Engineering Mechanics

Chapter 26 : Helical Springs and Pendulums „„„„„ 551

Example 26.19. A conical pendulum 1·5 m long is revolving at 30 revolutions per minute.
Find the angle which the string will make with the vertical, if the bob describes a circle of 500 mm
radius.

Solution. Given : Length of pendulum (l) = 1·5 m ; Angular speed of the pendulum (N) = 30
r.p.m and radius of the circle (r) = 500 mm = 0·5 m

Let θ = Angle which the string will make with the vertical.

We know that angular velocity of the bob,

2230

rad/s

60 60

ππ×N

ω= = =π

∴

(^22) 0·5
tan 0·5036
9·8
r
g
ωπ×
θ= = =
or θ = 26.7° Ans.
EXERCISE 26.3

1. A uniform rod of mass 1 kg is 1 m long. The rod is pivoted about a point 100 mm from one
   end. Find the frequency of the rod about the pivot, when it turns freely in the vertical
   plane. (Ans. 0·64 Hz)


2. A body of mass 2·5 kg oscillates about an axis at a distance of 500 mm from its centre of
   gravity. Find the length of the equivalent simple pendulum, if the mass moment of inertia
   parallel to the axis rotation and about the centroidal axis is 0·4 kg-m^2 .(Ans. 0·82 m)


3. A body of mass 1·5 kg is oscillating about an axis, which is at a distance of 0·5 m from the
   centre of gravity of the body. If the radius of gyration of the body about its centroidal axis
   is 0·8 m, find (i) mass moment of inertia of the body, (ii) length of the equivalent simple
   pendulum, and (iii) time period. (Ans. 0·96 kg-m^2 ; 1·78 m ; 2·68 s)


4. A conical pendulum consists of a 500 gm bob and 1 m long string. If the bob describes a
   horizontal circle of radius 250 mm, find the time taken by the bob for one revolution.
   (Ans. 1·97 s)


5. A conical pendulum 2 m long has a bob of 1 kg mass. If the bob describes a circle of 1 m
   radius at 45 r.p.m, find (i) the angle which the string will make with the vertical ; and (ii)
   tension in the string. (Ans. 68.2º ; 11.32 N)

QUESTIONS

1. Derive an expression for the period of oscillation of a weight, when attached to the helical
   spring.


2. What is a simple pendulum? Under what conditions its motion is regarded as simple
   harmonic.


3. Differentiate the equation for the stiffness of two springs, when they are arranged in series
   and parallel.


4. How does the change in length or gravity affects the time period of a simple pendulum.
   Justify your answers with the help of derivations.


5. Obtain a relation for the change of no. of oscillations due to change in the position of a
   pendulum.


6. What is a compound pendulum? Derive an expression for the time period of a compound
   pendulum.


7. What is meant by centre of oscillation? Describe its importance.