Engineering Mechanics

(^468) „„„„„ A Textbook of Engineering Mechanics
∴ Angular velocity of connecting rod,
6.65 11.1 × 60
11.1 rad/s = 106 r.p.m.
0.6 2
bc
BC
ω= = = =
π
Ans.
Now locate a point d at the centre of bc. Join ad. By measurement, we also find that velocity of D,
vD = ad = 7.43 m/s Ans.
22.9. MOTION OF A ROLLING WHEEL WITHOUT SLIPPING
Consider a wheel rolling along a straight path, in such a way that there is no slipping of the
wheel on its path. A little consideration will show, that the centre of the wheel O moves with some
linear velocity. And each particle on the periphery of the wheel rotates with some angular velocity.
Thus the motion of any particle on the periphery of the
wheel is a combination of linear and angular velocity as
shown in Fig. 22.17
Let v= Linear velocity of the centre of the
wheel
ω= Angular velocity of the wheel, and
r= Radius of the wheel
Now consider any particle P on the periphery of
the wheel making an angle θ with the vertical through the
centre of the wheel as shown in Fig. 22.17. We know that
particle (P) is subjected to the following two motions
simultaneously.

1. Linear velocity (v) acting in the horizontal direction.


2. Tangential velocity (equal to linear velocity, such that v = ωr) acting at right angles to OP.
   From the geometry of the figure, we find that the angle between these two velocities is equal
   to θ. The resultant of these velocities (R) will act along the bisector of the angle between the two
   forces, such that

Rv2cos 2

⎛⎞θ

= ⎜⎟

⎝⎠

We know that angular velocity of P about A

P

v

AP

=

We know that from the triangle AOP,

22cos

2

AP AC r

⎛⎞θ

== ⎜⎟

⎝⎠

∴ Angular velocity of C about A

2 cos 2 cos

22

2cos

2

vv

v

APrr

⎛⎞ ⎛⎞θθ

⎜⎟ ⎜⎟⎝⎠ ⎝⎠

===

⎛⎞θ

⎜⎟⎝⎠

It is thus obvious, that any point on the wheel rotates about the lowest point A (which is in

touch with the ground) with the same angular velocity ()

v

r

ω=.

Notes. 1. The particle A is subjected to the following two velocities :

(i) linear velocity (v) towards P, and

(ii) tangential velocity (v = ωr) towards Q

Fig. 22.17. Motion of a rolling wheel without

slipping.