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Math  /  Geometry

QuestionIf the resultant of the loads shown passes through point B, determine the equivalent force-couple system at O.

Studdy Solution

STEP 1

1. The beam is in static equilibrium.
2. The moment and forces can be summed to find the resultant force and moment.
3. The equivalent force-couple system at point O must produce the same external effect as the given system of forces and moment.
4. The distances are measured horizontally from point O.

STEP 2

1. Calculate the resultant force.
2. Calculate the resultant moment about point O.
3. Determine the equivalent force-couple system at point O.

STEP 3

Calculate the resultant force by summing up all the vertical forces.
FR=−8 kip+3 kip−10 kip F_R = -8 \text{ kip} + 3 \text{ kip} - 10 \text{ kip}
FR=−15 kip F_R = -15 \text{ kip}
The resultant force is 15 kip downward.

STEP 4

Calculate the moment about point O due to each force and the applied moment. Use the sign convention that counterclockwise moments are positive.
MO=−8 kip×0 ft+3 kip×4.7 ft−10 kip×8.6 ft+M M_O = -8 \text{ kip} \times 0 \text{ ft} + 3 \text{ kip} \times 4.7 \text{ ft} - 10 \text{ kip} \times 8.6 \text{ ft} + M
Calculate each term separately:
MO=0+3×4.7−10×8.6+M M_O = 0 + 3 \times 4.7 - 10 \times 8.6 + M
MO=14.1−86+M M_O = 14.1 - 86 + M
MO=−71.9+M kip-ft M_O = -71.9 + M \text{ kip-ft}
The resultant moment about point O is −71.9 kip-ft+M-71.9 \text{ kip-ft} + M.

STEP 5

Determine the equivalent force-couple system at point O. The equivalent system consists of the resultant force FRF_R and the resultant moment MOM_O about point O.
The equivalent force-couple system at point O is: - Force: 1515 kip downward. - Moment: −71.9+M-71.9 + M kip-ft (clockwise if MM is positive).

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