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1.

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

Answer» Magnification of the microscope = 100 Average width of the hair in the field of view of the microscope = 3.5 mm `:.` Actual thickness of the hair is`3.5/100= 0.035 mm`
Discussion
2.

Precise measurements of physical quantities are a need of science. For example to escertain the speed of an aircraft, one must have an accurate methi=od to find its positions at closely separated instants of time. This was the actual motivation behind the discovery of radar in World War II. think of different examples in modern science where precise measurements of length, time, mass etc, arc needed. Also, whereever you can, give a quantitative idea of the precision needed.

Answer» Extremely precise measurements are needed in modem science. As an example, while launching a satellite using a space launch rocket system we must measure time to a precision of 1 micro second. Again working with lasers we require length measurements to an angstrom unit `(1 A^(@)=10^(-10) m)` or even a fraction of it. for estimating nuclear sizes we require a precision of `10^(-15)m`. To measure atomic masses using mass spectrograph we require a precision of `10^(-30)`kg and so on.
Discussion
3.

Answer the following : (a) You are given a tread 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 arbitratily 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 measurement only ?

Answer» (a) Wrap the thread on a uniform smooth rod in such a way that the coils thus formed are very close to each other. Measure the length of the thread using a metre scale. The diameter of the thread is given by the relation,
Diameter=`("Length of thread")/("Number of turns")`
(b) It is not possible to increase the accuracy of a screw gauge by increasing the number of divisions of the circular scale. Increasing the number divisions of the circular scale will increase its accuracy to a certain extent only.
(c) A set of 100 measurements is more reliable than a set of 5 measurements because random errors involved in the former are very less as compared to the latter.
Discussion
4.

A new unit of length is chosen such that the speed of light in vecuum is unity. What is the distance between the sun and the earth in terms of the new unit, if light takes 8 min and 20 sec. to cover the distance ?

Answer» Distance between the Sun and the Earth: = Speed of light × Time taken by light to cover the distance Given that in the new unit, speed of light = 1 unit Time taken, t = 8 min 20 s = 500 s `:.` Distance between the Sun and the Earth = 1 × 500 = 500 units
Discussion
5.

Consider a simple pendulum having a bob attached to a string that oscillates under the action of a force of gracity. Suppose that the period of oscillation of the simple pendulum depends on its length (I), mass of the bob (m) and acc. Due to gravity (g). Derive the expression for its time period using method of dimensions.

Answer» The dependence of time period T on the quantities `l, g` and `m` as a product may be written as :
`T=K l^(x) g^(y) m^(z)`
Where k is dimensionless constant and x, y and z are the exponents.
By considering dimensions on both sides, we have
`[L^(@)M^(@)T^(1)]=[L^(1)]^(x)[L^(1)T^(-2)]^(y) [M^(1)]^(z)`
`= L^(x+y) T^(-2y) M^(z)`
On equation the dimensions on both sides, we have
`x+y=0, -2y=1,` and `z=0`
So that `x=1/2, y=-1/2, z=0`
Then, `T=K l^(1//2) g^(-1///2)`
or, `T=ksqrt(l/g)`
Note that value of constant k can not be obtained by the method of dimensions. Here it does not matter if some number multiplies the right side of this formula, because that does not affect its dimensions.
Actually, `k=2pi` so that `T=2pi sqrt(l/g)`
Discussion