Scribd
Upload
20 views2 pages

HW 1

The document contains homework problems related to acoustics and the physics of sound, focusing on mass-spring systems and their oscillatory behavior. It includes calculations for elongation due to added mass, mechanical resistance, and the effects of sinusoidal excitation on displacement and velocity. Additionally, there is a bonus problem involving a simple pendulum and its oscillation characteristics.

Uploaded by

Pedro Vilaplana López
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
20 views2 pages

HW 1

The document contains homework problems related to acoustics and the physics of sound, focusing on mass-spring systems and their oscillatory behavior. It includes calculations for elongation due to added mass, mechanical resistance, and the effects of sinusoidal excitation on displacement and velocity. Additionally, there is a bonus problem involving a simple pendulum and its oscillation characteristics.

Uploaded by

Pedro Vilaplana López
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 2

Acoustics and the Physics of Sound 2024 Homework #1 Problem 1

Problem 1
A mass of 500 g is attached to a massless spring. (6 p.)

1. How much elongation does an extra mass of 200 g cause on the spring, if the undamped natural
frequency f 0 of the original system without the extra mass is 5 Hz?

2. What is the mechanical resistance of the original system without the extra mass, if the frequency of
its damped oscillation f d is 4 Hz?

The extra mass is removed and this sets the system to oscillate.

1. How long does it take for the amplitude to halve?

2. What is the initial amplitude of the motion?

3. What is the initial phase of the motion?

Problem 2
In the following figures the stiffness of all springs is K and the mass of the objects is m. The springs are
assumed to be massless. Solve for the natural undamped angular frequency in each case. (10 p.)

K K K K K K K

m
K K
m m m

m
K K K
m

(a) (b) (c) (d)

1
Acoustics and the Physics of Sound 2024 Homework #1 Problem 3

Problem 3
A mass (m = 10 kg) is attached to a spring (K = 1 N/m). The system is excited with a sinusoidal excitation
of constant force. Calculate the frequency of maximum magnitude for i) the displacement and ii) velocity of
the mass, when the mechanical resistance is R = 1 kg/s and R = 500 kg/s. Check if the oscillation condition
(α < ω0 ) is met for each case. (4 p.)

Hints:
i) start with the expression of the transfer function of the complex displacement of a system driven by a
sinusoidal excitation of magnitude F : ||x̃|
F̃ |
=√ 2
1
2 2
(K−ω m) +(ωR)
ii) start with the expression of the transfer function of the complex velocity of a system driven by a sinusoidal
excitation of magnitude F : ||ṽ|
F̃ |
=√ 2 1 2
R +(ωm−K/ω)

Bonus
Consider a simple pendulum with a length L and a mass m, as shown in the figure below. The pendulum
is displaced and then released from an angle θ and thus oscillates in simple harmonic motion. We consider
small oscillation amplitudes. (5 p.)
1. What are the forces applied to the mass? Give their expression.
2. Use Hooke’s law for springs to find the expression for the force constant k.
3. Express the period of oscillation T of the simple pendulum.
4. In the light of your previous answer, what would be the natural frequency of the motion if the angle
of displacement θ was doubled while still considering small oscillation amplitudes?

Hints:
- For small angles: sin θ ≈ θ.
- For questions 1. and 2., you could also write the differential equation for the motion of the pendulum using
Newton’s second law.

You might also like