Suppose a diving board with no one on it bounces up and down in a simple harmonic motion with a frequency of 3.00 Hz. The board has an effective mass of 10.0 kg. What is the frequency of the simple harmonic motion of a 62.0 kg diver on the board?
m₁=10 kg,
m₂=m₁+62=72 kg,
f₁= ω₁/2π = sqrt(k/m₁)/2 π ,
f₂=ω₂/2π = sqrt(m₂/k)/2 π,
f₂ =f₁•sqrt(m₁/ m₂)=3sqrt(10/72)=1.12 Hz.
To solve this problem, we can use the principle of conservation of energy. The total energy of the system remains constant in simple harmonic motion, and it is equal to the sum of the potential energy and the kinetic energy.
The potential energy of the diving board is given by the formula:
PE = 1/2 k x^2
The kinetic energy of the diving board is given by the formula:
KE = 1/2 m v^2
where k is the effective spring constant, x is the displacement from the equilibrium position, m is the effective mass, and v is the velocity.
Since the diving board is bouncing up and down, the maximum displacement x is given by:
x = A
The maximum velocity v is given by:
v = Aω
where A is the amplitude of the motion and ω is the angular frequency.
The angular frequency ω is related to the frequency f by the formula:
ω = 2πf
Given that the frequency of the diving board is 3.00 Hz, we can calculate the angular frequency:
ω = 2π(3.00 Hz) = 6π rad/s
Now, we can use the conservation of energy principle to find the frequency of the diver on the board.
For the diving board, the total energy is given by:
E1 = PE + KE = 1/2 k A^2 + 1/2 m A^2 ω^2
For the diver on the board, the total energy is given by:
E2 = PE + KE = mgh + 1/2 m v^2
Since the total energy must be the same for both cases, we can equate E1 and E2:
1/2 k A^2 + 1/2 m A^2 ω^2 = mgh + 1/2 m v^2
Now, we can substitute the expression for v in terms of A and ω:
1/2 k A^2 + 1/2 m A^2 ω^2 = mgh + 1/2 m (Aω)^2
Simplifying the equation:
1/2 k A^2 + 1/2 m A^2 ω^2 = mgh + 1/2 m^2 A^2 ω^2
Now, we can solve for the frequency of the diver on the board. We know that the diving board has a mass of 10.0 kg and the diver has a mass of 62.0 kg.
Plugging in the values:
1/2 k A^2 + 1/2 (10.0 kg + 62.0 kg) A^2 ω^2 = (62.0 kg)(9.8 m/s^2)h + 1/2 (62.0 kg)^2 A^2 ω^2
Simplifying further:
1/2 k A^2 + 1/2 (72.0 kg) A^2 ω^2 = (607.6 kg·m/s^2)h + 1/2 (3844 kg^2) A^2 ω^2
Since both sides of the equation have the same units, we can equate the coefficients of A^2 ω^2:
1/2 k + 1/2 (72.0 kg) = 1/2 (3844 kg^2) ω^2
Now we can solve for the angular frequency ω:
ω^2 = (1/2 k + 36.0 kg) / (3844 kg^2)
ω = sqrt[(1/2 k + 36.0 kg) / (3844 kg^2)]
Finally, we can calculate the frequency f of the diver on the board:
f = ω / (2π) = [sqrt(1/2 k + 36.0 kg) / (3844 kg^2)] / (2π)
Now, you can plug in the value of the spring constant k to find the frequency of the diver on the board.
To find the frequency of the simple harmonic motion for the 62.0 kg diver on the board, we need to consider the effect of the diver's mass on the system. The natural frequency of the diving board without any load is given as 3.00 Hz.
The frequency of simple harmonic motion for a mass-spring system can be calculated using the formula:
f = 1 / (2π) * sqrt(k / m)
Where:
- f is the frequency of the simple harmonic motion
- k is the spring constant of the system
- m is the effective mass of the system
Since the diving board is the spring in this case and has an effective mass of 10.0 kg, we need to calculate the spring constant first. The effective mass of the board includes the mass of the board itself and any additional masses attached to it.
Once we have the spring constant, we can calculate the frequency of the diver on the board.
Here are the steps to find the frequency:
Step 1: Calculate the spring constant (k) of the board.
- Since the effective mass (m) is given as 10.0 kg and the frequency (f) is given as 3.00 Hz, we can rearrange the formula to solve for k:
k = (2πf)^2 * m
Step 2: Calculate the frequency (f') of the diver on the board.
- Now, using the same formula, with the diver's mass (m') as 62.0 kg, we can solve for the frequency of the diver on the board:
f' = 1 / (2π) * sqrt(k / m')
By plugging in the calculated value of k and the given values of m' and π, we can solve for the desired frequency f'.