Question

Suppose a diving board with no one on it bounces up and down in a simple harmonic motion with a frequency of 6.90 Hz. The board has an effective mass of 10.0 kg. What is the frequency of the simple harmonic motion of a 67.6 kg diver on the board?

Answer 1

Frequency = f = 2.33 Hz

Step-by-step explanation:

We will use the equation of mass spring system and find out spring constant first and then using the same constant we will be finding out the frequency required in the question statement.

f1 = (1/2π) √(k/m)

`k=(2*3.14*f1)^2 * (m)\\k=(2*3.14*6.9)^2 * 10\\k= 18776.6`

Now using the same equation with diver on board.

mass = mass of board + mass of diver = 10 + 67.6 = 77.6 kg

Now f2 = (1/2π) √(k/m)

`f2=(1/2*3.14)√((18776.6/77.6)) \\f2 = 0.1592 * √(241.96) \\f2=2.33 Hz`

Answer 2

The frequency of the simple harmonic motion of a 67.6 kg diver on the board = 2.48 Hz

Step-by-step explanation:

The frequency in simple harmonic motion is related to spring constant and mass causing the motion through the relation

f = (1/2π) √(k/m)

When mass = 10 kg, f = 6.90 Hz,

6.9 = (1/2π) √(k/10)

(√(k/10) = 6.9×2π = 43.354

k/10 = 43.354² = 1879.57

K = 18795.7 N/m

When a diver of mass 67 kg climbs the diving board, the total mass on the diving board now becomes (10+67.6) = 77.6 kg

Spring constant of the diving board doesn't change,

So, the frequency is then given by

f = (1/2π) √(18795.7/77.6)

f = 2.48 Hz