A weight of 30.0 N is suspended from a spring that has a force constant of 180 N/m. The system is undamped and is subjected to a harmonic driving force of frequency 11.0 Hz, resulting in a forced-motion amplitude of 5.00 cm. Determine the maximum value of the driving force.
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Get the natural frequency of the system
wo = sqrt (k/m)
m = 30N/g = 3.06 kg
wo= 7.67 rad/s
The forced vibration frequency is
w = 2*pi*f = 17.28 rad/s
For oscillations of this type, without damping, the amplitude A is related to the force amplitude F by
A = F/[k(1 - (w/wo)^2)]
= -0.245 F/m
You should do some reading on the subject of resonance to confirm the above formula.
Substitute 0.05m for A and solve for F
To determine the maximum value of the driving force, we can use the concept of resonance in a harmonic motion system.
The equation for the amplitude of steady-state forced motion in a driven harmonic oscillator is given by:
Amplitude (A) = Fmax / (m * ω^2)
where Fmax is the maximum value of the driving force, m is the mass of the object being oscillated, and ω is the angular frequency of the driving force.
In this case, the given weight of 30.0 N is acting as the driving force, and the system is undamped. We can consider the weight as the force exerted by the driving force.
From Newton's second law, we know that the force exerted by a weight is given by:
Force (F) = m * g
where m is the mass and g is the acceleration due to gravity.
Rearranging this equation, we can find the mass:
m = F / g
Given that the weight is 30.0 N and the acceleration due to gravity is approximately 9.81 m/s^2, we can calculate:
m = 30.0 N / 9.81 m/s^2 ≈ 3.06 kg
Next, we need to find the angular frequency (ω) of the driving force, given its frequency (f):
ω = 2πf
Given that the frequency is 11.0 Hz, we can calculate:
ω = 2π * 11.0 Hz ≈ 69.1 rad/s
Now, substituting the values of mass (m) and angular frequency (ω) into the amplitude equation, we can solve for the maximum value of the driving force:
Amplitude (A) = Fmax / (m * ω^2)
Rearranging the equation, we have:
Fmax = A * m * ω^2
Given that the amplitude (A) is 5.00 cm (which is equivalent to 0.05 m), substituting the values:
Fmax = 0.05 m * 3.06 kg * (69.1 rad/s)^2 ≈ 65.9 N
Therefore, the maximum value of the driving force is approximately 65.9 N.