The ultrasonic transducer used in medical ultrasound imaging device is a very thin disk m = .10g driven back and forth in SHM at 1.0 MHz by an electromagnetic coil.
a.) The maximum restoring force is 40,000 N. What is the maximum oscillation amplitude that won't rupture disk?
b.) What is the disk's maximum speed at this amplitude?
a.) Well, we don't want the disk to rupture into a million tiny pieces, do we? That would be quite a sight in the ultrasound lab. To find the maximum oscillation amplitude that won't cause a disaster, we can use the equation for the maximum restoring force:
F_max = mω²A_max
Here, F_max is the maximum restoring force (40,000 N), m is the mass of the disk (0.10 g), ω is the angular frequency (which is equal to 2πf, where f is the frequency in Hz), and A_max is the maximum amplitude we want to find.
Let's plug in the values and solve for A_max:
40,000 N = (0.10 g)(2π(1.0 MHz))²A_max
Simplifying the calculation, we get:
A_max = 40,000 N / ((0.10 g)(2π(1.0 MHz))²)
So, A_max is the maximum oscillation amplitude that won't rupture the disk. Just be sure to note that the frequency needs to be converted from MHz to Hz.
b.) Now, let's find the maximum speed of the disk at this amplitude. It's like a rollercoaster ride, but without the screaming and expensive snacks.
The maximum speed of an object undergoing Simple Harmonic Motion can be found using the formula:
V_max = ωA_max
Here, V_max is the maximum speed we want to find, ω is the angular frequency, and A_max is the maximum amplitude we found earlier.
Substituting the values:
V_max = (2π(1.0 MHz))(A_max)
Now you can calculate V_max, the disk's maximum speed. Just remember to convert the frequency from MHz to Hz. Enjoy the rollercoaster ride!
To solve these questions, we can use the equations of simple harmonic motion (SHM) and the properties of the ultrasonic transducer.
a.) The maximum restoring force can be related to the maximum oscillation amplitude using the formula:
F = mω²A
where F is the restoring force, m is the mass of the disk (0.10 g = 0.00010 kg), ω is the angular frequency (given by 2πf, where f is the frequency), and A is the amplitude.
Given:
F = 40,000 N
m = 0.10 g = 0.00010 kg
f = 1.0 MHz = 1.0 × 10^6 Hz
First, we need to find ω by substituting f into the formula:
ω = 2πf = 2π(1.0 × 10^6 Hz) = 2π(1.0 × 10^6/s)
Next, we can rearrange the equation to solve for A:
A = √(F / (mω²))
Substituting the given values:
A = √(40,000 N / (0.00010 kg × (2π(1.0 × 10^6/s))²))
Calculating this will give us the maximum oscillation amplitude that won't rupture the disk.
b.) The maximum speed of the disk can be related to the angular frequency and amplitude using the formula:
v_max = ωA
Given that we already have ω and A from the previous calculations, we can substitute them to find the maximum speed of the disk.
Now, let's calculate the answers step-by-step:
a.)
ω = 2π(1.0 × 10^6/s)
A = √(40,000 N / (0.00010 kg × (2π(1.0 × 10^6/s))²))
Calculating the above equation will give the maximum oscillation amplitude (A) in meters.
b.)
v_max = (2π(1.0 × 10^6/s)) × amplitude (found in part a.)
Calculating the above equation will give the maximum speed (v_max) in meters per second.
To answer these questions, we need to use some principles from physics, specifically related to simple harmonic motion (SHM). Let's break down each question step by step:
a.) The maximum restoring force is given as 40,000 N. In SHM, the restoring force is directly proportional to the displacement from the equilibrium position.
We can use Hooke's law, which states that the restoring force (F) in SHM is equal to the negative product of the spring constant (k) and the displacement (x):
F = -kx
In this case, the spring constant is not provided directly. However, we know that the disk is driven back and forth in SHM at 1.0 MHz, so we can relate the frequency (f) and the angular frequency (ω) using the formula:
ω = 2πf
The angular frequency is also related to the spring constant and the mass (m) of the disk:
ω = √(k/m)
By rearranging this equation, we can find the spring constant:
k = mω²
Now we can substitute this into Hooke's law to find the maximum oscillation amplitude (Amax) that won't rupture the disk:
Fmax = -kAmax
Rearranging again, we find:
Amax = -Fmax/k
Substituting the given values (Fmax = 40,000 N and m = 0.10 g = 0.0001 kg) and calculating ω, we can find the value of Amax.
b.) The maximum speed of the disk occurs at the equilibrium position, where the acceleration is at a maximum. In SHM, the maximum acceleration (amax) is directly proportional to the maximum speed (vmax) and the angular frequency (ω).
The equation relating these variables is:
amax = ω²Amax
Given the value of ω, we can find amax. Since amax = vmax × ω, we can rearrange this equation to solve for vmax:
vmax = amax / ω
Substituting the calculated values of amax and ω, we can find the maximum speed of the disk.