How does the length of a pendulum affect the frequency and wave speed?

To solve, let's consider the period of the pendulum. The formula of period is:

`T= 2pisqrt(L/g)`

where T is the period, L is the length of pendulum and g is the acceleration due to gravity.

Take note that the frequency is the reciprocal of period.

`f=1/T`

Plugging in the formula of T, it becomes:

`f=1/(2pi) sqrt(g/L)`

Based on this, the frequency is inversely proportional to the length.

`f prop 1/L`

This means that as the length increases, the frequency decreases.

For the speed of the wave, let's consider the relationship between linear and angular velocity.

`v = r* omega`

Take note that in a pendulum, the angular velocity is equal to angular frequency, which is `2pif` . And the radius refers to the length of the pendulum.

`v= r*omega`

`v=L*2pif`

Plugging in the formula of frequency, it becomes:

`v = L*2pi * 1/(2pi)sqrt(g/L)`

And it simplifies to:

`v = Lsqrt(g/L)`

`v=sqrt(Lg)`

Based on this, the speed of the wave is directly proportional to the length of the pendulum.

`v prop L`

So, as the length of the pendulum increases, the speed of the wave increases too.

Therefore, as the length of the pendulum increases, the frequency decreases while the speed of the wave increases.