The maximum speed of a 3.7-{\rm kg} mass attached to a spring is 0.66 m/s, and the maximum force exerted on the mass is 11 N.
What is the amplitude of motion for this mass?
What is the force constant of the spring?
What is the frequency of this system?
To find the answers to these questions, we need to use the formulas related to the motion of a mass-spring system. Let's go step by step.
1. Amplitude of Motion:
The amplitude of motion, represented by A, is the maximum displacement from the equilibrium position of the mass. In this case, since the mass is attached to a spring, it undergoes simple harmonic motion.
To find the amplitude, we can use the formula for maximum velocity (vmax) and angular frequency (ω):
vmax = Aω
Given vmax = 0.66 m/s, and knowing that ω = 2πf (where f is the frequency), we can rearrange the formula to solve for A:
A = vmax / ω
2. Force Constant of the Spring:
The force constant, represented by k, is a measure of the stiffness of the spring. It relates the displacement of the mass to the force exerted by the spring according to Hooke's Law:
F = -kx
Given that the maximum force exerted on the mass is 11 N, we can rearrange Hooke's Law to solve for k:
k = -Fmax / xmax
3. Frequency of the System:
The frequency, represented by f, is the number of complete cycles per unit of time. It is related to the angular frequency ω by the formula:
ω = 2πf
Now that we have found the amplitude (A), force constant (k), and frequency (f), let's substitute the given values into the formulas to find the answers.