1.1:  A light year is the distance light travels in one year. How many metres are there in one light year: (speed of light = 3.0 x 10⁸ms⁻¹). (Ans: 9.5 x 10¹⁵m)

(Click for verified Solution)


1.2: 

a) How many seconds are there in 1 year?
b) How many nanoseconds in 1 year?
c) How many years in 1 second?

[Ans. ( a) 3.1536 x 10⁷s, (b) 3.1536 x 10¹⁶ns (c) 3.1 x 10⁻⁸ yr]



1.3: The length and width of a rectangular plate are measured to be 15.3 cm and 12.80 cm, respectively. Find the area of the plate. (Ans: 196 cm²)


1.4: Add the following masses given in kg upto appropriate precision. 2.189, 0.089, 11.8 and 5.32. (Ans: 19.4 kg)



1.5: Find the value of 'g' and its uncertainty using T = 2π `sqrt\frac {l}{g}` from the following measurements made during an experiment.
Length of simple pendulum l = 100 cm. Time for 20 vibrations = 40.2 s.
Length was measured by a metre scale of accuracy upto 1 mm and time by stop watch of accuracy upto 0.1 s.
(Ans: 9.76 ± 0.06 ms⁻²)


1.6: What are the dimensions and units of gravitational constant G in the formula 
F = G`\frac {m₁m₂}{r^2}` 
(Ans: [M⁻¹L³T⁻²], Nm² kg⁻²)



1.7: Show that the expression of vf =vi +at is dimensionally correct, where vi is the velocity at t =0 , a is acceleration and vf is the velocity at time t. 


1.8:  The speed v of sound waves through a medium may be assumed to depend on (a) the density 𝝆 of the medium and (b) its modulus of elasticity E which is the ratio of stress to strain. Deduce by the method of dimensions, the formula for the speed of sound. (Ans: v = `sqrt \frac {E}{⍴}` )



1.9: Show that the famous "Einstein equation" E = mc² is dimensionally consistent.


1.10: Suppose, we are told that the acceleration of a particle moving in a circle of radius r with uniform speed v is proportional to some power of r, say `r^n`, and some power of v, say `v^m` , determine the powers of r and v ? (Ans: n = -1, m = 2)