A weight of 50.0 N is suspended from a spring that has a force constant of 190 N/m. The system is undamped and is subjected to a harmonic driving force of frequency 10.0 Hz, resulting in a forced-motion amplitude of 4.00 cm. Determine the maximum value of the driving force.
N
Force = m a
w = 2 pi f = 62.8 radians/second
Forcing function -kx = m d^2x/dt^2
let forcing function = F sin wt
so F sin w t = k x + m d^2x/dt^2
let resulting motion be x = a sin wt + b cos w t
then
d^2x/dt^2 = - a w^2 sin wt - bw^2 cos wt
so (s is sin and c is cos)
Fsin wt=k(a s wt+b c wt)-m(a w^2 s wt-b w^2 c wt)
sin terms
F = k a - m a w^2
F = a (k-w^2 m)
but
a = .04 meters
k = 190 N/m
m = 50/9.8 = 5.1 kg
w^2 = 3944 rad^2/s^2
so
F = .04(190 - 3944*5.1)
= -797N
the cos terms give you the natural frequency motion at w = sqrt(k/m)
note we are driving at about 60 rad/sec whereas the natural frequency is about 6 rad/sec so it is not going to move much for a large force.
To determine the maximum value of the driving force, we need to consider the relationship between the force exerted by the spring and the displacement of the weight from its equilibrium position.
The force exerted by the spring is given by Hooke's Law:
F_spring = -kx
where F_spring is the force exerted by the spring, k is the force constant of the spring, and x is the displacement from the equilibrium position.
In this case, the weight is suspended from the spring, so the force exerted by the weight is given by:
F_weight = mg
where F_weight is the force exerted by the weight, m is the mass of the weight, and g is the acceleration due to gravity.
Since the weight is in equilibrium, the force exerted by the spring and the force exerted by the weight must be equal:
F_spring = F_weight
-kx = mg
Given that the weight is 50.0 N, we can rearrange the equation to solve for x:
x = -(mg)/k
Now, we can determine the maximum value of the driving force. The equation for the forced-motion amplitude of a driven oscillator is given by:
A = (F_d)/(mω^2)
where A is the amplitude of the forced motion, F_d is the driving force, m is the mass of the weight, and ω is the angular frequency of the driving force.
In this case, the amplitude of the forced motion is given as 4.00 cm (or 0.04 m), and the angular frequency of the driving force is 10.0 Hz. Rearranging the equation, we can solve for the driving force:
F_d = Amω^2
Plugging in the values:
F_d = (0.04)(50.0)(10.0^2)
Simplifying the equation, we get:
F_d = 200 N
Therefore, the maximum value of the driving force is 200 N.