Exam 1 (21).pdf - 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

Exam 1 (21).pdf - Question A small 200-g collar C can slide...

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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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