty = yi + vit + ½ at2

vi and yi are the initial velocity and position respectively and a is the acceleration.

The first step is to choose a suitable free fall vertical frame of reference or axis, which could be pointing upward or downward. If we choose the direction downward, the initial velocity is then +12m/s and the acceleration of gravity is taken as +g = + 9.8m/s2 If we choose the upward direction, the initial velocity is taken as -12m/s and the acceleration as - 9.8m/s2

Choosing a Suitable Free Fall Frame of Reference (Axis).- A stone is dropped from a balloon that is descending at a uniform rate of 12 m/s when it is 1000 m from ground. Calculate the velocity and position of the stone after 10 s and the time it takes the stone to hit the ground.

Your turn

it seems convenient a vertical axis pointing downward with the origin at 1000 m over ground. Of course, a different reference axis and origin could be chosen. The result will be consistent with any inertial frame.

The data is then taken as: a=+9.8 m/s2, vi=+12m/s, yi=0, t=10sv=12 m/s+9.8 m/s2·10s=110 m/sy=12 m/s·10s+½·9.8m/s2·(10s)2=610 m or 390 m from ground. The time to hit the ground is obtained doingy = 1000 m in y = yi + vit + ½ at2:1000 m = 12 m/s·t + ½ ·9,8m/s2·t2 or0 = 4,9t2 + 12t - 1000, t in secondsSolving this quadratic equation results t = 13.1 s