Question:
A small 200-g collar
C
can slide on a semicircular rod that is made
to rotate about the vertical
AB
at the constant rate of 6 rad/s.
Determine the minimum required value of the coefficient of static
friction between the collar and the rod if the collar is not to slide
when (
a
)
Î¸
= 90
Â°
, (
b
)
Î¸
= 75
Â°
, (
c
)
Î¸
= 45
Â°
. Indicate in each case
the direction of the impending motion.
Fig. P12.128
Answer :
Given
Mass of the collar
C
,

The rate of rotation,
.
Let us draw the figure.
We have,

a) Given
The angle,
.
So,
Now, draw the figure.
Take the condition

.
Now,
.
Now, take the condition
.
So,

Therefore, the minimum required value of the coefficient of static friction is
and the direction of the impending motion is downward.
b)
For an arbitrary value of
, the direction is not known.
To consider both possibilities for each value of
, let
corresponds to
impending motion downward and
correspond to impending motion upward.
Now, consider the downward impending motion.
Take the condition
.

On rearranging,
And,
.
Take the condition
.
But,
So,
Substitute the known values.

Rearrange and simplify for

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