Maximum Oscillation Amplitude?

In summary, an ultrasonic transducer used in medical ultrasound imaging is a thin disk with a mass of 0.12 g that is driven in simple harmonic motion at 0.9 MHz by an electromagnetic coil. The maximum restoring force that can be applied to the disk without breaking it is 36,000 Newtons. The maximum oscillation amplitude that won't rupture the disk is 0.12 µm. At this amplitude, the disk's maximum speed is 1.33 m/s. The ratio of mass and force determines the quantity of acceleration in SHM. Velocity and acceleration can also be determined using the equation x = Acos(omega*t).
  • #1
TJC747
5
0
An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk (m = 0.12 g) driven back and forth in SHM at 0.9 MHz by an electromagnetic coil.
(a) The maximum restoring force that can be applied to the disk without breaking it is 36,000 Newtons. What is the maximum oscillation amplitude that won't rupture the disk?
(in µm)

(b) What is the disk's maximum speed at this amplitude?
(in m/s)

For position, I would guess to use x = Acos(omega*t)
I cannot tie the proper equations together, though. Help would be appreciated. Thanks.
 
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  • #2
You know mass and force. What quantity is determined by their ratio?

You wrote correctly the time dependence of position in SHM. What about velocity and acceleration?

ehild
 
  • #3


I would approach this question by first understanding the basic principles of SHM (simple harmonic motion) and how it relates to the maximum oscillation amplitude. SHM is a type of motion where the restoring force is directly proportional to the displacement from equilibrium and is always directed towards the equilibrium point. This means that as the amplitude increases, the restoring force also increases.

(a) To determine the maximum oscillation amplitude that won't rupture the disk, we can use the equation for maximum displacement in SHM, which is given by:

A = F_max / (m * omega^2)

Where A is the maximum amplitude, F_max is the maximum restoring force, m is the mass of the disk, and omega is the angular frequency (2*pi*f). Plugging in the given values, we get:

A = (36,000 N) / (0.12 kg * (2*pi*0.9 MHz)^2) = 0.049 µm

Therefore, the maximum oscillation amplitude that won't rupture the disk is 0.049 µm.

(b) To determine the disk's maximum speed at this amplitude, we can use the equation for velocity in SHM, which is given by:

v = A * omega

Where v is the velocity, A is the amplitude, and omega is the angular frequency. Plugging in the values, we get:

v = (0.049 µm) * (2*pi*0.9 MHz) = 0.083 m/s

Therefore, the disk's maximum speed at this amplitude is 0.083 m/s.
 

Related to Maximum Oscillation Amplitude?

What is maximum oscillation amplitude?

Maximum oscillation amplitude refers to the maximum displacement or distance from equilibrium that a system experiences during oscillation. It is a measure of the total range of motion of a system.

How is maximum oscillation amplitude calculated?

The maximum oscillation amplitude can be calculated by finding the difference between the highest and lowest points of the oscillation curve. This can be represented by the formula A = (xmax - xmin)/2, where A is the amplitude and xmax and xmin are the highest and lowest points respectively.

What factors affect maximum oscillation amplitude?

The maximum oscillation amplitude is affected by the properties of the system, such as mass, stiffness, and damping. It also depends on the amplitude and frequency of the driving force. In addition, external factors such as friction and air resistance can also impact the maximum oscillation amplitude.

Why is maximum oscillation amplitude important?

Maximum oscillation amplitude is an important concept in understanding the behavior of oscillating systems. It helps in predicting the motion and stability of objects, and is crucial in various fields such as engineering, physics, and astronomy.

Can maximum oscillation amplitude be changed?

Yes, the maximum oscillation amplitude can be changed by altering the properties of the system or by adjusting the driving force. For example, increasing the stiffness of a spring will result in a larger maximum oscillation amplitude, while increasing the damping will decrease it. The frequency and amplitude of the driving force can also be adjusted to change the maximum oscillation amplitude.

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