4.6: Chapter 4 Homework Problems

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Exercise \(\PageIndex{1}\)

Determine if the two systems below are statically equivalent.

A horizontal rod held off the ground by two supports: one at the right end, and the other 10 feet to the left of the first. A downwards point force of 60 lbs is applied at the midpoint between these supports, and another downwards point force of 40 lbs is applied 3 feet to the left of the support on the left. — Figure \(\PageIndex{1}\): part 1 of the problem diagram for Exercise \(\PageIndex{1}\). A horizontal bar held off the ground by two supports experiences two point forces at different points along its length.

A horizontal rod held off the ground by two supports: one at the right end, and the other 10 feet to the left of the first. A downwards point force of 100 lbs is applied to the rod, 2 feet to the right of the support on the left. — Figure \(\PageIndex{2}\): part 2 of the problem diagram for Exercise \(\PageIndex{1}\). A horizontal bar held off the ground by two supports experiences a single point force partway along its length.

Solution: No, they are not equivalent.

Exercise \(\PageIndex{2}\)

Determine if the set of forces in A is statically equivalent to the set of forces and moments in B.

System A (on the left) consists of a uniform cross composed of two 60-cm rods joined at their midpoints, with an upwards force of 200 N applied to the end of the cross's left arm, a leftwards force of 200 N applied to the end of the upper arm, an upwards force of 200 N applied to the midpoint of the right arm, and a rightwards force of 100 N applied to the end of the lower arm. System B (on the right) consists of an identical cross with an upwards force of 200 N applied to the end of the left arm, a leftwards force of 100 N applied to the end of the upper arm, an upwards force of 200 N applied to the end of the right arm, a rightwards force of 200 N applied to the lower arm, and a clockwise moment of 30 N-m about the cross center. — Figure \(\PageIndex{3}\): problem diagram for Exercise \(\PageIndex{2}\). Two uniform crosses of equal dimensions experience a set of forces (system A, left) and a set of forces and moments (system B, right).

Solution: No, they are not equivalent.

Exercise \(\PageIndex{3}\)

Resolve the force shown below into a force and a couple acting at point A. Draw this force and couple on a diagram of the L-shaped beam.

An L-shaped beam, with a 5-meter-long horizontal arm and a 3-meter-long vertical arm joined at the lower right corner of the diagram, is held above the ground by a pin joint attached to the left end of the horizontal arm (point A). A horizontal leftwards point force of 150 N is applied to the end of the vertical arm of the L. — Figure \(\PageIndex{4}\): problem diagram for Exercise \(\PageIndex{3}\). An L-shaped beam is held parallel to the ground by a pin joint attached at one end (point A), with a force applied to the other free end of the L shape.

Solution

\(F_A = 150 \, N\) to the left

\(M_A = 450 \, Nm\)

Exercise \(\PageIndex{4}\)

Find the equivalent force couple system acting at point A for the setup shown below. Draw this force and couple on a diagram of the L-shaped beam.

An L-shaped beam with a vertical arm 3 feet long and a horizontal arm 7 feet long, with beams instersecting at the lower left corner of the diagram (point A). A force of 60 lbs is applied at the end of the vertical arm, pushing down and to the left at a 25-degree angle above the horizontal. Two downwards forces of magnitude 60 lbs each are applied on the horizontal arm, one at the very end and the other 5 feet to the right of point A. — Figure \(\PageIndex{5}\): problem diagram for Exercise \(\PageIndex{4}\). An L-shaped beam, whose arms intersect at the point A, experiences several forces acting at different points along the arms.

Solution

\(F_A = 155.2 \, lbs, \, 69.5\)° below the negative \(x\)-axis

\(M_A = -556.9 \, ft \, lbs\)

Exercise \(\PageIndex{5}\)

A helicopter is hovering with the wind force, the force from the tail rotor, and the moment due to drag shown below. Determine the equivalent force couple system at point C. Draw the final force and moment on a new diagram of the helicopter.

Top-down view of a hovering helicopter, aligned with the nose at the top of the diagram and the tail at the bottom. The central rotor hub, point C, experiences a wind force of 300 N, pointing downwards and to the left to make an angle of 55 degrees above the horizontal. Point C also experiences a counterclockwise moment due to drag, of magnitude 3 kN-m. The tail rotor, 5 meters behind C, applies a leftwards force of magnitude 650 N. — Figure \(\PageIndex{6}\): problem diagram for Exercise \(\PageIndex{5}\). A hovering helicopter with point C at the central hub of its main rotor experiences a force applied at C, a moment about C, and a force applied at a distance from C.

Solution

\(F_{eq} = 858.01 \, N\) acting at \(16.6\)° below the negative \(x\)-axis

\(M_{eq} = -250 \, N \, m\)

Exercise \(\PageIndex{6}\)

Determine the equivalent point load (magnitude and location) for the distributed force shown below, using integration.

A horizontal bar 5 meters long is attached to a wall at its left end. Starting at the point 2 meters to the right of the wall, it experiences a downwards distributed force over the rest of its length that varies linearly in magnitude: starting at 50 N/m and decreasing to 20 N/m at the right end. — Figure \(\PageIndex{7}\): problem diagram for Exercise \(\PageIndex{6}\). A horizontal bar attached to a wall at one end experiences a distributed force, which varies linearly, over part of its length.

Solution

\(F_{eq} = 105 \, N\)

\(x_{eq} = 3.29 \, m\) (measured from wall)

Exercise \(\PageIndex{7}\)

Determine the equivalent point load (magnitude and location) for the distributed force shown below, using integration.

A horizontal bar 18 feet long is attached to a wall at its left end. It experiences a distributed force whose magnitude starts at 0 lb/ft at the left end, increases linearly tp 600 lbs/ft at the point 6 feet to the right of the wall, and decreases linearly to 0 lb/ft at the right end of the bar. — Figure \(\PageIndex{8}\): problem diagram for Exercise \(\PageIndex{7}\). A horizontal bar attached to a wall experiences a distributed force over its length, with magnitude varying linearly according to a piecewise force function.

Solution

\(F_{eq} = 5400 \, lbs\)

\(x_{eq} = 8 \, ft\) (measured from wall)

Exercise \(\PageIndex{8}\)

Use the method of composite parts to determine the magnitude and location of the equivalent point load for the distributed force shown below.

A horizontal bar 6 meters long is attached to a wall at its left end. The bar experiences a downwards distributed force, whose magnitude increases linearly from 0 at the wall to 3 kN/m at the point 1.5 meters to the right of the wall, remains constant at 3 kN/m for 2 meters, and then decreases linearly to 0 over the last 2.5 meters. — Figure \(\PageIndex{9}\): problem diagram for Exercise \(\PageIndex{8}\). A horizontal bar attached to a wall experiences a distributed force over its length, with magnitude varying linearly according to a piecewise force function.

Solution

\(F_{eq} = 12 \, kN\)

\(x_{eq} = 2.79 \, m\) (measured from wall)