A device called a rolamite is used in various ways to replace slipping motion with rolling motion. If the belt, which wraps between the rollers, is subjected to a tension of 15 N, determine the reactive forces N of the top and bottom plates on the rollers so that the resultant couple acting on the rollers is equal to zero.
Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.
Solution:
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There are two forces creating two coupling moments in the rolamite. The N force and the T force both create coupling moments. Let us assume clockwise moments to be positive and write an equation to determine the value of N. To do so, we need to use trigonometry to figure out the perpendicular distances between our forces.
The green line represents the hypotenuse in the right angle triangle we can create. The pink dashed line is the perpendicular distance between the N forces. The blue dashed line is one segment of the total perpendicular distance between the T forces. Using trigonometry, we can now write a moment equation.
To clarify where these values came from, let us take a look at the diagram again. The hypotenuse has a length of 25 + 25=50 mm. The length of the pink dashed line is then 50\cos30^0 and the blue dashed line has a value of 50\sin30^0. When calculating the perpendicular distance between the T forces, remember to add the top and bottom radius of the rolamite as well, which gives us 25+25+50\sin30^0. Also note that the coupling moment created must equal 0 as stated by the question, thus the left side of our equation equals 0.
Solving gives us:
N=25.98 N
Final Answer: