Question

A visitor to a lighthouse wishes to determine the height of the tower. She ties a spool of thread to a small rock to make a simple pendulum, which she hangs down the center of a spiral staircase of the tower. The period of oscillation is 9.15 is. The acceleration of gravity is 9.8 m/s2 . What is the height of the tower

Answer 1

The height of the tower is 20.81 metres.

Step-by-step explanation:

Extracting the key information from the question:-

*** The height of a tower needs to be determined.

*** A spool of thread is tied to a small rock to create the principle of simple pendulum.

*** The thread with the rock is hung down the centre of a staircase of the tower.

*** The period of the oscillation of the pendulum is 9.15 seconds.

*** Acceleration due to gravity will be taken as 9.8 metres per second.

*** We are simply required to calculate the height of the tower.

Now, since the period of the oscillation of the pendulum is 9.15 seconds, we will then first determine the angular frequency of the pendulum using the formula:-

w = 2π/T

Where, "w" is the angular frequency.

π will be taken as 3.14

T is the period of oscillation (9.15 seconds).

We will then solve for the angular frequency:

w = (2 × 3.14)/9.15

w = 6.28/9.15

w = 0.686 rad/sec

With the knowledge of the angular frequency of the pendulum, we can now calculate the height of the tower using the formula:

w = √(g/L)

Where, w = angular frequency

g = acceleration due to gravity

L = length (height of the tower).

Since "w" = 0.686 rad/sec and acceleration due to gravity is 9.8 metres per second, we can then determine/calculate the height of the tower by substituting appropriately and then making "L" the subject of the formula. That is:

√(g/L) = w

We then square both sides of the equation.

g/L = w^2

Cross multiply

L×(w^2) = g

L = g/(w^2)

We then substitute:

L = 9.8/(0.686^2)

L = 9.8/0.471

L = 20.81 metres

Therefore the height of the tower is 20.81 metres.

Answer 2

The height of the tower is 20.764 m.

Step-by-step explanation:

The angular frequency of the pendulum is:

`\omega = (2\pi)/(T)`

`\omega = (2\pi)/(9.15\,s)`

`\omega = 0.687\,(rad)/(s)`

The height of the lighthouse is:

`\omega = \sqrt{(g)/(l)}`

`\omega^(2) = (g)/(l)`

`l = (g)/(\omega^(2))`

`l = ((9.800\,(m)/(s^(2)) ))/((0.687\,(rad)/(s))^(2) )`

`l = 20.764\,m`

The height of the tower is 20.764 m.