Physics: Conversion of Units

Welcome (1 of 2)

Introduction

Welcome to Physics 1010! Physics is the study of physical phenomena in nature. The concepts of physics such as force, motion, matter, and energy are involved in every activity. Physics can help you understand common occurrences of daily life, such as a rainbow, a spherical drop of water, lightning, and your reflection in a mirror. This course aims at conveying some of the concepts of physics to help you understand the common physical phenomena.

A core concept this course will highlight is that nature obeys some fundamental laws. It’s amazing how so many seemingly unrelated phenomena obey a common set of laws. One of the quests of physics is unification—to be able to explain diverse phenomena with the same set of laws.

Physics and math go hand in hand, and for this course, you will use algebra (learned in MTH1010 College Math) as a tool to solve quantitative problems. The lectures cover key concepts, and they are supplemented by reading assignments, practice exercises, and experiments. Online discussions give you the opportunity to exchange your views and learning with your classmates.

Physics is a vast subject and six weeks is a short time to study it. You will need to do a lot of reading from the lectures, textbooks, and Web sites. You should plan to spend about three hours a day for this course.

Welcome (2 of 2)

Weekly Overview

Physics also deals with physical quantities such as length, time, mass, speed, velocity, and their interrelationships. This week’s lectures will discuss the units and standards of physical quantities, specifically how they are measured. You will also learn to convert units from the imperial and US customary units to the metric system and vice versa.

This week will also cover the classification of physical quantities as vectors or scalars and the differences between vectors and scalars. Because of these differences, the method to add vectors is different from the method used to add scalars. You will learn how to add vectors and find their components along the direction of any coordinate axis.

Finally, you will learn about kinematics—the study of the relationship between the position, velocity, and acceleration of an object. You will analyze motion along a straight line and a curved path and calculate the velocity and distance traveled when acceleration is constant. In addition, you’ll explore how gravity affects objects that fall freely or objects that are thrown vertically upwards.

Ready to start? Let’s go!

Physical Quantities

Units and Standards

Imagine a customer going to a grocery store and asking for five sugar or one milk! The clerk will be confused because the customer hasn’t specified the units. A quantity is always specified with a unit such as pound or gallon. In the above situation, the customer could have asked for five pounds of sugar and a gallon of milk.  Note that units tell us two important pieces of information about a measurement. They tell us what type of measurement is involved, as well as its scale. For example, if given a measurement of three meters, the “meters” indicates that we are dealing with a length. It also informs us of the scale. For instance, three inches is also a length, but it is a different scale than three meters.

Before specifying or using a unit, it needs to be defined; before measuring length in feet, you must know how much one foot is. Once defined, a unit becomes a standard for all subsequent measurements.

A standard should be precise, invariable, and accessible. A widely used system of units is the international system of units, also called the SI system or metric system, which comprises seven base quantities (See the SI Base Quantities table). Each quantity has a unit for which a primary standard is carefully defined.

Measurement (1 of 2)

The Importance of Measurement

Once units are defined, we can use them to measure quantities. Measurements are important to us in many ways. Even in daily life, we frequently measure quantities. Here are some examples from daily life where we measure quantities:

· Measuring body temperature

· Checking blood pressure

· Checking the pressure inside your car tires

· Monitoring the body weight to analyze the effectiveness of a new diet and exercise regimen

· Purchasing goods on the basis of their weights or volumes

· Measuring the distance traveled by a car or the car’s velocity

Measurement is central to physics, because any new theory that is proposed must be tested by experiment, which requires careful measurements.

Conversion of Units

Often, we require converting a quantity from one unit to another unit. Look at the following examples:

Vectors (1 of 2)

Scalars and Vectors

When you measure quantities, some quantities can be described by just specifying their magnitude. For example, if you want to describe how heavy a box is, you’ll simply say that its mass is four kg. Such quantities are called scalar quantities; other examples of scalars quantities are area, volume, and temperature.

However, there are other quantities for which you need to specify the direction in addition to magnitude. For example, to describe the displacement of an object, you say that the object is displaced by three m in the northeast direction. Such quantities are called vector quantities; other examples of vector quantities are force, velocity, and acceleration.

In this course, whenever we define a new quantity, we’ll ask ourselves whether the quantity is a scalar or a vector. Classifying quantities as vectors and scalars will help you understand them better.

The Addition of Vectors

Adding vectors is not as simple as adding scalars. To add two scalar quantities you just need to add their magnitude. For example, the total mass of two objects with masses 2 kg and 3 kg is 2 kg + 3 kg = 5 kg.

To add two vectors, the direction of the quantities has to be taken into account in addition to their magnitude. For example, if you apply two forces on a body, one of 2 N and another of 3 N, the resultant force depends on the direction of the two forces.

The addition of vectors is done by the triangle rule of addition. This rule will be clear when you read the following examples.

Vectors (2 of 2)

Components of a Vector

Just as two vectors can be added up by the triangle rule, it is possible to break up a vector into two component vectors. This helps determine the effect of a quantity in a particular direction.

Of special interest are components that are perpendicular to each other. Let there be a vector A, and let’s draw X and Y coordinate axes. (Vectors are denoted in bold font and scalars in normal font.) Then, you can draw two components of vector A, one parallel to the X-axis and the other parallel to the Y-axis. As it can be seen, the two components add up to vector A.

To replace a vector with its component vectors, you need to find the magnitude and direction of the components. Let’s see how to find the components of vector A, parallel to the coordinate axes. Choose the positive direction of the X-axis as pointing to the right and the positive direction of the Y-axis as pointing upward.

When vector A is along the positive X direction, the X-component is vector A itself, and the Y-component is zero. This is because a vector cannot have a component perpendicular to itself.

When the vector makes an acute angle (less than 90°) with the positive X-axis, then the component along the X-axis is in the direction of the positive X-axis. The Y component points either along + Y or – Y, depending on whether the arrow representing A is in quadrant I or in quadrant IV.

When A is parallel to the Y-axis, the X component is zero and the Y component is the vector itself.

When A makes an obtuse angle (more than 90°) with the positive X-axis, the component points in the negative X-axis direction. The Y component may be again positive or negative depending on the direction in which the vector is pointing.

Kinematics (1 of 6)

Kinematics is the study of the relationship between the position, velocity, and acceleration of an object. Kinematics is concerned with the motion of an object, and not with the forces that cause this motion. The relation between force and motion will be covered later under dynamics.

There are various quantities that describe the motion of an object; the prime ones being displacement, velocity, and acceleration. Let’s begin our study on kinematics by first looking at displacement.

Displacement

Displacement is the change in the position of an object, and it is a vector quantity. Consider an example of displacement. Sally walks every day from home to her office, which is at a distance of two km. Sally actually walks more than two km to reach her office, because there’s no straight road connecting her home to the office. In addition, she does not take the same route every day, but her initial and final points are the same. To describe her position at the two points, an arrow OA can be drawn from the initial point to the final point. The arrow represents a vector quantity, because it has a magnitude that is at a distance of two km, and its direction is from Sally’s house to her work place. This vector is called the displacement vector.

Kinematics (2 of 6)

An object can move in a straight path or a curved path; it can also move in the horizontal or vertical direction. Let’s study about each type of motion.

Motion Along a Straight Path

A particle may sometimes move along a straight path. Examples of straight-line motion are:

· An object thrown vertically upward

· An object sliding or rolling straight down an incline

· A car traveling along a straight stretch of road

The simplest motion is when a particle moves in a straight path uniformly without any change in its speed. The distance it travels is given by distance = velocity * time or d = vt, and the acceleration of the particle is zero.

Uniform Acceleration

We know that acceleration is a measure of how quickly the velocity of an object changes. If the velocity of an object changes at a constant rate, we say that the object is accelerating uniformly.

It is possible to calculate the velocity of an object if it is accelerating uniformly. The equation vf = vf + at can be used for this purpose. Note that this equation can be solved for acceleration and gives:

E1: a = (vf – vi)/t   (acceleration is change in velocity over time)

In this equation:

· vf is the final velocity

· vi is the initial velocity

· t is the time

· a is the acceleration

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Deceleration

An object moving along a straight path with a constant acceleration can have acceleration either in the direction of motion or opposite to the direction of motion. Acceleration in a direction opposite to the direction of motion is also called deceleration. Deceleration happens when the final velocity of an object is lower than the initial velocity; for example, when you apply brakes to stop a moving car.

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Finding the Distance Traveled

How do you find the distance that the car travels when it is accelerating? When velocity is constant, the distance traveled is d = vt.

When the car is accelerating, the velocity of the car is continuously changing and you cannot use this formula. Instead, you can take the average of the initial and final velocities.

E3: vavg = (1/2)(vf + vi)

Thus, the distance traveled can be expressed as:

E4: d = vavgt = (1/2)(vf + vi)t

However, sometimes it’s best to express distance in terms of initial velocity and acceleration. To find this expression, replace the vf with vf = at + vi.

d = (1/2)(vf + vi)t
d = (1/2) (at + vi + vi)t
d = (1/2) (at +2vi)t
d = (1/2)at2 + vit

Expressed this way, the first term, (1/2)at2 tells you the distance traveled due to the acceleration, and the second term, vit tells you the distance traveled due to the initial velocity. Adding the two terms gives the total distance traveled.

Kinematics (6 of 6)

Motion Along a Curved Path

Let’s now look into the motion of an object in a curved path, which is the most general case of an object in motion. For simplicity’s sake, we consider the motion of an object only in a single plane. An object moving along a curve always has acceleration, because the velocity of the particle changes in direction even if it is traveling at a constant speed.

At any given instant, the velocity of the particle is along the tangent to the curve. (The tangent at a point to a curve is a line that touches the curve at that point and nowhere else.) A roller coaster is an example of a particle moving in a curved path. Even if the roller coaster is moving with constant speed, its velocity is changing in direction, and therefore, it is accelerating. The acceleration is directed perpendicular to the tangent towards the center of the curve. If the path is a circle of radius r, the acceleration is towards the center of the circle. This type of acceleration is called centripetal acceleration, and is given by ac = v2/R .

Summary

This brings us to the end of the first week of the course. The lectures covered basic concepts of science such as units, standards, and the importance of measurements. You learned to classify quantities as vectors and scalars. The week also covered the concepts of kinematics, which explains the motion of objects in straight and curved paths. In the lecture on kinematics, you learned about displacement, velocity, and acceleration.