Unit 1 Engineering Physics

26 CHAPTER 1. PROPERTIES OF MATTER

Worked out Example 1.7.1

Data obtained in a torsional pendulum experiment are given below.

Mean radius of the suspension wire (r) = 0.3 mm = 0. 3 ◊ 10 ≠^3 m

Length of the wire(l) = 70.4 cm = 70. 4 ◊ 10 ≠^2 m

Mass of one of the identical cylinders (m)=100g= 100 ◊ 10 ≠^3 kg

Distancesd 1 andd 2 between suspension wire & massmare:

d 1 = 2.5 cm = 2. 5 ◊ 10 ≠^2 m,d 2 = 4.5 cm = 4. 5 ◊ 10 ≠^2 m

Periods of oscillations:T 0 =10. 6 s,T 1 =11. 2 s,T 2 =11. 7 s

Calculate the moment of inertia of the disc and Rigidity Modulus of the suspension

wire.

Solution:

Moment of inertia of the disc

I =

2 m(d^22 ≠d^21 )T 02

T 22 ≠T 12

=

(2)(100◊ 10 ≠^3 kg) [(4. 5 ◊ 10 ≠^2 m)^2 ≠(2. 5 ◊ 10 ≠^2 m)^2 ](10.6s)^2

(11.7s)^2 ≠(11.2s)^2

=2. 7 ◊ 10 ≠^3 kg.m^2

Rigidity Modulus of the suspension wire

n =

8 fiIl

T 02 r^4

=

(8)(3.14)(2. 747 ◊ 10 ≠^3 kg.m^2 )(70. 4 ◊ 10 ≠^2 m)

(10.6s)^2 (0. 3 ◊ 10 ≠^3 m)^4

=4. 8 ◊ 1010 Nm≠^2

1.8 Bending of Beams................................

Beam theory began with Galileo Galilei (1564-1642), who investigated the behavior of
various types of beams. His work in mechanics of materials is described in his famous
bookTwo New Sciences, first published in 1638.

A structural member that is designed to resist forces acting laterally to its axis is
called a beam. Beams dier from bars in tension and torsion, primarily because of the
directions of the loads that act upon them. A bar in tension is subjected to loads directed
along the axis, and a bar in torsion is subjected to torques having their vectors along the
axis. By contrast, the loads on a beam are directed normal to the axis.

Before a load is applied, the longitudinal axis of the beam is a straight line. Lateral
loads acting on a beam cause the beam to bend, or flex, thereby deforming the axis of
the beam into a curve, called the deflection curve of the beam.

We consider beams which are symmetric about a plane (usually taken as the plane of
the paper). This plane is called a plane of symmetry of the beam. Therefore, the vertical

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