1) A long rope hangs down from the top of a cliff near a waterfall and has a seat at the bottom . Suppose you hop on the seat at an angle of 22 degrees and swing out and back from the waterfall in 10 seconds. Estimate the length of the rope and the speed which you pass through the waterfall with.
2) Why do soldiers break cadence when they cross bridges?
1) Here, we are examining the mechanics of a simple pendulum with respect to its period. To do this, we'll consider torques. the total torque on the pendulum will be equivalent to the following by a modified version of Newton's second law:
`tau = I ddottheta = ml^2 ddottheta`
The only torque involved here will be the restorative tangential component of gravitational force (because it is directed downward it is negative) multiplied by the length, as follows:
`tau = l*-mgsintheta`
Since this is the only torque acting on the system, we equate the two terms to get our differential equation:
`ml^2ddottheta = -lmgsintheta`
`ddottheta = -g/lsintheta`
```ddottheta + g/l sintheta = 0`
This differential equation would be quite difficult to solve, so we will make an approximation based on the series expansion of sine because 22 degrees (0.38 radians) is pretty small. Thus, we will say `sintheta = theta` for our purposes:
`ddottheta + g/l theta = 0`
This gives us an equation that looks pretty close to that for a spring with the following known solution:
`theta(t) = theta_0sin(sqrt(g/l)t)`
From this equation, we can deduce the period to be:
`T = 2pisqrt(l/g)`
Now, we can solve for length with the given period of 10 seconds:
`10 = 2pisqrt(l/9.8)`
`1.59 = sqrt(l/9.8)`
`2.53 = l/9.8`
`24.8 = l`
Thus, our length is 24.8 meters.
To calculate the speed, we simply need to calculate the derivative of the equation at the equilibrium point and multiply by the length of the rope (remember, theta must be in radians to make these equations work!):
`theta(t) = 0.38cos(sqrt(9.8/24.8)t)`
`theta(t) = 0.38cos(0.63t)`
`dottheta(t) = -0.24sin(0.63t)`
Because of the nature of the sinusoid, we know that it will reach the bottom of its swing when sin(0.63t) = 1 (also, this is the point at which the absolute value of `dottheta` is maximized:
`|dottheta_max| = 0.24`
Now we just multiply by the length to get the speed:
`s = l|dottheta_max|=24.8xx0.24 = 5.92 m/s`
So, our speed at the bottom of the swing is 5.92 m/s.
2) Why do troops break cadence on a bridge?
They don't want to hit the resonant frequency of the bridge to cause its collapse. A Mythbusters episode called this into question.