Mechanical Engineering Principles

FORCE, MASS AND ACCELERATION 149

constant at 34 km/h, determine the change

in acceleration when leaving one arc and

entering the other. [1.49 m/s^2 ]

3. An object is suspended by a thread
   400 mm long and both object and thread
   move in a horizontal circle with a constant
   angular velocity of 3.0 rad/s. If the tension
   in the thread is 36 N, determine the mass
   of the object. [10 kg]

13.4 Rotation of a rigid body about a

fixed axis

A rigid body is said to be a body that does not
change its shape or size during motion. Thus, any
two particles on a rigid body will remain the same
distance apart during motion.
Consider the rigidity of Figure 13.4, which is
rotating about the fixed axisO.

O

y

x

at

r

ΔFt= Δm.ar

Δm

a

Figure 13.4

In Figure 13.4,

α=the constant angular acceleration

m=the mass of a particle

r=the radius of rotation ofm

at=the tangential acceleration ofm

Ft=the elemental force on the particle

Now, forceF=ma

or Ft=mat

=m(αr)

Multiplying both sides of the above equation byr,

gives:

Ftr=mαr^2

Sinceαis a constant

∑

Ftr=α

∑

mr^2

or T=Ioα ( 13. 3 )

where T=the total turning moment

exerted on the rigid body

=

∑

Ftr

andIo=the mass moment of inertia (or second

moment) aboutO(inkgm^2 ).

Equation (13.3) can be seen to be the rotational

equivalent ofF =ma (Newton’s second law of

motion).

Problem 11. Determine the angular

acceleration that occurs when a circular disc

of mass moment of inertia of 0.5 kg m^2 is

subjected to a torque of 6 N m. Neglect

friction and other losses.

From equation (13.3), torqueT=Iα,

from which,angular acceleration,

α=

T

I

=

6Nm

0 .5kgm^2

=12 rad/s^2

13.5 Moment of inertia (I)

The moment of inertia is required for analysing

problems involving the rotation of rigid bodies. It

is defined as:

I=mk^2 =mass moment of inertia(kg/m^2 )

wherem=the mass of the rigid body

k=its radius of gyration about the point

of rotation (see Chapter 7).

In general,I=

∑

mr^2 where the definitions of

Figure 13.1 apply.