# Lecture 11 Units and Measurement

**Exercise 2.7** A student measures the thickness of a human hair by looking at it through a microscope of magnification 100. He makes 20 observations and finds that the average width of the hair in the field of view of the microscope is 3.5 mm. What is the estimate on the thickness of hair ?

**Exercise 2.8** Answer the following : (a)You are given a thread and a metre scale. How will you estimate the diameter of the thread ?

(b)A screw gauge has a pitch of 1.0 mm and 200 divisions on the circular scale. Do you think it is possible to increase the accuracy of the screw gauge arbitrarily by increasing the number of divisions on the circular scale ?

(c) The mean diameter of a thin brass rod is to be measured by vernier callipers. Why is a set of 100 measurements of the diameter expected to yield a more reliable estimate than a set of 5 measurements only ?

**Exercise 2.9** The photograph of a house occupies an area of 1.75 cm^{2} on a 35 mm slide. The slide is projected on to a screen, and the area of the house on the screen is 1.55 m^{2}. What is the linear magnification of the projector-screen arrangement.

**Exercise 2.10** State the number of significant figures in the following :

(a) 0.007 m^{2}

(b) 2.64 × 10^{24 }kg

(c) 0.2370 g cm^{–3}

(d) 6.320 J

(e) 6.032 N m^{–2}

(f) 0.0006032 m^{2}

**Exercise 2.11** The length, breadth and thickness of a rectangular sheet of metal are 4.234 m, 1.005 m, and 2.01 cm respectively. Give the area and volume of the sheet to correct significant figures.

**Exercise 2.12** The mass of a box measured by a grocer’s balance is 2.30 kg. Two gold pieces of masses 20.15 g and 20.17 g are added to the box. What is (a) the total mass of the box, (b) the difference in the masses of the pieces to correct significant figures ?

**Exercise 2.13** A physical quantity P is related to four observables *a*, *b*, *c* and *d* as follows : \( P = \frac{a^3 b^2}{\sqrt{c} d}\) The percentage errors of measurement in a, b, c and d are 1%, 3%, 4% and 2%, respectively. What is the percentage error in the quantity P ? If the value of P calculated using the above relation turns out to be 3.763, to what value should you round off the result ?

**Exercise 2.14** A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion :

(a) *y* = *a* sin 2π *t/T*

(b) *y* = *a* sin *vt*

(c) *y* = (*a*/T) sin *t/a*

(d) \( y = (a \sqrt{2}) ((sin 2 \pi t/ T + cos 2 \pi t / T ))\)

(a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.

**Exercise 2.15** A famous relation in physics relates ‘moving mass’ m to the ‘rest mass’ *m*_{0} of a particle in terms of its speed v and the speed of light, *c*. (This relation first arose as a consequence of special relativity due to Albert Einstein). A boy recalls the relation almost correctly but forgets where to put the constant c. He writes : \( m =\frac{m_0}{(1-v^2)^{\frac{1}{2}}}\) Guess where to put the missing c.