Physics Measurement PDF

1.0 MEASUREMENT Physics is a science based upon exact measurement of physical quantities. Therefore it is essential that student first becomes familiar with the various methods of measurement and the units in which these measurements are expressed. A unit is a value quantity or magnitude in term...

1.0 MEASUREMENT Physics is a science based upon exact measurement of physical quantities. Therefore it is essential that student first becomes familiar with the various methods of measurement and the units in which these measurements are expressed. A unit is a value quantity or magnitude in terms of which other values, quantities or magnitudes are expressed. 1.1 FUNDAMENTAL QUANTITIES AND UNITS A fundamental quantity also known as base quantity is a quantity which cannot be expressed in terms of any other physical quantity. The units in which the fundamental quantities are measured are called fundamental units. In mechanics (study of the effects of external forces on bodies at rest or in motion), the quantities length, mass and time are chosen as fundamental quantities. Fundamental Quantity Fundamental Unit Unit Symbol Length Meter m Mass Second s Time Kilogram kg 1.2 SYSTEM OF UNITS The following systems of units have been in use – (i) The French or C.G.S (Centimeter, Gramme, Second) System; (ii) The British or F.P.S (Foot, Pound, Secons) System; (iii) The M.K.S (Metre, Kilogram, Second) System; and (iv) The S.I. (International System of Units). 1.3 THE INTERNATIONAL SYSTEM OF UNITS (S.I.) The S.I. is the latest version of the system of units and only system likely to be used all over the world. This system consists of seven base or fundamental units from which we can derive other possible quantities of science. They are S.No Physical Quantity Unit Unit Symbol 1. Length Meter M 2. Mass Kilogram Kg 3. Time Second S Page 2 of 40  4. Electric Current Ampere A 5. Temperature Kelvin K 6. Amount of Mole Mol Substance 7. Luminous Intensity Candela Cd 1.4 CONCEPT OF DIMENSION Dimension of a physical quantity simply indicates the physical quantities which appear in that quantity and gives absolutely no idea about the magnitude of the quantity. In mechanics the length, mass and time are taken as the three base dimensions and are expressed by as letter [L], [M] and [T] respectively. Hence, a formula which indicates the relation between the derived unit and the fundamental units is called dimensional formula. Example 1: Deduce the dimensional formula for the following physical quantities: (a) Velocity, (b) acceleration, (c) force, (d) pressure, (e) work, and (f) power. Solution: [ ] ( ) [ ] [ ] [ ] [ ] ( ) [ ] [ ] [ ] ( ) [ ] [ ] [ ] [ ] ( ) [ ] [ ] Students to attempt ( ) and( ). Example 2: Deduce the dimensional formula for (a) modulus of elasticity ( ), and (b) coefficient of viscosity( ). Page 3 of 40 Solution: ⁄ ( ) ⁄ [ ] [ ] [ ] [ ] [ ] ( ) ( ) ( ⁄ ) [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1.4 USES OF DIMENSIONAL EQUATIONS (a) To check the homogeneity of a derived physical equation (i.e. to check the correctness of a physical equation). Example 3: Show that the following relation for the time period of a body executing simple harmonic motion is correct. √ where and are the displacement and acceleration due to gravity respectively. Solution: For the relation to be correct, the dimension of L.H.S must be equal to dimension of R.H.S [ ] ⁄ [ ] ⁄ [ ] ⁄ [ ] ⁄ [ ] ⁄ [ ]⁄ Page 4 of 40  Since , the relation is dimensionally correct. (b) To derive a relationship between different physical quantities. Example 4: If the frequency of a stretched string depends upon the length of the string, the tension in the string and the mass per unit length of the string. Establish a relation for the frequency using the concept of dimension. Solution: ( ) ( ) [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ⁄ ⁄ √ ( ) Equation ( ) is the required relation. (c) To derive the unit of a Physical Quantity Page 5 of 40  Example 5: The viscous drag F between two layers of liquid with surface area of contact A in a region of velocity gradient ⁄ is given by ⁄ where is the coefficient of viscosity of the liquid. Obtain the unit for. Solution: ⁄ ⁄ [ ] [ ] [ ] [ ] [ ][ ] [ ][ ] 1.5 LIMITATIONS OF DIMENSIONAL ANALYSIS (i) The method does not provide any information about the magnitude of dimensionless variables and dimensionless constants. (ii) The method cannot be used if the quantities depend upon more than three-dimensional quantities: M,L and T. (iii) The method is not applicable if the relationship involves trigonometric, exponential and logarithmic functions. EXERCISE 1) Show on the basis of dimensional analysis that the following relations are correct: a. , where is the initial velocity, is final velocity, is acceleration of the body and is the distance moved. b. ⁄ where is the density of earth, is the gravitational constant, is the radius of the earth and is acceleration due to gravity. 2) The speed of sound in a medium depends on its wavelength , the young modulus , and the density , of the medium. Use the method of dimensional analysis to derive a formula for the speed of sound in a medium. (Unit for Young Modulus : ) Page 6 of 40 2.0 VECTORS Physical quantities can generally be classified as (i) Scalars and (ii) Vectors. Scalars are physical quantities which possess only magnitude and no direction in space. Examples are mass, time, temperature, volume, speed etc. On the other hand, Vectors are physical quantities which have both magnitude and direction in space. Examples are force, velocity, acceleration, etc. 2.1 RESOLUTION OF VECTORS A two dimensional vector can be represented as the sum of two vectors. Consider figure A above, the vector ⃗ can be expressed as ⃗ ⃗ ⃗ ⃗ ̂ ̂ Vectors ⃗ and ⃗ are called vector components of ⃗. ̂ and ̂ are unit vectors along the axis and axis respectively. A unit vector is a vector that has a magnitude of exactly 1 and specify a particular direction. Let be the angle which the vector ⃗ makes with the positive -axis, then we have | ⃗| √ In three dimensions, a vector ⃗ can be expressed as Page 7 of 40  ⃗ ⃗ ⃗ ⃗ ⃗ ̂ ̂ ̂ Here, the magnitude is expressed as | ⃗| √ 2.2 ADDITION/SUBTRACTION OF VECTORS BY COMPONENTS Example: If ⃗ ̂ ̂ ̂ and ⃗⃗ ̂ ̂ ̂. Find the magnitude of ( ⃗ ⃗⃗) and ( ⃗ ⃗⃗). Solution: ( ⃗ ⃗⃗) ( ̂ ̂ ̂) ( ̂ ̂ ̂) ̂ ̂ ̂ ( ⃗ ⃗⃗) ( ̂ ̂ ̂) ( ̂ ̂ ̂) ̂ ̂ ̂ Hence their magnitudes will be | ⃗ ⃗⃗| √( ) ( ) ( ) √ | ⃗ ⃗⃗| √( ) ( ) ( ) √ 2.3 MULTIPLICATION OF VECTORS (i) Dot Product or Scalar Product The scalar product of two vectors is defined as the product of the magnitude of two vectors and the cosine of the smaller angle between them. ⃗ ⃗⃗ | ⃗|| ⃗⃗| where is a scalar quantity. Example: Find the angle between the two vectors ⃗ ̂ ̂ ̂ and ⃗⃗ ̂ ̂ ̂. Solution: ⃗ ⃗⃗ | ⃗|| ⃗⃗| ⃗ ⃗⃗ ( ) | ⃗| √ √ | ⃗⃗| √ ( ) √ ⃗ ⃗⃗ | ⃗|| ⃗⃗| √ √ Page 8 of 40  (ii) Cross Product or Vector product The vector product of two vectors is defined as a vector having a magnitude equal to the product of the magnitudes of the two vectors and the sine of the angle between them and is in the direction perpendicular to the plane containing the two vectors. Thus if ⃗ and ⃗⃗ are two vectors, then their vector product (or cross product), written as ⃗ ⃗⃗, is a vector ⃗ defined as ⃗ ⃗ ⃗⃗ | ⃗|| ⃗⃗| where is the angle between the two vectors and is the unit vector perpendicular to the plane of vectors ⃗ and ⃗⃗. 3.0 KINEMATICS Kinematics is the study of the motion of objects without referring to what causes the motion. Motion is a change in position in a time interval. 3.1 MOTION IN ONE DIMENSION/MOTION ALONG A STRAIGHT LINE A straight line motion or one-dimensional motion could either be vertical (like that of a falling body), horizontal, or slanted, but it must be straight. 3.1.1 POSITION AND DISPLACEMENT The position of an object in space is its location relative to some reference point, often the origin (or zero point). For example, a particle might be located at , which means the position of the particle is in positive direction from the origin. Meanwhile, the displacement is the change from one position to another position. It is given as ( ) 3.1.2 AVERAGE VELOCITY AND AVERAGE SPEED The average velocity is the ratio of the displacement tht occurs during a particular time interval to that interval. It is a vector quantity and is expressed as: ( ) ( ) Page 9 of 40 The average speed is the ratio of the total distance covered by a particle to the time. It is a scalar and is expressed as: ( ) 3.1.3 INSTANTANEOUS VELOCITY AND SPEED The instantaneous velocity describes how fast a particle is moving at a given instant. It is expressed as: ( ) Speed is the magnitude of instantaneous velocity; that is, speed is velocity that has no indication of direction either in words or via an algebraic sign. For example, a velocity of or is associated with a speed of. Example 6: The position of a particle moving on an axis is given by , with in meters and in seconds. What is its velocity at ? Is the velocity constant, or is it continuously changing? Solution: ( ) ( )( ) ( ) At , ( ) ( ) ( ) 3.1.4 ACCELERATION When a particle’s velocity changes, the particle is said to undergo acceleration. The average acceleration over a time interval is where the particle has velocity at time and then velocity at time. The instantaneous acceleration (or simply acceleration) is the derivative of velocity with respect to time: Page 10 of 40 Equation can also be written as ( ) ( ) In words, the acceleration of a particle at any instant is the second derivative of its position ( ) with respect to time. Example 7: A particle’s position on the -axis is given by , with in meters and in seconds. (a) find the particle’s velocity function ( ) and acceleration ( ). (b) Is there ever a time when ? Solution: ( ) ( ) ( ) ( ) ⁄ 3.2 MOTION IN TWO AND THREE DIMENSIONS The concept of position, velocity and acceleration for motion along a straight line is similar to that two and three dimension, but more complex. 3.2.1 Position and Displacement The position vector ⃗ is used to specify the position of a particle in space. It is generally expressed in the unit vector notation as ⃗ ̂ ̂ ̂ ( ) where ̂ ̂ and ̂ are the vector components of ⃗, and the coefficients and are its scalar components. Page 11 of 40  ⃗ If a particle moves from point 1 to point 2 during a certain time interval, the position vector changes from ⃗ to ⃗ , then the particle’s displacement is ⃗ ⃗ ⃗ ( ) Example 8: The position vector of a particle is initially ⃗ ( ) ̂ ( ) ̂ ( ) ̂ and then later is ⃗ ( ) ̂ ( ) ̂ ( ) ̂. What is the particle’s displacement ⃗ from ⃗ to ⃗ ? Solution: ⃗ ⃗ ⃗ [ ( )] ̂ [ ]̂ [ ]̂ ⃗ ( )̂ ( )̂ Example 9: A rabbit runs across a parking lot. The coordinates of the rabbit as a function of time are: and with in seconds and and in meters. At , what is the rabbit’s position vector ⃗ in unit vector notation and as a magnitude and an angle? Solution: ⃗ ̂ ̂ At ( ) ( ) ( ) ( ) ⃗ ( )̂ ( )̂ The magnitude of the position vector ⃗ is √ √( ) ( ) Page 12 of 40 The angle of ⃗ is ( ) 3.2.1 Average Velocity and Instantaneous Velocity If a particle moves through a displacement ⃗ in a time interval , then its average velocity ⃗ is ⃗ ̂ ̂ ̂ ⃗ ̂ ̂ ̂ When we speak of the velocity of a particle, we usually mean the particle’s instantaneous velocity ⃗⃗. It is expressed as ⃗ ⃗ ̂ ̂ ̂ ̂ ̂ ̂ Example 10: Find the velocity ⃗ at time of the rabbit in example 9 in unit vector notation and as a magnitude and angle. Solution: The components of the velocity are and ( ) ( ) ( ) ( ) ( ) ( ) ⃗ ̂ ̂ ⃗ ( )̂ ( )̂ ⃗ √ √( ) ( ) ( ) ( ) 3.2.2 Average Acceleration and Instantaneous Acceleration The average acceleration is defined as the change in velocity from ⃗ to ⃗ in a time interval. It is mathematically expressed as ⃗ ⃗ ⃗ ⃗ Page 13 of 40 The instantaneous acceleration ⃗ (or acceleration) is the limit of average velocity as approaches zero. It is expressed mathematically as ⃗ ⃗ ⃗ We note that ⃗ ⃗ ( ⃗) ( ̂ ̂ ̂) ⃗ ̂ ̂ ̂ ⃗ ̂ ̂ ̂ 3.2.3 Constant Acceleration In many types of motion, the acceleration is either constant or approximately so. The following equations describe the motion of a particle with constant acceleration: s/n Equation 1. 2. 3. 4. ( ) 5. Note that these equations are only applicable if acceleration is constant. EXERCISE: A driver spotted a police car and he braked from a speed of to a speed of during a displacement of , at a constant acceleration. ( ) ( ) Example 11: Find the acceleration ⃗ at time of the rabbit in example 10 in unit vector notation and as a magnitude and angle. Solution: ( ) ( ) Page 14 of 40  ( ) ( ) ⃗ ( )̂ ( )̂ ⃗ √( ) ( ) ( ) EXERCISE A particle with velocity ⃗ ̂ ̂ in meters per second at undergoes a constant acceleration ⃗ of magnitude at an angle from the positive direction of the axis. What is the particle’s velocity ⃗ at , in unit vector notation and as a magnitude and an angle? ⃗ ̂ ̂( ) 3.3 PROJECTILE MOTION A projectile is a particle which moves in a vertical plane with some initial velocity ⃗ but its acceleration is always the free fall acceleration ⃗ which is downward. Example of a projectile is a golf ball. The motion of a projectile is called projectile motion. Consider the figure below. ⃗ ⃗ When an object is object is projected at an angle to the horizontal, as shown in figure above, the following should be noted: (i) the horizontal component of the velocity is constant; (ii) the vertical component of the velocity changes; (iii) the vertical component of the velocity is zero at the highest point; (iv) the vertical acceleration throughout the motion, while the horizontal acceleration is zero. Page 15 of 40 In projectile motion, the horizontal motion and the vertical motion are independent of each other; that is, neither motion affects the other. Vertical motion: Initial velocity , acceleration Let be the maximum height reached. At maximum height. ( ) ( ) Let the time taken to reach the maximum height be. ( ) When the object landed on the ground, the vertical displacement. Let be the total time of flight. ( ) Horizontal Motion: Initial velocity , To obtain the range (horizontal displacement) of the projectile, we use The range of the projectile is Page 16 of 40  ( )( ) ( ) The maximum value of is and it occurs when , or. Therefore the maximum range is obtained if the object is projected at an angle to the horizontal. Example 12: A pirate ship is from a military island base. A military cannon (large gun on wheels) located at sea level fires balls at initial speed. ( ) At what angle from the horizontal must a ball be fired to hit the ship? ( ) How far should the pirate ship be from the cannon if it is to be beyond the maximum range of the cannonballs? Solution: ( ) ( ) ( ) ( ) ( ) ( ) Page 17 of 40 Example 13: A projectile shot at an angle of above the horizontal strikes a building away at a point above the point of projection. Find the magnitude and direction of the velocity of the projectile as it strikes the building. Solution: Building ⃗ Horizontal motion Vertical motion ( ) ( ) ( ) Page 18 of 40