PHYSICS AT A GLANCE DIRECT THEORY NOTES UNITS AND DIMENSIONS Physics is the science of natural phenomena It is the science of observation and measurement It is the science of interpretation of results To measure a quantity we need units To define and establish units, we require physical standards The important requisites of a physical standard are accessibility, accuracy and invariance Such requisites are possessed by modern standards An example of physical standard is the wavelength of light, which is used to define metre In the international system used today in science and engineering, we have seven base units and three supplementary units The system is called SI Table 1.1 gives ten fundamental quantities, i.e seven base units and three supplementary units, their symbols and what they represent TABLE 1.1 FUNDAMENTAL QUANTITIES Quantity symbol What it represents metre m Length kilogram kg Mass second s Time kelvin K Temperature candela cd Luminous intensity ampere A Electric current mole mol Amount of substance radian rad Angle in plane steradian sr Solid angle curie Ci Radioactivity Conventions Followed in Using SI units Symbols should not be used with plural, e.g 10 m is correct, but 10 ms in wrong A void using bars in writing units, e.g write m/s2 as ms-2 As far as possible multiples of 103n of base units are to be used Each multiple has a prefix E.g when n = 1, the prefix is kilo Symbol is ‘k’ When n = 2, prefix is mega; symbol M When n = -1, prefix is ‘milli’ and symbol m n = -2, micro etc Dimensions: Physical quantities are classified into fundamental and derived quantities The five fundamental quantities in physics are mass (M), length (L), time (T), temperature (θ) and current (I) All others are derived quantities Dimensions are the relation between fundamental and derived quantities If we write any derived quantity as Ma Lb Tc θd Ie, then this is called a dimensional formula and the powers a,b,c,d,e are called dimensions The dimension of the quantities will be given along with their definitions MOTION IN ONE AND TWO DIMENSIONS Scalars and Vectors Physical quantities are classified into scalars and vectors A scalar is fully defined and understood, when its magnitude or value is given, e.g mass, length, distance, speed, etc A vector quantity is fully defined only when both its magnitude and direction are given.e.g., force, acceleration, magnetic field etc All quantities having magnitude and direction will not be vectors They must also obey the rules of geometric addition Vector Algebra Two vectors can be added by (i) triangle method, (ii) parallelogram method, and (iii) analytical method The sum of two vectors A and B at an angle α is the vector C, where the magnitude of C is given by the analytical method as C= A + B + 2AB cos α and the direction of C with vector A, θ is given by Physics for IIT-JEE Screening Test B sin α A + B cos α To subtract a vector we reverse the vector to be subtracted and add with the other, e.g A-B = A+(-B) To add a number of vectors we use the law of polygon of vectors If we represent the vectors as the sides of a polygon, the side that completes the polygon in the opposite direction is the sum Multiplication of vectors: Two vectors A and B can be multiplied in two ways If the product is scalar, it is scalar multiplication or dot product The dot product of two vectors a and B is defined as A.B = A B Cos α where α is the angle between them If the product is a vector, it is called a vector product or cross product The cross product of A and B is A x B = AB sin α nˆ is unit vector perpendicular to the plane of A and B Resolution of a vector: A vector F inclined at angle α with a given direction can be resolved as two rectangular components, as F cos α along the direction and F sin α perpendicular to it Unit vectors: A vector F can be written F = ˆiF + ˆj F + kˆF where ˆi , ˆj and kˆ are known as unit vectors tan θ = x y z along the three axes of coordinates The magnitude of each is one unit Fx, Fy, Fz are the x,y and z components of the vector F Types of Motion One-dimensional motion means a particle moves such that its position can be represented by one coordinate Two- dimensional motion (motion in a plane) means the position of a particle can be represented by two coordinates (motion of a projectile) In three-dimensional motion, the position of a particle can be represented by three coordinates (motion of molecules of a gas) Displacement is the change in the position of a body in a given direction It is a vector The rate of displacement is velocity The rate of travelling a distance is speed Unit : ms-1 Dimensions : M0L1T-1 Acceleration is the rate of change of velocity Unit: ms-2 Dimensions: M0L1T-2 Equations of motion are equations connecting various parameters of a body’s motion For simplicity we use conventional letters If ‘u’ is the initial velocity, ‘v’ the final velocity, ‘a’ the uniform acceleration, ‘s’ the distance travelled, and ‘t’ the duration of travel and ‘sn’ the distance travelled in n th second, then v = u + at ⎡u + v⎤ s=⎢ ⎥t ⎣ ⎦ s = ut + at 2 v2 = u2 + 2as 1⎤ ⎡ sn = u + a ⎢n − ⎥ 2⎦ ⎣ Equations of motion under gravity (bodies moving vertically up and down under gracvity) are got by substituting a = + or – g depending on whether the body is travelling downwards or upwards g is usually taken as 9.8 ms-2 For rough calculations it can be taken as 10 Projectile motion: A projectile is a body thrown at an angle so that it moves in a vertical plane under the action of gravity If h is the maximum height, R is the horizontal range, T is the time of flight, Rm is the maximum range, u is the initial velocity of projection and α is the angle of projection of the body u sin α 2g 2u sin α T= g h= R= u sin 2α g u2 g Uniform circular motion: A body in uniform circular motion has constant speed and varying velocity Rm = The acceleration towards the centre of the body is v2 or ω2 r, where v is its velocity and ω angular r velocity The force towards centre acting on the body is the centripetal force = mv2 or mω2r r Physics at a Glance Centrifugal force: An observer in a circular motion feels a radially outward force This force is called the centrifugal force Its value is equal to the centripetal force It is pseudo a force The properties of pseudo force are given in indirect theory notes In a non-uniform circular motion, speed and velocity change An example such a motion is a stone tied to the end of a string and whirled in a vertical circle At the highest point, when the string has minimum tension (T1) mv1 − mg r where v1 is velocity of the stone at the highest point If the tension is zero, velocity of the stone is T1 = v1 = rg At the lowest point it has maximum tension T2 given by mv 2 T2 = mg + r where v2 is velocity of stone at the lowest point The velocity of the stone at the lowest point when the tension at the highest point is zero, is v2 = 5rg The difference in tension at the highest and lowest points is T2 – T1 = 6mg i.e., times the weight of the stone LAWS OF MOTION Frame of Reference: A frame of reference is a coordinate system It has an origin and axes of coordinates Two frames, which move with respect to each other in uniform speed in a straight line, are called inertial frames Newton’s first law (law of inertia) is obeyed in such frames Two frames having relative acceleration are non-inertial frames Newton’s second law is obeyed in such frames Newton’s Laws of Motion: First law: A body at rest or moving with uniform speed in a straight line continues so until an external force acts on it Second law: The rate of change of momentum is directly proportional to the unbalanced (resultant) force acting on the body Third law: For every action there is an equal and opposite reaction Newton’s first law defines force while the second law measures it as f = ma, where m is mass and a the acceleration Unit of force: newton (N) Dimensions of force: MLT-2 Weight of a body is a force exerted by the earth on the body: W = mg It is measured by the reaction, resistance or tension In a freely falling body, only the force of gravity acts That is, the reaction or resistance is zero Hence a freely falling body experiences weightlessness Law of Conservation of Momentum: A closed system is one, in which no external force acts The total momentum of a closed system remains constant Examples: Rocket propulsion, recoil of a gun, explosion of a shell and collision For recoil of a gun, mv = MV, where m is the mass of the shot, M is the mass of the gun, v is the velocity of the shot and V is the recoil velocity of the gun More accurate formula will be mv = (M-m)V For m