Question

In the 2001 film Shrek, Shrek and Donkey are on their way to the castle to save Princess Fiona. In one scene, they come to a rope bridge that spans across a lake of lava. Donkey is afraid to go on, but Shrek, er, encourages him to do so. The bridge is made of rope that’s secured (fixed) at both ends. Shrek is listed as 6’9”, 355 lbs. Donkey has a mass of 90 ??. The total mass of the bridge is 150 ? long and, including planks, has a total mass of 700 ??. Based on some approximations, the rope has a linear mass density of ? = 1.4 ??⁄?. Because of the way the bridge is secured, the tension in the rope would have to hold up about ten times the total weight on/of the bridge. However, because there are four ropes, each attached at two ends, the tension in each rope is only 1.25 the total weight on/of the bridge.

The expression for the wave speed in a rope is: ? = √??/?

where ?? is the tension in the rope and ? is the linear mass density of the rope.

Find the speed of a wave in this rope, in ?⁄?.

At one point, Shrek stomps on the bridge, effectively creating a pulse that travels down the rope. If he was standing at the halfway point of the bridge (“I know that half is safe!”), how long would it take the pulse to return to him, and would it return upright or inverted?

Answer 1

Total mass on bridge (including Shrek and donkey), M = 700 kg + 90 kg + 355 lbs = 700 kg + 90 kg + 161.025 kg = 951.025 kg

Total tension in all ropes, T = Mg = (951.025 kg) (9.81 m/s²) = 9329.555 N

Tension is each rope, F_T = T/4 = = 2332.389 N

Given, linear mass density of rope,   = 1.4 kg/m

The expression for the wave speed in a rope is,  

Thus, speed of a wave in this rope is 40.817 m/s.

At one point, Shrek stomps on the bridge while standing at the halfway point of the bridge, effectively creating a pulse that travels down the rope.

The distance that the wave has to travel to return to Shrek is, d = 150 m (half the bridge length while going to other end and half the bridge length while returning)

Time taken by wave to cover this distance is ,

Thus, it would take 3.675 s for the pulse to return to Shrek. And it would return inverted.