Try this without calculus (fixed)
A beam of mass and length is supported at each end by two supports A and B. The semicircles and rectangle all have uniform mass densities of . Calculate , where and are the vertical reactions at A and B (in Newtons) respectively. Enter your answer to one decimal place.
Additional information: (also in diagram)
- The semicircles are identical and each have a radius of .
- The rectangle has a length of .
- The horizontal distance from A to the center of the rectangle is .
- is the gravity constant, .
The answer is 119.2.
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1 solution
Center of mass of each semicircle is at a distance of 3 π 4 r = 3 π 8 from it's diameter. So, the distances of the centers of mass of the left semicircle, the rectangle, the plank, and the right semicircles are 6 − 3 π 8 , 1 0 , 1 1 and 1 4 + 3 π 8 respectively. Masses of the semicircles and the rectangle are 6 π kg. and 9 6 kg. respectively. Hence, the Force Balance Equation gives
V A + V B = ( 1 0 6 + 1 2 π ) g
Moment Balance Equation about the point A gives
V B × 2 2 = ( 6 π ( 6 − 3 π 8 ) + 9 6 × 1 0 + 1 0 × 1 1 + 6 π ( 1 4 + 3 π 8 ) ) g = ( 1 2 0 π + 1 0 7 0 ) g ⟹ V B = 1 1 5 ( 1 2 π + 1 0 7 ) g ⟹ V A = 1 1 1 ( 7 2 π + 6 3 1 ) g ⟹ V A − V B = 1 1 1 2 ( π + 8 ) g ≈ 1 1 9 . 2 3 5 2 9 8 8 N.