General Physics Chapter 1 PDF

‟General Physics” AHST-104 (Chapter one) Presented by: Dr. Walaa Mahmoud Tawfik  Physics and Measurements ❑ Measurement in everyday life. ❑ Need for measurement in physics. ❑ Physical Quantity. ❑ Units for measurement. ❑ Systems of units. ❑ International Syste...

‟General Physics” AHST-104 (Chapter one) Presented by: Dr. Walaa Mahmoud Tawfik  Physics and Measurements ❑ Measurement in everyday life. ❑ Need for measurement in physics. ❑ Physical Quantity. ❑ Units for measurement. ❑ Systems of units. ❑ International System of units. ❑ Dimensions. ❑ Fundamental dimension. ❑ Derived dimensions. ❑ Dimensions analysis[Example]. ❑ SI prefixes. ❑ Vector and Scalar quantities. ❑ The components of a vector. ❑ Sign Convention. ❑ Unit-Vector notation.  Measurement in everyday life Measurement of mass Measurement of volume Measurement of length Measurement of temperature  Need for measurement in physics ❑ Physics is a quantitative science, based on careful measurements of quantities such as mass, length, and time. ❑ The laws of physics allow us to gain a deeper understanding of the world in which we live, for example, the rockets sent to the moon, the chips embedded in computers, the lasers used to perform eye surgery. ❑ In short, everything in nature - from atoms and subatomic particles to solar systems and galaxies - obeys the laws of physics. ❑ To understand any phenomenon in physics we have to perform experiments. ❑ Experiments require measurements, and we measure several physical properties like length, mass, time , temperature, pressure etc. ❑ When we begin our study of physics, one of the most important fundamentals of physics is the system of units that we use when we make these measurements.  Physical Quantity ❑ A physical property that can be measured and described by a magnitude and unit. Examples: Mass of a person is 65 kg. Length of a table is 3 m. Area of a hall is 100 𝒎𝟐. Temperature of a room is 300 K Physical Quantities Fundamental quantities Derived quantities The physical quantities that do not depend The physical quantities which depend on any other physical quantities in their on one or more fundamental quantities measurements, and on the base of which for their measurements. other quantities are expressed. Mass Length Time Temperature Area Volume Speed Force  Units for measurement ❑ The standard used for the measurement of a physical quantity is called a unit. Examples: meter, foot, inch for length. kilogram, pound for mass. second, minute, hour for time. Fahrenheit, kelvin for temperature. ❑ Systems of units: ✓ CGS system. ✓ MKS system. ✓ FPS system. ✓ International System of units (SI).  Systems of units ❑ CGS system: This system was first introduced in France. It is also known as Gaussian system of units. ❑ MKS system: This system was also introduced in France. It is also known as French system of units.  Systems of units ❑ FPS system: This system was introduced in Britain. It is also known as British system of units. ❑ International System of units (SI): In 1971, General Conference on Weight and Measures held its meeting and decided a system of units for international usage. This system is called international system of units and abbreviated as SI. The SI unit consists of seven fundamental units and two supplementary units. International System of units[Fundamental units]  International System of units (SI) ❑ Meter (m): is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. ❑ Kilogram (kg): is the mass of a particular platinum- iridium alloy cylinder at the International Bureau of Weights and Measures in Sèvres , France. ❑ Seconds(s): is the time taken by 9,19,26,31,770 oscillations of the light emitted by a cesium–133 atom. International System of units [Supplementary units] ❑ Radian (rad): It is used to measure plane angle. It is the measure of the central angle whose arc length is the same as the radius of the circle. θ = 1 radian ❑ Steradian (sr): It is used to measure solid angle. It is the solid angle subtended at the center of a sphere by its surface whose area is equal to the square of the radius of the sphere. Ω = 1 steradian  Dimensions ❑ The unique quality or type of every physical quantity which distinguishes it from all other quantities , i.e. it refers to the qualitative nature of a physical quantity. ❑ Whether a distance is measured in units of feet or meter, it is still a distance ,and we say its dimension is length. ❑ We can convert between different units with the same physical quantity (e.g. feet into meter) but we can't convert one dimensions into another (e.g. you cannot convert length into time). ❑ It is common to use brackets [ ] to denote the dimensions of a physical quantity (e.g. dimensional symbols for Length is [L]. Dimension Fundamental or basic dimension: Derived dimensions: dimensions that are measured independently and dimensions that are products or quotients of fundamental enough to express essential physical quantities. dimensions. Fundamental dimension Derived dimensions Derived dimensions Derived dimensions Derived dimensions Derived dimensions  Dimensions analysis ❑ It’s a simple mathematical method which used to: ✓ Check the correctness of a physical equation. This means that each term in the equation must have the same dimensions. ✓ Derive a physical law. Example 1.1 A body is moving in a straight line with an initial velocity v0 and an acceleration a. After time t, its velocity v is given by v = v0 + at, check the correctness of this equation. Solution Finding the dimensions of both sides of the equation and if the LHS and RHS have the same dimensions, the physical equation is correct. LHS: [v] = 𝑳𝑻−𝟏 RHS:[v0] = 𝑳𝑻−𝟏 [at]= [(𝑳𝑻−𝟐 ) (T)] = [𝑳𝑻−𝟏 ] i.e. [LHS] = [RHS] and the equation is correct  Dimensions analysis [Example] Einstein's relation between the total energy, E, of a body and its mass m is given by: E = mc2 Where, c is the free space velocity of light. Check the correctness of this equation. Solution LHS: [E] = [𝑴𝑳𝟐 𝑻−𝟐 ] 𝟐 RHS: [mc2] = (𝑴) 𝑳𝑻−𝟏 = 𝑴𝑳𝟐 𝑻−𝟐 i.e. [LHS] = [RHS] and the equation is correct.  Dimensions analysis [Example] The speed of waves v in shallow water depends only on the acceleration due to gravity; g, and on the water depth h. Which of the following formulas for the wave speed v could be correct? 𝟏 𝟐 𝒂) 𝒗 = 𝒈𝒉 𝒃) 𝒗 = 𝒈𝒉 𝟐 Solution 𝒂) 𝑳𝑯𝑺: 𝒗 = 𝑳𝑻−𝟏 𝟏 𝟐 𝑹𝑯𝑺: 𝒈𝒉 = 𝑳𝑻−𝟐 𝑳𝟐 = 𝑳𝟑 𝑻−𝟐 𝟐 As clear , the dimensions of both sides are not the same , and the formula is not correct. 𝐛) 𝐋𝐇𝐒: 𝒗 = 𝑳𝑻−𝟏 𝟏 𝟏 𝟏 𝑹𝑯𝑺: 𝒈𝒉 = −𝟐 𝑳𝑻 𝟐 𝑳 𝟐 = 𝟐 −𝟐 𝑳 𝑻 𝟐 = 𝑳𝑻−𝟏 As clear , the dimensions of both sides are the same , and the formula could be correct.  Dimensions analysis [Example] How does the frequency f of a simple pendulum depend on the mass m, the length of the string ℓ, and the acceleration due to gravity g? Solution 𝒇 = 𝑲𝒎𝒙 𝒍𝒚 𝒈𝒛 [ K is a dimensionless constant and x , y and z are to be determined] LHS: 𝒇 = 𝑻−𝟏 𝒛 𝑹𝑯𝑺: 𝑲𝒎𝒙 𝒍𝒚 𝒈𝒛 = 𝑴𝒙 𝑳𝒚 𝑳𝑻−𝟐 = 𝑴𝒙 𝑳𝒚+𝒛 𝑻−𝟐𝒁 Since the dimensions of both sides must be the same , it follows that: 𝑻−𝟏 = 𝑴𝒙 𝑳𝒚+𝒛 𝑻−𝟐𝒛 𝑴𝟎 𝑳𝟎 𝑻−𝟏 = 𝑴𝒙 𝑳𝒚+𝒛 𝑻−𝟐𝒛 Equating the powers of each dimension on either side of equation , we obtain: 𝑥=0 𝑦+𝑧 =0 −2𝑧 = −1 𝟏 𝟏 ∴ x =0 & 𝒚 = − 𝟐 & 𝒛= 𝟐 𝟏 𝟏 −𝟐 𝟐 𝒈 𝒇= 𝑲𝒎𝒙 𝒍𝒚 𝒈𝒛 = 𝟎 𝑲𝒎 𝒍 𝒈 𝒇=𝑲 𝑳  Dimensions analysis [Example] Consider a string of length ℓ and mass m stretched under a tension T. If a transverse mechanical wave is propagating along it with velocity v, how does v depends on ℓ, m, and T. Solution 𝒗 = 𝑲𝒍𝒙 𝒎𝒚 𝑻𝒁 where K is a dimensionless constant. Writing the dimensions of both sides: 𝒛 𝑳𝑻−𝟏 = 𝑳𝒙 𝑴𝒚 𝑴𝑳𝑻−𝟐 𝑳𝟏 𝑴𝟎 𝑻−𝟏 = 𝑳𝒙+𝒛 𝑴𝒚+𝒛 𝑻−𝟐𝒛 Thus , equating the powers of each dimension on either side of the dimensional equation 𝒙+𝒛=𝟏 𝒚+𝒛=𝟎 −𝟐𝒛 = −𝟏 𝟏 𝟏 𝟏 ∴x= 𝒚=− 𝒛= 𝟐 𝟐 𝟐 𝟏 𝟏 −𝟐 𝟏 𝒍𝑻 𝑻 𝒗= 𝑲 𝒍𝒙 𝒎𝒚 𝑻𝒁 =K𝒍 𝒎 𝟐 𝑻 𝟐 𝒗=𝑲 =𝑲 𝒎 𝒎 𝒍 𝑻 𝒗=𝑲 𝒎 𝝁 [𝝁 = is the mass per unit length]. 𝒍  Dimensions analysis [Example] The gravitational force between two masses m1 and m2 separated by a distance r is given by: 𝒎𝟏 𝒎𝟐 𝑭=𝑮 𝒓𝟐 Where, G is the gravitational constant. Find the dimensions of G. Solution The dimensions of both sides are as follows: 𝑳𝑯𝑺 = 𝑭 = 𝑴𝑳𝑻−𝟐 𝒎𝟏 𝒎𝟐 𝑴𝑴 𝑹𝑯𝑺 = 𝑮 = 𝑮 = 𝑮 𝑳−𝟐 𝑴𝟐 𝒓𝟐 𝑳𝟐 𝑀 𝐿 𝑇 −2 −1 𝐿3 𝑇 −2 ∴ 𝑴𝑳𝑻−𝟐 = 𝑮 𝑳−𝟐 𝑴𝟐 𝐺 = = 𝑀 𝐿−2 𝑀2  Dimensions analysis [Example] The magnitude of the force of attraction or repulsion between two-point charges 𝒒𝟏 and 𝒒𝟐 separated by a distance r is given by: 𝒒𝟏 𝒒𝟐 𝑭=𝑲 𝒓𝟐 where K is the constant of proportionality. Find the dimensions of K. Solution 𝒒𝟏 𝒒𝟐 𝑭 = 𝑲 𝒓𝟐 𝑸𝑸 𝑴𝑳𝑻−𝟐 = 𝑲 = 𝑲 𝑳−𝟐 𝑸𝟐 𝑳𝟐 𝑴𝑳𝑻−𝟐 𝟑 𝑻−𝟐 𝑸−𝟐 𝑲 = = 𝑴𝑳 𝑳−𝟐 𝑸𝟐 SI prefixes  Use of SI prefixes 3 milliampere = 3 mA = 3 x 𝟏𝟎−𝟑 A 5 microvolt = 5 μV = 5 x 𝟏𝟎−𝟔 V 8 nanosecond = 8 ns = 8 x 𝟏𝟎−𝟗 s 6 picometre = 6 pm = 6 x 𝟏𝟎−𝟏𝟐 m 5 kilometer = 5 km = 5 x 𝟏𝟎𝟑 m 7 megawatt = 7 MW = 7 x 𝟏𝟎𝟔 W Problems  Vector and Scalar quantities ❑ Scalar quantities are measured with numbers and units. length temperature time (e.g. 16 cm) (e.g. 102 °C) (e.g. 7 s) ❑ Vector quantities are measured with numbers and units, but also have a specific direction. acceleration displacement force (e.g. 30 m/s2 upwards) (e.g. 200 miles northwest) (e.g. 2 N downwards)  The components of a vector ❑ Any vector A lying in the xy plane can be represented by a vector 𝑨𝒙 lying along the x axis and by a vector 𝑨𝒚 lying along the y axis. If a vector A makes an angle θ with the x-axis as shown in figure: A ✓ The horizontal component is A sin  𝑨𝑿 = A cos  ✓ The vertical component is  𝑨𝒀 = A sin  A cos  ❑ the resultant force : 𝑨= 𝑨𝟐𝒙 + 𝑨𝟐𝒚 𝑨𝒚 𝑨𝒚 𝑨𝒙 𝜽 = 𝐭𝐚𝐧−𝟏 = 𝐬𝐢𝐧−𝟏 = 𝒄𝒐𝒔−𝟏 𝑨𝒙 𝑭 𝑭  Sign Convention ❑ The signs of the components of a vector A depend on the quadrant in which the vector is located. ✓ If θ = 120°, then Ax is negative, and Ay is positive. ✓ If θ = 225°, then both Ax and Ay are negative.  Unit-Vector notation ❑ A unit vector is a dimensionless vector having a magnitude of exactly 1. i → unit vector in the positive x direction. j → unit vector in the positive y direction. k → unit vector in the positive z direction. ❑ The magnitude of each unit vector equals 1; that is, | i | = | j | = | k | = 1. ❑ The unit-vector notation for the vector A lying in the xy plane: A=𝑨𝒙 𝒊 + 𝑨𝒚 j ✓ 𝑨𝒙 𝒊 , is a vector of magnitude |Ax| lying on the x axis. ✓ 𝑨𝒚 j , is a vector of magnitude |Ay| lying on the y axis.  Example: the components of a vector A force vector F has a magnitude of 10 N directed 𝟓𝟑° above the negative x axis as shown in figure. Find the x and y components of this force. Solution 𝑭𝒙 = −𝑭 𝑪𝑶𝑺 𝟓𝟑° = - (10 N) (0.6) = -6 N 𝑭𝒚 = 𝑭 𝐬𝐢𝐧 𝟓𝟑° = (10N)(0.8) = 8N  Example A certain vector A has components 𝑨𝒙 = 5 and 𝑨𝒚 = -3. Find the magnitude and direction of A. Express A in unit vector notation. Solution ❑ The magnitude of a vector A A = 𝑨𝟐𝒙 + 𝑨𝟐𝒚 = (𝟓)𝟐 +(−𝟑)𝟐 = 5.83 ❑ The direction of a vector A 𝑨𝒚 (−𝟑) 𝜽 = 𝒕𝒂𝒏−𝟏 = 𝒕𝒂𝒏−𝟏 = 𝒕𝒂𝒏−𝟏 −𝟎. 𝟔 = −𝟑𝟏° 𝑨𝒙 𝟓 ❑ A in unit vector notation. A = 𝑨𝒙 𝒊 + 𝑨𝒚 j = 5i – 3j  Example The centripetal force (F) acting on a particle (moving uniformly in a circle) depends on the mass (m) of the particle, its velocity (v) and radius (r) of the circle. Derive dimensionally formula for force (F). Ref.: Physics, James Walker, Chapter 1, page 1-18 THANK YOU