13.3: Chapter 13 Homework Problems

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Exercise \(\PageIndex{1}\)

An impact-testing device consists of a 20-kilogram box supported by two slender 5-kilogram rods. The two rods are set up in parallel so that the box remains level as it swings. If the whole system is released from the horizontal position shown below, what is the velocity of the box after it has traveled 90°?

A long rectangular box hanging in the air is supported by 2 vertical rods, each 0.75 meters long with their top ends attached to supports jutting out from a ceiling. This is the mechanism's final position, having rotated 90° clockwise after being released from a position where the top of the box and the two rods were all parallel to the ceiling. The rods are not rigidly fixed to the box, but are able to swivel while keeping the box level. — Figure \(\PageIndex{1}\): problem diagram for Exercise \(\PageIndex{1}\). An impact-testing device composed of a pair of rods swivel-mounted on a heavy box and attached to the ceiling, allowing for the box to be raised close to the ceiling and to swing downwards when released, remaining parallel throughout.

Solution: \(v = 3.97 \ m/s\)

Example \(\PageIndex{2}\)

A 40-lb door with a width of 36 inches is attached to a spring with an unstretched length of 4 inches, designed to close the door when left open. The spring is anchored as shown below when closed (solid outline is closed, dotted outline is open 90°). If we want the door to have an angular velocity of 0.2 rad/s upon closing when released from rest at the 90° open position, what should the spring constant of the spring be? (This is the top view of the door below)

Top-down view of a door, represented as a thin bar 36 inches long. The solid-outline bar, which is horizontal and has a hinge at its left end, represents the door in its shut position. A diagonal spring is attached to this, with one end fixed on the point on the bar 4 inches to the right of the hinge and the other end being 3 inches above the hinge's location. The dotted-outline bar, which is vertical and has the hinge at its top end, represents the door in its open position. — Figure \(\PageIndex{2}\): problem diagram for Exercise \(\PageIndex{2}\). Top-down view of the closed and open positions a door designed to close when left open, due to the action of a spring. The unstretched spring is shown attached to the door and the wall beside the hinge, in the closed position.

Solution: \(k = 2.68 \ lb/ft = 0.224 \ lb/in.\)