SOLUTION
| length |
Period |
(Period)2 |
| L (m) |
T (s) |
T2 (s2 ) |
| 0.20 |
1.00 |
1.00 |
| 0.40 |
1.34 |
1.80 |
| 0.60 |
1.61 |
2.59 |
| 0.80 |
1.84 |
3.39 |
| 1.00 |
2.03 |
4.12 |
( The student should plot a graph of T2 against
L and determine its slope s)
Slope s = 4.0 s2m-1
k = (sg)½ = 6.26 ( note k is about 2p)
k = 2p
T = 2pÖ (L/g)
Problem: A simple pendulum was suspended from the ceiling
of a laboratory. The following readings for the period of
oscillations T of the pendulum were obtained for various
lengths of the pendulum. The length was not measured directly,
but the height x of the bob above the floor was recorded.
| x (cm) |
10 |
40 |
80 |
120 |
160 |
| Period T(s) |
3.38 |
3.20 |
2.95 |
2.66 |
2.34 |
By a graphical method, find the value of the acceleration due
to gravity and the height of the laboratory.
Hint: L = (H - x) where H is the height of the laboratory.
( Answers: g = 9.86 ms-2, H = 2.95m)
Student Exercise
1.(a)(i) Explain the meaning of dimensions of a physical
quantity.
(ii) The velocity v of waves of wavelength l, on
the surface on the pool of liquid, whose surface tension
g and density r respectively is given by
v2 = lg/2p + 2pg/lr where g is the acceleration
due to gravity.
Show that the above equation is dimensionally correct.
(iii) A sphere of radius, a, moving through a density
r with high velocity v experiences a retarding
force F given by
F = k ax ry vz where k is
a non-dimensional coefficient.
Use the method of dimensions to find the values of x, y and
z.
(b)(i) Define coefficient of viscocity h and obtain
its dimensons.
(ii) The viscous drag F on a solid sphere moving through
a viscous medium may be considered to depend on the velocity
v of the sphere, its radius r and the coefficient
of viscocity h of the medium.
F = k va rb hc where a,
b and c are numbers and k is a numerical constant.
Use dimensional analysis to solve for a, b and c.
2.(a) Assuming conditions of streamline flow, the volume
rate of flow (V/t) of a liquid issuing from the tube will
depend on the pressure gradient (P/L) along the tube, the
radius r of the tube and the coefficient of viscocity h of
the liquid.
Show that (V/t) = kP r4/(hL) where k is some numerical
constant.
(b) The characteristic of wave motion in deep water is such
that
v = [ l ( A + ( 4p2 g)/l2r
) ] x
2p
where A is a constant which has dimensions, v is the velocity
of the wave, l is its wavelength.
g is the surface tension and r is the density.
Using a method of dimensions, obtain a value for x and obtain
the dimensions of A
(c) Use dimensional analysis to show how the velocity of
transverse vibrations of a stretched string depend on its
length (L), mass (m) and the tensional force (F) in the string.
Answers:
1. (a)(ii) x = 2, y = 1, z = 2 (b)(ii)
a = 1, b = 1, c = 1
2. (b) x = ½, [A] = LT-2
(c) v = k Ö (FL)/m
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