Question

An opera singer has two precisely identical glasses. The singer produces as pure a tone as possible and shatters the first glass at a frequency of $808\mathrm{Hz}$. She then sings a frequency of $838\mathrm{Hz}$ in the presence of the second glass. The second glass will likely: (A) $$ shatter after a longer amount of time because the applied frequency is higher. (B) shatter after a shorter amount of time because the applied frequency is higher. $(C)$ not shatter because the applied frequency is not equal to the natural frequency of the glass. (D) $$ not shatter because higher-frequency sounds are associated with more attenuation.

Short answer

Option C. The glass will not shatter because the applied frequency is not equal to the natural frequency of the glass.

Step-by-step solution

Understand the Concept of Resonance

When a material vibrates strongly at its natural frequency due to an external force matching that frequency, it is experiencing resonance. Shattering the first glass with a frequency of 808 Hz implies that this frequency is the natural frequency of both glasses.

Analyze the New Frequency

The singer sings a new frequency of 838 Hz. This frequency is different from the natural frequency of 808 Hz. For resonance to occur, the external frequency must match the natural frequency of the object.

Determine the Result on the Second Glass

Since the new frequency (838 Hz) does not match the natural frequency of the glass (808 Hz), resonance will not occur. Without resonance, the second glass will not shatter.

Conclude the Correct Answer

With the understanding that the glass will not experience resonance because the frequency is different, option C is the correct answer: 'not shatter because the applied frequency is not equal to the natural frequency of the glass.'

Key concepts

Natural Frequency

Natural frequency is the specific frequency at which an object tends to vibrate when it is disturbed without any external force acting on it. For example, the opera singer's first glass has a natural frequency of 808 Hz. This means, if you gently tap the glass, it will naturally vibrate at this frequency. Objects have unique natural frequencies based on their physical properties like size, shape, and material. When an external force acts on an object at its natural frequency, the object will absorb this energy and begin to vibrate more intensely.

Resonance

Resonance occurs when an external force acts at the same frequency as the natural frequency of an object, causing it to vibrate with greater amplitude. This is why the glass shattered when the opera singer hit the exact natural frequency of 808 Hz. When the frequency of the external force and the natural frequency match, the energy transferred is maximized, leading to increased vibrations. In the case of the second glass, the singer used a frequency of 838 Hz, which did not match the natural frequency (808 Hz) of the glass. As a result, resonance did not occur, and the glass remained intact.

Vibration

Vibration is the rapid back-and-forth motion of an object. Every object that vibrates does so at its natural frequency when not influenced by an external force. For example, the glass vibrates at 808 Hz when its natural frequency is matched. When an external force, such as sound waves from the opera singer, causes the glass to vibrate, the motion's intensity depends on whether the force matches the object's natural frequency. Different frequencies cause different levels of vibration amplitude. In summary, when the singer's pitch was 838 Hz, the glass vibrated, but not at the intense level required for resonance.