The maximum speed of a 3.68-kg mass attached to a spring is 0.688 m/s, and the maximum force exerted on the mass is 11.6 N. What is the amplitude of motion for this mass?
(in m)
What is the force constant of the spring?
(in N/m)
What is the frequency of this system?
(in Hz)
To find the amplitude of motion for the mass, we can use the equation for maximum speed of an object attached to a spring:
v_max = Aω
where:
v_max is the maximum speed (0.688 m/s in this case),
A is the amplitude of motion,
ω is the angular frequency.
Since we are given v_max, we can rearrange the equation to solve for A:
A = v_max / ω
To find the force constant of the spring, we can use Hooke's Law:
F = -kx
where:
F is the force exerted on the mass (11.6 N in this case),
k is the force constant of the spring,
x is the displacement of the mass from its equilibrium position.
Since we are given F, let's rearrange the equation to solve for k:
k = -F / x
To find the frequency of the system, we can use the formula for angular frequency:
ω = 2πf
where:
ω is the angular frequency,
f is the frequency of the system.
Given the values of ω and A, we can rearrange the equation to solve for f:
f = ω / 2π
Now, let's substitute the given values and calculate the answers.
Given:
mass (m) = 3.68 kg
maximum speed (v_max) = 0.688 m/s
maximum force (F) = 11.6 N
1. Amplitude of motion (A):
To find A, we need to know the angular frequency (ω). We can calculate ω using the maximum speed equation:
v_max = Aω
0.688 = Aω
Rearranging the equation to solve for ω:
ω = v_max / A
ω = 0.688 / A
2. Force constant of the spring (k):
To find k, we need to know the displacement of the mass (x). Unfortunately, this information is not provided in the question. Without knowing x, we cannot determine the force constant of the spring.
3. Frequency of the system (f):
To find f, we need to know the angular frequency (ω). We can calculate ω using the maximum speed equation:
ω = 2πf
Rearranging the equation to solve for f:
f = ω / 2π
Now, let's substitute the given values and calculate the answers:
Amplitude of motion (A):
A = v_max / ω
A = 0.688 / (0.688 / A)
A = 1 (since ω cancels out)
Force constant of the spring (k):
Information not provided, so we cannot determine k.
Frequency of the system (f):
f = ω / 2π
f = (0.688 / 1) / (2π)
f = 0.109 Hz (approximately)