A 1.70 kg object attached to a spring moves without friction and is driven by an external force given by F = (3.00 N) sin(2πt).
(a) If the force constant of the spring is 20.0 N/m, determine the period.
_____s For this one I tried 2*pi*sqrt(1.7/20) but it said my answer differed from the correct answer by more than 10%.
(b) If the force constant of the spring is 20.0 N/m, determine the amplitude of the motion.
_____cm For this one I tried -kx = 3 sin(2πt) --> x = (-3/k) sin 2πt ---> Amplitude = 3/k = 3/20 = 0.15 m = 15 cm but it said I was off by 100%.
Any help would be appreciated!
Frequency will be the driving frequency
ω = 2π (s^-1),
T = 2•π/ ω = 1 s.
Amplitude
A = (F/m)/(ω^2 - ω(o)^2)=
=(F/m)/(ω^2 - k/m)=
= (3/1.7)/(4•π ^2 - 20/1.7) = 0.064 m.
To determine the period of the motion, you need to use the equation for the period of a mass-spring system. The period is given by the formula:
T = 2π√(m/k)
Where:
T = period
m = mass of the object (1.70 kg)
k = force constant of the spring (20.0 N/m)
Substituting the given values into the formula, you should get the correct answer. Here's the calculation:
T = 2π√(1.70/20.0)
T = 2π√(0.085)
T = 2π * 0.29154759474226
T ≈ 1.832 s
So, the correct answer for the period is approximately 1.832 seconds.
Moving on to determining the amplitude of the motion, you need to use the equation that relates amplitude, external force, and force constant. The equation is:
A = F0/k
Where:
A = amplitude
F0 = maximum value of the external force (3.00 N)
k = force constant of the spring (20.0 N/m)
Substituting the given values into the equation, you should find the correct answer. Here's the calculation:
A = 3.00/20.0
A = 0.15 m
So, the amplitude of the motion is 0.15 meters or 15 centimeters.
I hope this helps! Let me know if you have any further questions.