Statics · General force systems and equilibrium of the rigid body
Condition: The theoretical background develops the couple, the moment of a force, reduction and the three equilibrium conditions on the guiding system of the wall-mounted slewing crane.
Open theoretical foundations →First solve each task the classic way on paper: sketch the system, draw the free-body diagram, set up the balances and solve by hand — exactly as in the exam later. The subtasks on this page are your check: enter your results and use them to check every step of your hand calculation individually.
Exercise 1 · Concept check
A force is moved parallel to itself. What must be added so that the effect on the rigid body stays the same?
Along its line of action a force may travel — across it, it may not. Moving it parallel by the distance additionally produces the moment ; only force and moment together are equivalent to the original force.
Match the terms to their statements.
Its effect depends not on the reference point
Magnitude and sense depend on the reference point
Perpendicular distance between reference point and line of action
Reducing a force group to and
The couple is not bound to a line of action — its moment is the same everywhere. For the moment of a force it is different: magnitude and sense follow from the lever arm, the perpendicular distance to the chosen reference point. The reduction combines an entire force group into one resultant force and one resultant moment.
A bell crank is pin-supported at . At the long arm the force vertically downwards; an applied moment acts at the short arm counterclockwise. Build the free-body diagram: where do you cut, which forces are released — and do not forget the moment.
The cut is made at support A: there the lever is separated from its surroundings. The two components of the support reaction are released — the support is two-valued, it holds in both directions but lets the lever rotate. There is therefore no support moment.
The applied force and the applied moment belong in the diagram unchanged. Because the couple is not bound to a line of action, the moment arrow may be placed anywhere on the lever — only its sense of rotation must be right.
Conclusion: force and moment are two quantities
For the rigid body the force alone is no longer enough. Alongside the two force balances comes the moment balance — three equations for the three degrees of freedom of the plane. The next two exercises use both: first the reduction of a force group, then full equilibrium.
Exercise 2 · Loaded mounting plate
Four forces act at the corners of a rectangular mounting plate (width , height ), four forces act at the corners:
- horizontally to the right, at the top left corner
- horizontally to the left, at the bottom right corner
- vertically bottom, at the top right corner
- vertically top, at the bottom left corner
Find the magnitude, direction and position of the line of action of the resultant. The reference point is the corner at the lower left (origin ); moments count positive counter-clockwise.
— only and have horizontal components.
— negative, the resultant points downwards.
Magnitude of the resultant, symbolically of which: and .
Allowed symbols: Rx, Ry
Notation help: sqrt(...) for the square root, ^ for powers.
— a 3-4-5 triangle, the magnitude comes out without a calculator.
Negative, because the resultant points to the right and bottom . Mind the quadrant: for only values between and the calculator only returns values between −90° and +90° — the signs of and decide the correct quadrant.
Only two forces have a lever arm about the origin: with and with . and pass through the axes and contribute nothing.
— negative, the group turns clockwise.
— by this perpendicular distance the line of action of the resultant must be shifted relative to the origin. The direction angle is .
First enter, for every force, its components and with their signs in a table — right and up are positive.
For the moment you need the point of application . Forces whose line of action passes through the reference point have zero lever arm.
The distance follows from — textbook equation (3.11).
Conclusion: three pieces of information describe the resultant
Magnitude , direction and position — only all three together fully replace the four individual forces. Without the position the moment would be left out.
Exercise 3 · Tailgate
A horizontal tailgate of length is pin-supported at is pin-jointed to the vehicle. It is homogeneous, its own weight acts at the centre. At the free end a rope holds it, pulling upwards at the angle to the gate.
Part step a: Build the free-body diagram of the gate. First decide where to cut, then add all forces at their points of application.
The gate is separated from the vehicle and from the rope. At the pin two forces are released — it has two reaction components. At the free end the rope force pulls along the rope; a rope can only pull, so its direction is fixed. The own weight acts at the centre of the gate and points vertically downwards.
Part step b: Assign to each balance the unknown it yields directly.
About both pin forces have zero lever arm — they drop out, and the moment balance yields on its own. That is exactly why the reference point is placed on the line of action of a force. With known, the two force balances then give and .
Part step c: rope force symbolically from the moment balance about .
Allowed symbols: G, L, alpha
Notation help: sin(alpha), fraction with /, brackets allowed.
The length cancels out — the rope force does not depend on how long the gate is, but only on the weight and the rope angle.
With the vertical component of the rope force becomes .
— here rope and pin share the weight exactly in half. Horizontally, .
First draw the free-body diagram: the gate on its own, with the rope force, the own weight and both pin forces.
The vertical component of the rope force is ; it alone has, about a lever arm — namely the full length .
The own weight acts at the centre, so its lever arm about is .
Conclusion: choose the balance first, then calculate
Anyone who sets up the moment balance about obtains the rope force in one equation — anyone starting with the force balances instead faces a system with three unknowns. Choosing the reference point is not a formality; it saves work.
Report
The completion status of all three tasks is exported as a PDF report with a verification token — checkable via the platform's report verification.
| Exercise | Status |
|---|---|
| Exercise 1: Concept check | Open |
| Exercise 2: Loaded mounting plate | Open |
| Exercise 3: Tailgate | Open |