A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion :
(a) $y = a \sin \frac{2\pi t}{T}$
(a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.

Fill in the blanks by suitable conversion of units
(c) 3.0 m s$^{–2}$ = .... km h$^{–2}$

One mole of an ideal gas at standard temperature and pressure occupies 22.4 L (molar volume). What is the ratio of molar volume to the atomic volume of a mole of hydrogen ? (Take the size of hydrogen molecule to be about 1 Å). Why is this ratio so large ?

A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion :
(b) $y = a \sin vt$
(a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.

Fill in the blanks
(a) The volume of a cube of side 1 cm is equal to .....m$^3$

Explain this statement clearly :
“To call a dimensional quantity ‘large’ or ‘small’ is meaningless without specifying a standard for comparison”. In view of this, reframe the following statements wherever necessary :
(f) the speed of sound is much smaller than the speed of light.

The unit of length convenient on the atomic scale is known as an angstrom and is denoted by Å: 1 Å = $10^{–10}$ m. The size of a hydrogen atom is about 0.5 Å. What is the total atomic volume in m$^3$ of a mole of hydrogen atoms ?

Explain this statement clearly :
“To call a dimensional quantity ‘large’ or ‘small’ is meaningless without specifying a standard for comparison”. In view of this, reframe the following statements wherever necessary :
(e) a proton is much more massive than an electron

Just as precise measurements are necessary in science, it is equally important to be able to make rough estimates of quantities using rudimentary ideas and common observations. Think of ways by which you can estimate the following (where an estimate is difficult to obtain, try to get an upper bound on the quantity) :
(e) the number of air molecules in your classroom.

Fill in the blanks by suitable conversion of units
(b) 1 m = ..... ly

Fill in the blanks
(b) The surface area of a solid cylinder of radius 2.0 cm and height 10.0 cm is equal to ...(mm)$^2$

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 :
$m = \frac{m_0}{(1 - v^2)^{1/2}}$.
Guess where to put the missing $c$.

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 ?

Fill in the blanks
(c) A vehicle moving with a speed of 18 km h$^{–1}$ covers....m in 1 s

State the number of significant figures in the following :
(b) $2.64 \times 10^{24}$ kg

Which of the following is the most precise device for measuring length :

Fill in the blanks by suitable conversion of units
(d) G = $6.67 \times 10^{–11}$ N m$^2$ (kg)$^{–2}$ = .... (cm)$^3$ s$^{–2}$ g$^{–1}$.

Fill in the blanks
(d) The relative density of lead is 11.3. Its density is ....g cm$^{–3}$ or ....kg m$^{–3}$.

A SONAR (sound navigation and ranging) uses ultrasonic waves to detect and locate objects under water. In a submarine equipped with a SONAR, the time delay between generation of a probe wave and the reception of its echo after reflection from an enemy submarine is found to be 77.0 s. What is the distance of the enemy submarine? (Speed of sound in water = 1450 m s$^{–1}$).

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.